# Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions II: The
Eisenstein Case

Chung-Hang Kwan<EMAIL_ADDRESS>University College London

###### Abstract.

This is a sequel to Kwan [Kw23]. We continue our investigations on the
spectral moments of $L$-functions of $GL(3)\times GL(2)$-type from the
perspective of period integrals. By manipulating an equality between two
different periods for the group $GL(3)$, we establish an exact identity of
Motohashi type connecting the shifted cubic moment of $GL(2)$ $L$-functions to
the shifted fourth moment of $GL(1)$ $L$-functions. Furthermore, we are able
to provide an intrinsic, automorphic explanation for the sources and
symmetries of the full sets of main terms for both moments. They are shown to
be in agreement with the CFKRS Moment Conjectures. This work presents a
strategy for accomplishing the aforementioned tasks, distinct from the
previous ones in the literature.

###### Key words and phrases:

Automorphic Forms, Automorphic $L$-functions, Rankin-Selberg $L$-functions,
Period Integrals, Maass forms, Eisenstein series, Poincaré series, Moments of
$L$-functions, CFKRS Moment Conjectures, Spectral Reciprocity, Motohashi
formula, Special Functions, Whittaker Functions, Hypergeometric Functions,
Barnes integrals

###### 2010 Mathematics Subject Classification:

11F55 (Primary) 11F72 (Secondary)

##  1\. Introduction

###  1.1. Background

In the last two decades, the cubic moment of automorphic $L$-functions for the
group $GL(2)$ has found plenty of striking applications in number theory.
These include the landmark result of Conrey-Iwaniec [CI00] and the vast recent
extensions by Petrow-Young [PY20, PY23] culminating in the full Weyl-strength
subconvexity for all primitive Dirichlet $L$-functions, i.e.,

$\displaystyle L\left(\frac{1}{2},\chi\right)\ \ll_{\epsilon}\
q^{\frac{1}{6}+\epsilon}$ (1.1)

for primitive characters $\chi\ (\bmod\ q)$. Readers should consult [Y17,
PY19, BK17, AW23, Iv01, Iv02] for applications in other directions — just to
name a few.

An unexpected structural feature the cubic moment of $GL(2)$ $L$-functions is
that it is dual to a moment of a very different shape and is of significant
interests in classical analytic number theory. This type of phenomena was
first discovered by Motohashi [Mo93, Mo97] in which an exact identity of the
shape

$\displaystyle\int\limits_{-\infty}^{\infty}\
\left|\zeta\left(\frac{1}{2}+it\right)\right|^{4}\ dt\ \ =\ \
\sum_{\begin{subarray}{c}f:\text{ level $1$ }\\\ \text{cusp
forms}\end{subarray}}\ L\left(\frac{1}{2},f\right)^{3}\ \ +\ \ (***)$ (1.2)

was established, where the weight functions for the moments are suppressed and
$(***)$ represents certain intricate polar and continuous contributions.

To study the cubic moment, one must perform a spectral inversion to the
identity (1.2). This was only obtained relatively recently by Petrow [Pe15] in
the level aspect and Frolenkov [Fr20] in the weight aspect. While [Pe15] was
built upon the foundation of [CI00] which starts with approximate functional
equations followed by relative trace formulae of Petersson/Kuznetsov type,
deducing a Motohashi-type identity turned out to require a decent amount of
refinements and they are too elaborate to be described here. The relevant
analytic machinery was developed in various ways and aspects by a number of
authors, e.g., [Bl12b, Li11, Y17, KPY19].

###  1.2. Moment Conjectures

A fundamental goal in any moment problem is to understand the sources and the
structures for the full sets of main terms. Throughout these years, this quest
has gained a good amount of interests in the literature and the outcomes have
proven to be fruitful. Most notably, there are two sets of heuristics leading
to very precise conjectures for the asymptotics of moments. Conrey et. al.
[CFKRS05] developed their heuristics (or the ‘Recipes’) based on the notion of
shifted moments and the use of approximate functional equations — their
conjectures are nonetheless intrinsic. They were able to draw surprising
connections to Random Matrix Theory which reveal many unexpected symmetries
and combinatorics behind moments. On the other hand, Diaconu et. al. [DGH03]
developed the framework of Multiple Dirichlet Series for moments of the
$\zeta$-function and the quadratic Dirichlet $L$-functions. Their predictions
for the asymptotic formulae are deduced from the (conjectural) analytic
continuation, polar structures and groups of functional equations of certain
carefully constructed multiple Dirichlet series.

Regarding the two moments to be examined in this article, i.e., the ones in
(1.2), it was a delicate task to extract the full structures behind the main
terms (which were suppressed as part of ‘$(***)$’ in (1.2)). For the fourth
moment of the $\zeta$-function, Heath-Brown [HB79] and Motohashi [Mo93] proved
the asymptotic formula

$\displaystyle\int_{T}^{2T}\
\left|\zeta\left(\frac{1}{2}+it\right)\right|^{4}\ dt\ \ =\ \ T\cdot
P_{2}(\log T)\ \ +\ \ O(T^{1-\delta}).$ (1.3)

for $\delta<1/8$ and $\delta<1/3$ respectively, where $P_{2}$ is a polynomial
of degree $4$. The shapes of $P_{2}$ obtained in these works (see also Conrey
[C96]) were unfortunately very complicated. It would be daunting to compute
all of the coefficients of $P_{2}$ using their formulae.

Motohashi [Mo93] considered a smooth and shifted variant of (1.3), i.e.,

$\displaystyle\int_{\mathbb{R}}\
\eta(t)\zeta\left(\tfrac{1}{2}+it+\alpha_{1}\right)\zeta\left(\tfrac{1}{2}+it+\alpha_{2}\right)\zeta\left(\tfrac{1}{2}-it+\beta_{1}\right)\zeta\left(\tfrac{1}{2}-it+\beta_{2}\right)\
dt,$ (1.4)

initially for $\operatorname{Re}\alpha_{1},\ \operatorname{Re}\alpha_{2},\
\operatorname{Re}\beta_{1},\ \operatorname{Re}\beta_{2}\gg 1$, where $\eta\in
C_{c}^{\infty}[T,2T]$. This variant plays a pivotal role in the intricate
analytic continuation argument of [Mo93]. Conrey et. al. [CFKRS05] takes one
step further by realizing (1.4) actually leads to an elegant intrinsic form of
the main terms which is far from obvious from the work [Mo93]. There are six
main terms in the formulation of [CFKRS05] from which the polynomial $P_{2}$
can be computed in full simply by letting the shifts
$\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\to 0$, see Section 5.1 of
[CFKRS05].

The ‘recipe’ of [CFKRS05] works equally well for the shifted cubic moment of
$GL(2)$ $L$-functions. Because of the differences in the families and their
underlying arithemetic, there will be eight different main terms this time.
The authors of [CI00] were unable to obtain an asymptotic formula but had
nevertheless expressed interests in accomplishing this task (pp. 1177
therein). We shall review these conjectures in Section 3 and examine the
technical difficulties in identifying these terms in the previous approaches
towards the end of Section 1.4.

###  1.3. Beyond the Petersson/ Kuznetsov paradigm: Recent Progress

The source of the ‘reciprocity’ in (1.2) is not entirely apparent from the
original work of Motohashi [Mo93]. In particular, there is no resaon to expect
the off-diagonal or the error terms should be nicely given by any moment of
$L$-functions. Having a conceptual understanding of this phenomenon, without
intermediate objects like Kloosterman sums, shifted convolution sums or the
Kuznetsov formulae, is hence of interest to this field.

This task was perhaps first picked up by Bruggeman-Motohashi [BM05]. To
construct the two distinct-looking moments in (1.2), their approach utilizes a
conjunction of spectral projections, Hecke operators and a two-fold limiting
process to an automorphic kernel. The choice of the automorphic kernel in
[BM05] is by no means straight-forward. Indeed, delicate smoothing and
regularization are essential, and one must handle various issues regarding
convergence, regularity, and exchanges of limits carefully. In turn, these
ingredients allow [BM05] deeper understanding of the integral transform
through the theories of Whittaker-Kirillov models and Bessel functions for the
group $PGL(2)$ (see [CPS90]).

Michel-Venkatesh [MV06] (Section 4), [MV10] (Section 4.5), and Reznikov [R08]
(pp. 453) took an extra step and asked for an even more intrinsic account than
the one offered by [BM05]. They envisioned a proof of Motohashi’s formula in
terms of a structural framework known as the strong Gelfand formation using a
‘regularized’ geodesic period for the product of two $GL(2)$ Eisenstein series
formally looks like

$\displaystyle\text{Reg}-\int_{\mathbb{R}^{\times}}\
E(iy,s_{1})E(iy,s_{2})y^{s_{3}-1/2}\ d^{\times}y.$ (1.5)

However, one must overcome numerous technical obstructions if one wishes to
implement their strategy. Indeed, the sources of the many main terms or
degenerate terms are rather unclear from (1.5), as well as a number of
integrals experience serious divergence issues due to non-compactness of
domains and non-square-integrability of Eisenstein series. Moreover, it has
become evident that one requires a more sophisticated regularization
techniques than that developed in [MV10]. See also Blomer et. al. [BHKM20] for
discussions on other intricacies related to this approach. Only very recently
Nelson [Ne20+] managed to make the period integral strategy of Michel-
Venkatesh and Reznikov completely rigorous with very impressive techniques.

Alternatively, Motohashi’s formula can be interpreted under another framework
due to Wu [Wu22]. Wu’s work is a generalization of [BM05] based on two
different (spectral and geometric) decompositions of a co-invariant tempered
distribution on the Schwartz space of $2$-by-$2$ matrices. The two sides can
then be recasted in terms of Godement-Jacquet’s $L$-functions which in turn
lead to the moments occured in (1.2). Further explications and analytic
subtlties are addressed in the follow-up work Balkanova-Frolenkov-Wu [BFW21+]
using many heavy inputs from the theory of special functions. The authors of
[BFW21+] also discussed the obstacles in obtaining an exact spectral inversion
of Motohashi’s result.

Overall speaking, the presence of these significant challenges in the
aforementioned approaches renders (1.2) a particularly intriguing instance of
‘Spectral Reciprocity’ and serves as a good testing ground for our period-
based approach (see below). Readers are invited to consult the references of
this subsection for more detailed technical discussions.

###  1.4. Our Main Results

This work aims to expand the scope of techniques developed in our prior work
[Kw23] for studying moments of $L$-functions. In [Kw23], we studied the
spectral first moment of $GL(3)\times GL(2)$ $L$-functions (of level $1$)
where the $GL(3)$ form was a fixed Maass cusp form. Incidentally, we uncovered
a Motohashi-type identity that serves as the essential structural input in the
celebrated convexity-breaking results of Li [Li11].

We incorporated the formalisms of period integrals and automorphic forms/
representations of $GL(3)$ to investigate various intrinsic aspects of the
moment we were interested in. The former lies outside the traditional domain
of analytic number theory, while the latter remains rather recent to the
field, with much of the research in this direction building upon the $GL(3)$
Voronoi formula developed in a series of works [MS06, GL06, GL08, MS11, IT13,
Li14, Co21]. Our works belong to the recent framework ‘Period Reciprocity’ in
which only a handful of other instances are known, see [MV10, Bl12a, Ne20+,
Nu20+, Za20+, Za21, JN21+, Mi21+]. We refer the readers to [Kw23] (Section
3-4) and Jana-Nunes [JN21+] for more in-depth discussions regarding the
implementations and technical features of this framework.

In this article, we shall provide a new proof of the spectral inversion of
Motohashi’s identity (1.2) using the period reciprocity of $GL(3)$ in the form
of [Kw23] which enables us to gain different perspectives of the Motohashi
phenomena and to surmount some of the technical difficulties encountered in
previous approaches. Furthermore, we wish to illustrate ways to harness fuller
advantages of the formalisms mentioned in the preceeding paragraph, elucidate
their rich connections to the Moment Conjectures of [CFKRS05], and streamline
the intricate sequence of analytic-arithmetic transformations found in the
existing literature.

To state our main results, it is convenient to introduce the Weyl group of
$GL_{3}(\mathbb{R})$ which consists of the matrices

$\displaystyle I\ $ $\displaystyle=\ \begin{pmatrix}1&&\\\ &1&\\\
&&1\end{pmatrix},\hskip 15.0ptw_{2}\ =\ -\begin{pmatrix}&1&\\\ 1&&\\\
&&1\end{pmatrix},\hskip 15.0ptw_{3}\ =\ -\begin{pmatrix}1&&\\\ &&1\\\
&1&\end{pmatrix}$ $\displaystyle w_{4}\ $ $\displaystyle=\
\begin{pmatrix}&1&\\\ &&1\\\ 1&&\end{pmatrix},\hskip 15.0ptw_{5}\ =\
-\begin{pmatrix}&&1\\\ 1&&\\\ &1&\end{pmatrix},\hskip 15.0ptw_{\ell}\ =\
-\begin{pmatrix}&&1\\\ &1&\\\ 1&&\end{pmatrix}.$ (1.6)

The corresponding Weyl actions on the Langlands parameters are given by:

$\displaystyle(\alpha_{1},\alpha_{2},\alpha_{3})^{I}\ $ $\displaystyle=\
(\alpha_{1},\alpha_{2},\alpha_{3}),\hskip
15.0pt(\alpha_{1},\alpha_{2},\alpha_{3})^{w_{2}}\ =\
(\alpha_{2},\alpha_{1},\alpha_{3}),\hskip
15.0pt(\alpha_{1},\alpha_{2},\alpha_{3})^{w_{3}}\ =\
(\alpha_{1},\alpha_{3},\alpha_{2}),$
$\displaystyle(\alpha_{1},\alpha_{2},\alpha_{3})^{w_{4}}\ $ $\displaystyle=\
(\alpha_{3},\alpha_{1},\alpha_{2}),\hskip
15.0pt(\alpha_{1},\alpha_{2},\alpha_{3})^{w_{5}}\ =\
(\alpha_{2},\alpha_{3},\alpha_{1}),\hskip
15.0pt(\alpha_{1},\alpha_{2},\alpha_{3})^{w_{\ell}}\ =\
(\alpha_{3},\alpha_{2},\alpha_{1}),$ (1.7)

where $(\alpha_{1},\alpha_{2},\alpha_{3})\in\mathbb{C}^{3}$ satisfies
$\alpha_{1}+\alpha_{2}+\alpha_{3}=0$. We adopt the following sets of
conventions in this article:

  * •

Denote by $\mathcal{C}_{\eta}$ ($\eta>40$) the class of holomorphic functions
$H$ defined on the vertical strip $|\operatorname{Re}\mu|<2\eta$ such that
$H(\mu)=H(-\mu)$ and has rapid decay

$\displaystyle\hskip 5.0ptH(\mu)\ \ll\ e^{-2\pi|\mu|}$

on the same domain.

  * •

For $H\in\mathcal{C}_{\eta}$, the integral transform
$\left(\mathcal{F}_{\Phi}H\right)\left(s_{0},s\right)$ is the same as the one
in [Kw23]. See Section 2.8 of this article for its definition and essential
analytic properties. In this article, we take $\Phi=(E_{\min}^{(3)})^{*}(\ *\
;\ \alpha)$ the minimal parabolic Eisenstein series of $SL_{3}(\mathbb{Z})$
(see Section 2.4). It is then more sensible to write
$\mathcal{F}_{\alpha}H:=\mathcal{F}_{\Phi}H$ since the transform only depends
on the Langlands parameters $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ of
$\Phi$.

  * •

Take $\epsilon:=1/100$. The Langlands parameters
$\alpha_{1},\alpha_{2},\alpha_{3}$ are assumed to be distinct and satisfy
$|\alpha_{1}|,|\alpha_{2}|<\epsilon/1000$.

######  Theorem 1.1.

Let $(\phi_{j})_{j=1}^{\infty}$ be an orthogonal basis of even, Hecke-
normalized Maass cusp forms of $SL_{2}(\mathbb{Z})$ which satisfy
$\Delta\phi_{j}=\left(1/4-\mu_{j}^{2}\right)\phi_{j}$. On the domain
$1/4+\epsilon<\operatorname{Re}s<3/4-\epsilon$, we have the following spectral
moment identity:

$\displaystyle\hskip 10.0pt\sum_{j=1}^{\infty}\ H(\mu_{j})\ $
$\displaystyle\frac{\Lambda(s-\alpha_{1},\phi_{j})\Lambda(s-\alpha_{2},\phi_{j})\Lambda(s-\alpha_{3},\phi_{j})}{\langle\phi_{j},\phi_{j}\rangle}$
$\displaystyle\ +\ \frac{1}{4\pi i}\ \int_{(0)}\ H(\mu)\
\frac{\prod\limits_{i=1}^{3}\Lambda(s+\mu-\alpha_{i})\Lambda(s-\mu-\alpha_{i})}{|\Lambda(1+2\mu)|^{2}}\
d\mu\ +\ (\mathcal{D}_{\alpha}H)(s)\ +\ (\mathcal{R}_{\alpha}H)(s)$
$\displaystyle\hskip 10.0pt\ =\ \frac{1}{2}\
\zeta(2s+\alpha_{1})\zeta(2s+\alpha_{2})\zeta(2s+\alpha_{3})\
\int_{(0)}\frac{H(\mu)}{|\Gamma(\mu)|^{2}}\cdot\prod_{i=1}^{3}\ \prod_{\pm}\
\Gamma_{\mathbb{R}}\left(s\pm\mu-\alpha_{i}\right)\ \frac{d\mu}{2\pi i}$
$\displaystyle\hskip 40.0pt\ +\
4\cdot\Gamma_{\mathbb{R}}\left(1+\alpha_{2}^{w}-\alpha_{3}^{w}\right)\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)$
$\displaystyle\hskip 83.0pt\cdot\sum_{w=w_{2},w_{4},w_{\ell}}\
\zeta\left(1+\alpha_{2}^{w}-\alpha_{3}^{w}\right)\zeta\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\zeta(2s+\alpha_{1}^{w}+\alpha_{2}^{w})(J_{\alpha}h)(s;w)$
(1.9) $\displaystyle\hskip 70.0pt\ +\ \frac{1}{2}\ \sum_{i=1}^{3}\
\zeta(2s-1+\alpha_{i})\prod_{\begin{subarray}{c}1\leq j\leq 3\\\ j\neq
i\end{subarray}}\zeta(1-\alpha_{i}+\alpha_{j})\left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{i},s\right)$
(1.10) $\displaystyle\hskip 100.0pt\ +\ \frac{1}{2}\
\zeta(2s-1+\alpha_{1})\zeta(2s-1+\alpha_{2})\zeta(2s-1+\alpha_{3})\left(\mathcal{F}_{\alpha}H\right)\left(2s-1,s\right)$
(1.11) $\displaystyle\hskip 130.0pt\ +\ \frac{1}{2}\
\int_{(\frac{1}{2})}\zeta\left(2s-s_{0}\right)\ \prod_{i=1}^{3}\
\zeta(s_{0}+\alpha_{i})\left(\mathcal{F}_{\alpha}H\right)\left(s_{0},s\right)\
\frac{ds_{0}}{2\pi i},$

where

$\displaystyle(\mathcal{D}_{\alpha}H)(s)\ :=\ 2\sum_{w=w_{3},w_{5},w_{\ell}}\
H(s-\alpha_{1}^{w})\
\frac{\Lambda\left(1-\alpha_{1}^{w}+\alpha_{2}^{w}\right)\Lambda\left(1-\alpha_{1}^{w}+\alpha_{3}^{w}\right)\Lambda\left(2s-\alpha_{1}^{w}-\alpha_{2}^{w}\right)\Lambda\left(2s-\alpha_{1}^{w}-\alpha_{3}^{w}\right)}{\Lambda(1+2(s-\alpha_{1}^{w}))},$
(1.13) $\displaystyle\hskip 60.0pt(\mathcal{R}_{\alpha}H)(s)\ :=\ 2\
\sum_{i=1}^{3}\ \ H(1-s+\alpha_{i})\
\frac{\prod\limits_{\begin{subarray}{c}1\leq j\leq 3\\\ j\neq
i\end{subarray}}\
\Lambda\left(1+\alpha_{i}-\alpha_{j}\right)\Lambda(2s-1-\alpha_{i}-\alpha_{j})}{\Lambda\left(3-2s+2\alpha_{i}\right)}.$
(1.14)

and the quantities $(J_{\alpha}h)(s;w)$,
$\left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{i},s\right)$, and
$\left(\mathcal{F}_{\alpha}H\right)\left(2s-1,s\right)$ can be evaluated
explicitly, see Theorem 1.2 below.

Recent works on Moment Conjectures (e.g., [Pe15] and [CIS12]) have realized
that it is both technically and aesthetically more pleasing to state and to
prove the results in terms of complete (instead of incomplete) $L$-functions.
Indeed, the totality of the main terms of a moment of $L$-functions is
predicted to preserve the full set of symmetries originating from the
applications of the family’s functional equation. (Incidentally, the root
number of the family is conveniently fixed in our theorem.) Nevertheless, the
admissible class of test functions $\mathcal{C}_{\eta}$ is sufficiently large
and it allows us to deduce the version of Theorem 1.1 for incomplete
$L$-functions. Such kind of set-up for Theorem 1.1 is also natural in view of
our use of period integrals.

Here is an extra benefit. The combinatorics of all of the main terms for the
cubic moment of $GL(2)$ can now be treated uniformly via certain archimedean
Rankin-Selberg calculations à la Bump [Bump88]. The exact computation
performed in [Bump88] only accounts for the diagonal term. To account for the
occurrence and symmetries of the rest of the seven off-diagonal main terms, we
have uncovered the following three groups of identities of (both single and
double) Barnes type.

######  Theorem 1.2 (Proposition 4.4, 4.9, & 4.10).

On $0<\sigma<1$ and for $w\in\left\\{w_{2},w_{4},w_{\ell}\right\\}$,
$i=1,2,3$, we have

$\displaystyle(J_{\alpha}h)(s;w)\ $ $\displaystyle=\
\frac{\pi^{-(s-\frac{\alpha_{3}^{w}}{2})}}{8}\
\Gamma\left(\frac{1+\alpha_{2}^{w}-\alpha_{3}^{w}}{2}\right)^{-1}\Gamma\left(\frac{1+\alpha_{1}^{w}-\alpha_{3}^{w}}{2}\right)^{-1}$
$\displaystyle\hskip 40.0pt\cdot\int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}\ \prod\limits_{\pm}\
\Gamma\left(\frac{s\pm\mu+\alpha_{1}^{w}}{2}\right)\Gamma\left(\frac{s\pm\mu+\alpha_{2}^{w}}{2}\right)\Gamma\left(\frac{1-s\pm\mu-\alpha_{3}^{w}}{2}\right)\
\frac{d\mu}{2\pi i},$ (1.15)
$\displaystyle\left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{i},s\right)\ $
$\displaystyle=\ \pi^{\frac{1}{2}-s}\
\frac{\Gamma\left(s-\frac{1}{2}+\frac{\alpha_{i}}{2}\right)}{\Gamma\left(1-s-\frac{\alpha_{i}}{2}\right)}$
$\displaystyle\hskip 60.0pt\cdot\int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}\ \prod_{\pm}\
\Gamma\left(\frac{s\pm\mu-\alpha_{i}}{2}\right)\prod_{\begin{subarray}{c}1\leq
j\leq 3\\\ j\neq
i\end{subarray}}\Gamma\left(\frac{1-s\pm\mu+\alpha_{j}}{2}\right)\
\frac{d\mu}{2\pi i},$ (1.16)
$\displaystyle\left(\mathcal{F}_{\alpha}H\right)\left(2s-1,s\right)\ $
$\displaystyle=\ \pi^{\frac{1}{2}-s}\ \prod_{i=1}^{3}\
\frac{\Gamma\left(s-\frac{1}{2}+\frac{\alpha_{i}}{2}\right)}{\Gamma\left(1-s-\frac{\alpha_{i}}{2}\right)}\cdot\int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}\ \prod\limits_{i=1}^{3}\
\prod\limits_{\pm}\ \Gamma\left(\frac{1-s\pm\mu+\alpha_{i}}{2}\right)\
\frac{d\mu}{2\pi i}.$ (1.17)

In our approach, the sources of the main terms are equally intrinsic and easy-
to-understand. Essentially, it suffices to know the $GL(3)$ Fourier-Whittaker
period/ expansion of $(E_{\min}^{(3)})^{*}(\ *\ ;\ \alpha)$ (see Lemma 2.15
and Proposition 2.20):

  * •

For the cubic moment of $GL(2)$, this is Proposition 3.1. The identities
(1.15), (1.16) and (1.17) are termed the ‘$1$-swap’, ‘$2$-swap’ and ‘$3$-swap’
identities in Section 3.1, and they correspond to the terms (1.9), (1.10) and
(1.11) respectively.

  * •

For the fourth moment of the $\zeta$-function, the full set of main terms (as
described in Section 3.2) originate from

    * –

The continuous spectrum: resulting in two ‘$1$-swap’ terms and one ‘$2$-swap’
term, see (1.14) and Lemma 4.13;

    * –

The non-constant degenerate part of the Fourier expansion: resulting in two
‘$1$-swap’ terms and one ‘$0$-swap’ term, see (1.13) and Proposition 4.7.

As noted by Young [Y11], Hughes-Young [HY10] and Wu [Wu22], and in alignment
with our earlier discussions in Section 1.2, the main terms obtained by
Motohashi’s method [Mo93] are not in the most convenient form. As anticipated,
their arguments are quite a bit distinct from [Mo93]. For example, [Y11]
(Section 5-8) and [HY10] (Section 5-6) applied versions of the $\delta$-method
(e.g., [DFI94]) to the shifted divisor sums which requires a prior use of the
approximate functional equations (with careful smoothing and truncations).
Most importantly, very curious cancellations and combinations of some
complicated-looking terms took place in both works which remarkably led to the
desired forms of main terms up to acceptable error. On the other hand, Petrow
[Pe15] (Section 2-3) studied the full set of main terms on the cubic moment
side with an eqaully elaborate argument compared to the ones of [Y11] and
[HY10].

Our approach to extracting the full sets of main terms is different from the
existing ones in the literature. We are able to handle and display the main
terms for both the cubic moment of $GL(2)$ and the fourth moment of the
$\zeta$-function in one go. Curiously, we do not need any delicate
combination/ cancellation of terms, nor any intermediate devices like the
shifted divisor sums and the $\delta$-symbol expansion. Our work might offer
some insights into the recent developments on the mean values of long
Dirichlet polynomials [HN22, BTB22+, CF22+] with the aim of identifying the
arithmetic sources of the main terms in various families of moments of
$L$-functions without the use of approximate functional equations. Readers are
also invited to consult [Ne20+] (Section 11) and [Wu22] (Section 1.5) for more
abstract treatment of these terms using certain regularized or degenerate
functionals from automorphic representation.

###  1.5. Sketch and Features of Our Argument

For the cuspidal analogue of Theorem 1.1 studied in [Kw23], we showed that the
corresponding Motohashi-type formula is a consequence of the seemingly trivial
observation

$\displaystyle\hskip 20.0pt\int_{\mathbb{Z}\setminus\mathbb{R}}\
\left[\int_{\mathbb{Z}^{\times}\setminus\mathbb{R}^{\times}}\
\Phi\begin{pmatrix}\begin{pmatrix}y_{0}&\\\
&y_{0}\end{pmatrix}\begin{pmatrix}1&u\\\ &1\end{pmatrix}&\\\ &1\end{pmatrix}\
d^{\times}y_{0}\right]\ $ $\displaystyle e(-u)\ du$ $\displaystyle=\ \
\int_{\mathbb{Z}^{\times}\setminus\mathbb{R}^{\times}}\
\left[\int_{\mathbb{Z}\setminus\mathbb{R}}\ \
\Phi\begin{pmatrix}\begin{pmatrix}1&u\\\
&1\end{pmatrix}\begin{pmatrix}y_{0}&\\\ &y_{0}\end{pmatrix}&\\\
&1\end{pmatrix}e(-u)\ du\right]\ d^{\times}y_{0}$ (1.18)

with $\Phi$ being a Maass cusp form of $SL_{3}(\mathbb{Z})$. In this article,
we intend to take $\Phi$ to be the Eisenstein series $(E_{\min}^{(3)})^{*}(\
*\ ;\alpha)\cong 1\ \boxplus\ 1\ \boxplus\ 1$.

As previously mentioned, while periods of Eisenstein series undertake
fundamental roles in many interesting instances of spectral reciprocity, it
can be a real challenge to handle them properly and no universal technique
exists at this moment. Apart from [Ne20+], let’s also mention [JN21+, Mi21+]
who applied the regularization method of Ichino-Yamana [IY15] (which in turn
based on [JLR99]), and [Za20+] who applied Michel-Venkatesh’s deformation and
regularization techniques.

The situation we encountered in this work is somewhat more favourable than the
previous ones. While (1.5) diverges when $\Phi\cong 1\ \boxplus\ 1\ \boxplus\
1$, the most simplistic treatment is sufficient for our strategy — subtract
the degenerate parts of $\Phi$, insert an extra factor $|\det*|^{s-1/2}$ to
(1.5), and perform analytic continuation in the variable $s$.

The presence of the unipotent integration in (1.5) is crucial to our approach.
Apart from serving the purpose discussed in the previous paragraph, it
connects the Whittaker models of $GL(2)$ and $GL(3)$ on the spectral and the
dual side of our reciprocity formula. Respectively, it ensures the abundance
of admissible test functions on the spectral side and the multiplicity-one
theorem of $GL(3)$ makes the dual integral moment immediately visible with an
explicit integral transform given by (2.57). The latter allows us to take
advantage of the equivariance of the Fourier-Whittaker periods, and to
circumvent complications in the Hecke combinatorics arising from the $GL(3)$
Voronoi formula in Kuznetsov-based approaches. Also, Theorem 1.2, and hence
the matching with the Moment Conjectures of [CFKRS05], are nice consequence of
this particular form of the integral transform. Readers may wish to consult
Section 4 of [Kw23] as well.

In addition, there is a simple way to further enlarge the class of test
functions by embedding the modified version of (1.5) into an inner product
construction (4.1). We have chosen this alternative for extra flexibility and
to facilitate the analytic continuation argument.

The focus of our work has been the spectral inversion of Motohashi’s formula
which complements [Mo93, BM05]. As a result, our test vector at $\infty$ is
taken to be the spherical one. However, our period integral set-up (1.5)
allows many other choices of test vectors. In particular, one should be able
to recover the results of [Mo93, BM05]. In this case, our analysis should be
simpler than the ones of [Mo93, BM05] — we do not require the ‘Atkinson
dissection’ (nor any dissection of geometric type), and the four-variable
continuation argument of [Mo93, BM05] can be replaced by a single-variable one
as the ‘shifts’ play no essential role in our analytic continuation argument
(just there to maintain simple residual evaluations). We shall leave all these
to a future work.

###  1.6. Additional Renarks and an Upcoming Work

It is clear that the techniques of this paper carry over to the case
$\Phi=\phi\ \boxplus\ 1$ with $\phi$ being a Maass cusp form of
$SL_{2}(\mathbb{Z})$, i.e., $\Phi$ is a maximal parabolic Eisenstein series of
$SL_{3}(\mathbb{Z})$. Theorem 1.1 now takes the shape

$\displaystyle\hskip 10.0pt\sum_{j=1}^{\infty}\ H(\mu_{j})\ $
$\displaystyle\frac{\Lambda\left(1/2,\phi\otimes\phi_{j}\right)\Lambda\left(1/2,\phi_{j}\right)}{\langle\phi_{j},\phi_{j}\rangle}$
$\displaystyle\ \ =\ \ \frac{1}{2\pi}\ \int_{\mathbb{R}}\
L\left(1/2+it,\phi\right)\left|\zeta\left(1/2+it\right)\right|^{2}(\mathcal{F}_{\Phi}H)\left(1/2+it,1/2\right)\
dt\ \ +\ \ (***).$ (1.19)

This type of mixed moments is also of interest to the literature as it has
applications to simultaneous non-vanishing, see [Li09, BBFR20, X11, Kh12,
Liu14, Liu15, Wu22, Ka21+] for a few examples.

In this article, we have not fully exploited the strength of our period
construction, particularly in the non-archimedean aspects. For example, we
shall establish a $GL(3)$ Motohashi’s formula that dualizes $GL(2)$ twists of
Hecke eigenvalues into $GL(1)$ twists by Dirichlet characters. When
specializing to the isobaric case, this gives a short proof the key spectral
identity of Young [Y11] and Blomer et. al. [BHKM20] on the fourth moment of
Dirichlet $L$-functions.

###  1.7. Outline

In Section 2, we collect the essential notions and results for later parts of
the article. An overview of the Moment Conjecture and its connection to our
period integral set-up can be found in Section 3.1. The proof of Theorem 1.1
is discussed in Section 4. The proof of Theorem 1.2 can be found in Section
4.2, 4.6 and Appendix B. We shall explicate the connections of Theorem 1.2 to
the Moment Conjecture of CFKRS in Section 5. We have made an effort in making
this article as self-contained as possible.

###  1.8. Acknowledgement

The reseach is supported by the EPSRC grant: EP/W009838/1. Part of the work
was completed during the author’s stay at the Chinese University of Hong Kong
and Queen’s University. The author would like to thank their generous
hospitality.

##  2\. Preliminary

In this section, we collect results and notions from the sources [Gold,
Bump84, Bu18] that are essential to our arguments. We have adjusted the
conventions to strive for consistency and to follow the recent shift of
conventions (closest to [Bu20]) which better align with those in the theory of
automorphic representation. Extra attention will be paid on the archimedean
calculations involving Whittaker functions as they will be important to this
article.

Some of the preliminary results can also be found in [Kw23], but we must
introduce a number of necessary modifications to account for the non-cuspidal
cases.

###  2.1. Notations and Conventions

Throughout this article, we use the following notations:
$\Gamma_{\mathbb{R}}(s):=\pi^{-s/2}\Gamma(s/2)$ $(s\in\mathbb{C})$;
$e(x):=e^{2\pi ix}$ $(x\in\mathbb{R})$; $\Gamma_{n}:=SL_{n}(\mathbb{Z})$
$(n\geq 2)$. We will be using both $\Gamma$ and $\Gamma_{\mathbb{R}}$ in this
article as the former is commonly used in our references whereas the latter
simplifies our formulae at several occasions. Without otherwise specified, our
test function $H$ lies in the class $\mathcal{C}_{\eta}$ and $H=h^{\\#}$. We
may use the same symbol to denote a function (in the variable $s$) and its
analytic continuation.

We will frequently encounter contour integrals of the shape

$\displaystyle\int_{-i\infty}^{i\infty}\ \cdots\ \int_{-i\infty}^{i\infty}\
(\cdots)\ \ \frac{ds_{1}}{2\pi i}\ \cdots\ \frac{ds_{k}}{2\pi i}$

where the contours involved should follow Barnes’ convention: they pass to the
right of all of the poles of the gamma functions in the form $\Gamma(s_{i}+a)$
and to the left of all of the poles of the gamma functions in the form
$\Gamma(a-s_{i})$.

We also adopt the following set of conventions:

  1. (1)

All Maass cusp forms will be simultaneous eigenfunctions of the Hecke
operators and will be either even or odd. Also, their first Fourier
coefficients are equal to $1$. In this case, the forms are said to be Hecke-
normalized. Note that there are no odd form for $SL_{3}(\mathbb{Z})$, see
Proposition 9.2.5 of [Gold].

  2. (2)

Our fixed Maass cusp form $\Phi$ of $SL_{3}(\mathbb{Z})$ is assumed to be
tempered at $\infty$, i.e., its Langlands parameters are purely imaginary.

  3. (3)

Denote by $\theta$ the best progress towards the Ramanujan conjecture for the
Maass cusp forms of $SL_{3}(\mathbb{Z})$. We have
$\theta\leq\frac{1}{2}-\frac{1}{10}$, see Theorem 12.5.1 of [Gold].

###  2.2. (Spherical) Whittaker Functions & Transforms

In this article, all Whittaker functions are referred to the spherical ones.
The Whittaker function of $GL_{2}(\mathbb{R})$ is given by

$\displaystyle W_{\mu}(y)\ :=\ 2\sqrt{y}K_{\mu}(2\pi y)$ (2.1)

for $\mu\in\mathbb{C}$ and $y>0$. Under this normalization, the following
holds:

######  Proposition 2.1.

For $\operatorname{Re}(w+\frac{1}{2}\pm\mu)>0$, we have

$\displaystyle\int_{0}^{\infty}W_{\mu}(y)y^{w}\ d^{\times}y\ =\
\frac{\pi^{-w-\frac{1}{2}}}{2}\
\Gamma\left(\frac{w+\frac{1}{2}+\mu}{2}\right)\Gamma\left(\frac{w+\frac{1}{2}-\mu}{2}\right).$
(2.2)

###### Proof.

Standard, see equation (2.5.2) of [Mo97] for instance. ∎

For the group $GL_{3}(\mathbb{R})$, we introduce the function

$\displaystyle I_{\alpha}(y_{0},y_{1})\ \ =\ \
I_{\alpha}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\ &&1\end{pmatrix}\ \ :=\ \
y_{0}^{1-\alpha_{3}}y_{1}^{1+\alpha_{1}}$ (2.3)

for $y_{0},y_{1}>0$ and
$\alpha\in\mathfrak{a}_{\mathbb{C}}^{(3)}:=\left\\{(\alpha_{1},\alpha_{2},\alpha_{3})\in\mathbb{C}^{3}:\alpha_{1}+\alpha_{2}+\alpha_{3}=0\right\\}$.
The Whittaker function for $GL_{3}(\mathbb{R})$, denoted by
$W_{\alpha}(y_{0},y_{1})=W_{\alpha}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}$, is defined in terms of the Jacquet integral:

$\displaystyle\prod_{1\leq j<k\leq
3}\Gamma_{\mathbb{R}}(1+\alpha_{j}-\alpha_{k})\
\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{\mathbb{R}}\
I_{\alpha}\left[\begin{pmatrix}&&1\\\ &-1&\\\
1&&\end{pmatrix}\begin{pmatrix}1&u_{1,2}&u_{1,3}\\\ &1&u_{2,3}\\\
&&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]\cdot e(-u_{1,2}-u_{2,3})\ du_{1,2}\ du_{1,3}\
du_{2,3}$ (2.4)

for $y_{0},y_{1}>0$ and $\alpha\in\mathfrak{a}_{\mathbb{C}}^{(3)}$. See
Chapter 5.5 of [Gold] for details.

Moreover, the Whittaker function of $GL_{3}(\mathbb{R})$ admits the following
Mellin-Barnes representation (also known as the Vinogradov-Takhtadzhyan
formula):

######  Proposition 2.2.

Assume $\alpha\in\mathfrak{a}_{\mathbb{C}}^{(3)}$ is tempered, i.e.,
$\operatorname{Re}\alpha_{i}=0$ ($i=1,2,3$). Then for any
$\sigma_{0},\sigma_{1}>0$,

$\displaystyle W_{-\alpha}(y_{0},y_{1})\ =\ \frac{1}{4}\
\int_{(\sigma_{0})}\int_{(\sigma_{1})}G_{\alpha}(s_{0},s_{1})y_{0}^{1-s_{0}}y_{1}^{1-s_{1}}\
\frac{ds_{0}}{2\pi i}\frac{ds_{1}}{2\pi i},\hskip 20.0pty_{0},y_{1}\ >\ 0,$
(2.5)

where

$\displaystyle G_{\alpha}(s_{0},s_{1})\ :=\
\frac{\prod\limits_{i=1}^{3}\Gamma_{\mathbb{R}}\left(s_{0}+\alpha_{i}\right)\Gamma_{\mathbb{R}}\left(s_{1}-\alpha_{i}\right)}{\Gamma_{\mathbb{R}}(s_{0}+s_{1})}.$
(2.6)

###### Proof.

See Chapter X of [Bump84]. ∎

######  Remark 2.3.

Notice the sign convention of the $\alpha_{i}$’s in formula (2.5) — it is
consistent with [Bu20] but is opposite to that of (6.1.4)$-$(6.1.5) in [Gold].
It is important to keep track of this convention, especially in Proposition
2.6, 4.10, 4.9 and Section 5.

######  Corollary 2.4.

For any $-\infty<A_{0},A_{1}<1$, we have

$\displaystyle\left|W_{-\alpha}(y_{0},y_{1})\right|\ \ll\
y_{0}^{A_{0}}y_{1}^{A_{1}},\hskip 20.0pty_{0},y_{1}\ >\ 0,$ (2.7)

where the implicit constant depends only on $\alpha,A_{0},A_{1}$.

###### Proof.

Follows directly from Proposition 2.2 by contour shifting. ∎

We will need the explicit evaluation of the $GL_{3}(\mathbb{R})\times
GL_{2}(\mathbb{R})$ Rankin-Selberg integral. It is a consequence of the Second
Barnes Lemma stated as follows.

######  Lemma 2.5.

For $a,b,c,d,e,f\in\mathbb{C}$ with $f=a+b+c+d+e$, we have

$\displaystyle\int_{-i\infty}^{i\infty}\frac{\Gamma(w+a)\Gamma(w+b)\Gamma(w+c)\Gamma(d-w)\Gamma(e-w)}{\Gamma(w+f)}\
$ $\displaystyle\frac{dw}{2\pi i}$ $\displaystyle\ =\
\frac{\Gamma(d+a)\Gamma(d+b)\Gamma(d+c)\Gamma(e+a)\Gamma(e+b)\Gamma(e+c)}{\Gamma(f-a)\Gamma(f-b)\Gamma(f-c)}.$
(2.8)

The contour must follow the Barnes convention (see Section 2.1).

###### Proof.

See Bailey [Ba64]. ∎

######  Proposition 2.6.

Let $W_{\mu}$ and $W_{-\alpha}$ be the Whittaker functions of
$GL_{2}(\mathbb{R})$ and $GL_{3}(\mathbb{R})$ respectively. For
$\operatorname{Re}s\gg 0$, we have

$\mathcal{Z}_{\infty}\left(s;\ W_{\mu},\ W_{-\alpha}\right)\ :=\
\int_{0}^{\infty}\int_{0}^{\infty}\ W_{\mu}(y_{1})\cdot
W_{-\alpha}(y_{0},y_{1})\cdot(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\
\frac{dy_{0}dy_{1}}{y_{0}y_{1}^{2}}\ =\ \frac{1}{4}\ \cdot\
\prod_{\pm}\prod_{k=1}^{3}\
\Gamma_{\mathbb{R}}\left(s\pm\mu-\alpha_{k}\right).$ (2.9)

###### Proof.

This was first proved in [Bump88]. To clear up any potential confusions
regarding conventions, we supply a short proof in Appendix A. ∎

The following pair of integral transforms plays an important role in the
archimedean aspect of this article.

######  Definition 2.7.

Let $h:(0,\infty)\rightarrow\mathbb{C}$ and
$H:i\mathbb{R}\rightarrow\mathbb{C}$ be measurable functions with
$H(\mu)=H(-\mu)$. Let $W_{\mu}(y):=2\sqrt{y}K_{\mu}(2\pi y)$. Then the
Kontorovich-Lebedev transform of $h$ is defined by

$h^{\\#}(\mu)\ :=\ \int_{0}^{\infty}h(y)\cdot W_{\mu}(y)\ \frac{dy}{y^{2}},$
(2.10)

whereas its inverse transform is defined by

$\displaystyle H^{\flat}(y)\ =\ \frac{1}{4\pi i}\int_{(0)}\ H(\mu)\cdot
W_{\mu}(y)\ \frac{d\mu}{\left|\Gamma(\mu)\right|^{2}},$ (2.11)

provided the integrals converge absolutely.

######  Definition 2.8.

Let $\mathcal{C}_{\eta}$ be the class of holomorphic functions $H$ on the
vertical strip $|\operatorname{Re}\mu|<2\eta$ such that

  1. (1)

$H(\mu)=H(-\mu)$,

  2. (2)

$H$ has rapid decay in the sense that

$\displaystyle H(\mu)\ \ll\ e^{-2\pi|\mu|}\hskip
50.0pt(|\operatorname{Re}\mu|\ <\ 2\eta).$ (2.12)

In this article, we take $\eta>40$ without otherwise specifying.

######  Proposition 2.9.

For any $H\in\mathcal{C}_{\eta}$, the integral (2.11) defining $H^{\flat}$
converges absolutely. Moreover, we have

$\displaystyle H^{\flat}(y)\ \ll\ \min\\{y,y^{-1}\\}^{\eta}\hskip 20.0pt(y\ >\
0).$ (2.13)

###### Proof.

See Lemma 2.10 of [Mo97]. ∎

######  Proposition 2.10.

Under the same assumptions of Proposition 2.9, we have

$\displaystyle(h^{\\#})^{\flat}(g)\ =\ h(g)\hskip 20.0pt\text{ and }\hskip
20.0pt(H^{\flat})^{\\#}(\mu)\ =\ H(\mu).$ (2.14)

###### Proof.

See Lemma 2.10 of [Mo97]. ∎

######  Lemma 2.11.

Suppose $H\in\mathcal{C}_{\eta}$ and $h:=H^{\flat}$. On the vertical strip
$-\frac{1}{2}<\operatorname{Re}w<\eta$, we have

$\displaystyle\widetilde{h}(w)\ :=\ \int_{0}^{\infty}h(y)y^{w}\ d^{\times}y\
=\ \frac{\pi^{-w-\frac{1}{2}}}{4}\int_{(0)}\
H(\mu)\cdot\frac{\Gamma\left(\frac{w+\frac{1}{2}+\mu}{2}\right)\Gamma\left(\frac{w+\frac{1}{2}-\mu}{2}\right)}{\left|\Gamma(\mu)\right|^{2}}\
\frac{d\mu}{2\pi i},$ (2.15)

###### Proof.

Follows directly from Propositions 2.10 and 2.2.

∎

###  2.3. Automorphic Forms of $GL(2)$

Let

$\displaystyle\mathfrak{h}^{2}:=\left\\{\begin{pmatrix}1&u\\\
&1\end{pmatrix}\begin{pmatrix}y&\\\ &1\end{pmatrix}:u\in\mathbb{R},\
y>0\right\\}$

with its invariant measure given by $y^{-2}dudy$. Let
$\Delta:=-y^{2}\left(\partial_{x}^{2}+\partial_{y}^{2}\right)$. An automorphic
form $\phi:\mathfrak{h}^{2}\rightarrow\mathbb{C}$ of
$\Gamma_{2}=SL_{2}(\mathbb{Z})$ satisfies
$\Delta\phi=\left(\frac{1}{4}-\mu^{2}\right)\phi$ for some
$\mu=\mu(\phi)\in\mathbb{C}$. We often identify $\mu$ with the pair
$(\mu,-\mu)\in\mathfrak{a}_{\mathbb{C}}^{(2)}$.

For $a\in\mathbb{Z}-\\{0\\}$, the $a$-th Fourier coefficient of $\phi$,
denoted by $\mathcal{B}_{\phi}(a)$, is defined by

$(\widehat{\phi})_{a}(y)\ :=\ \int_{0}^{1}\ \phi\left[\begin{pmatrix}1&u\\\
&1\end{pmatrix}\begin{pmatrix}y&\\\ &1\end{pmatrix}\right]e(-au)\ du\ =\
\frac{\mathcal{B}_{\phi}(a)}{\sqrt{|a|}}\cdot W_{\mu(\phi)}(|a|y).$ (2.16)

In the case of the Eisenstein series of $\Gamma_{2}$, i.e.,

$\displaystyle\phi\ =\ E(z;\mu)\ :=\ \frac{1}{2}\ \sum_{\gamma\in
U_{2}(\mathbb{Z})\setminus\Gamma_{2}}I_{\mu}(\operatorname{Im}\gamma z)\hskip
20.0pt(z\ \in\ \mathfrak{h}^{2}),$ (2.17)

where $I_{\mu}(y):=y^{\mu+\frac{1}{2}}$, it is well-known that $\Delta
E(*;\mu)\ =\ \left(\frac{1}{4}-\mu^{2}\right)E(*;\mu)$ and the Fourier
coefficients $\mathcal{B}(a;\mu)$ of $E(*;\mu)$ is given by

$\displaystyle\mathcal{B}(a;\mu)\ =\
\frac{|a|^{\mu}\sigma_{-2\mu}(|a|)}{\Lambda(1+2\mu)},$ (2.18)

where

$\displaystyle\Lambda(s)\ :=\ \pi^{-s/2}\Gamma(s/2)\zeta(s)\hskip 15.0pt\text{
and }\hskip 15.0pt\sigma_{-2\mu}(|a|)\ :=\ \sum_{d\mid a}d^{-2\mu}.$

The series (2.17) converges absolutely for $\operatorname{Re}\mu>1/2$ and it
admits a meromorphic continuation to $\mathbb{C}$.

###  2.4. Automorphic Forms of $GL(3)$

Let

$\displaystyle\mathfrak{h}^{3}\ :=\
\left\\{\begin{pmatrix}1&u_{1,2}&u_{1,3}\\\ &1&u_{2,3}\\\
&&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}:u_{i,j}\in\mathbb{R},\ y_{k}>0\right\\}$

and $\Phi:\mathfrak{h}^{3}\rightarrow\mathbb{C}$ be a Maass cusp form of
$\Gamma_{3}$ as defined in Definition 5.1.3 of [Gold]. In particular, there
exists $\alpha=\alpha(\Phi)\in\mathfrak{a}_{\mathbb{C}}^{(3)}$ such that for
any $D\in Z(U\mathfrak{gl}_{3}(\mathbb{C}))$ (the center of the universal
enveloping algebra of the Lie algebra $\mathfrak{gl}_{3}(\mathbb{C})$), we
have

$\displaystyle D\Phi\ =\ \lambda_{D}\Phi\hskip 20.0pt\text{ and }\hskip
20.0ptDI_{\alpha}\ =\ \lambda_{D}I_{\alpha}$

for some $\lambda_{D}\in\mathbb{C}$. The triple $\alpha(\Phi)$ is said to be
the Langlands parameters of $\Phi$.

######  Definition 2.12.

Let $m=(m_{1},m_{2})\in(\mathbb{Z}-\\{0\\})^{2}$ and
$\Phi:\mathfrak{h}^{3}\rightarrow\mathbb{C}$ be a Maass cusp form of
$SL_{3}(\mathbb{Z})$. For any $y_{0},y_{1}>0$, the integral defined by

$\displaystyle(\widehat{\Phi})_{(m_{1},m_{2})}\begin{pmatrix}y_{0}y_{1}&&\\\
&y_{0}&\\\ &&1\end{pmatrix}\ $ $\displaystyle:=\
\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\Phi\left[\begin{pmatrix}1&u_{1,2}&u_{1,3}\\\
&1&u_{2,3}\\\ &&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]\cdot e\left(-m_{1}u_{2,3}-m_{2}u_{1,2}\right)\
du_{1,2}\ du_{1,3}\ du_{2,3}.$ (2.19)

is said to be the $(m_{1},m_{2})$-th Fourier-Whittaker period of $\Phi$.
Moreover, the $(m_{1},m_{2})$-th Fourier coefficient of $\Phi$ is the complex
number $\mathcal{B}_{\Phi}(m_{1},m_{2})$ for which

$\displaystyle\hskip
40.0pt(\widehat{\Phi})_{(m_{1},m_{2})}\begin{pmatrix}y_{0}y_{1}&&\\\
&y_{0}&\\\ &&1\end{pmatrix}\ \ =\ \
\frac{\mathcal{B}_{\Phi}(m_{1},m_{2})}{|m_{1}m_{2}|}\
W_{\alpha(\Phi)}^{{}^{\operatorname{sgn}(m_{2})}}\begin{pmatrix}(|m_{1}|y_{0})(|m_{2}|y_{1})&&\\\
&|m_{1}|y_{0}&\\\ &&1\end{pmatrix}$ (2.20)

holds for any $y_{0},y_{1}>0$.

######  Remark 2.13.

The multiplicity-one theorem of Shalika (see Theorem 6.1.6 of [Gold])
guarantees the well-definedness of the Fourier coefficients for $\Phi$.

We will also need certain explicit calculations of Fourier-Whitttaker periods
for the Eisenstein series of $\Gamma_{3}:=SL_{3}(\mathbb{Z})$, e.g.,
repackaging (3.1) into a suitable period, extract the remaining $1$-swap terms
and their corresponding identities, and perform the necessary analytic
continuation.

The standard reference is Chapter VII of [Bump84] and the relevant
calculations can be found therein, see Theorem 7.2 in particular with
equations (7.7)-(7.8) will be the most useful to us, as well as equations
(3.10)-(3.15), (3.40) & (3.45) in Chapter III of [Bump84] for the
normalizations of the degenerate (spherical) Whittaker functions of
$GL_{3}(\mathbb{R})$. It is also helpful to consult the reformulation by
[Bu18] (Section 4) which is more streamlined and up-to-date. However, readers
should beware that the Whittaker functions used in [Bu18] are the incomplete
ones whereas the ones for this article and [Bump84] are the complete ones.

######  Definition 2.14.

The minimal parabolic Eisenstein series of $\Gamma_{3}$ is defined by

$\displaystyle E_{\min}^{(3)}(g;\alpha)\ \ :=\ \ \sum_{\gamma\in
U_{3}(\mathbb{Z})\setminus\Gamma_{3}}\ I_{\alpha}(\gamma g)\hskip 20.0pt(g\
\in\ \mathfrak{h}^{3})$ (2.21)

which converges absolutely on the domain
$\alpha\in\mathfrak{a}_{\mathbb{C}}^{(3)}$ with
$\operatorname{Re}(\alpha_{1}-\alpha_{2})>1$ and
$\operatorname{Re}(\alpha_{2}-\alpha_{3})>1$ (see [Bu18] Section 4.1).

A fundamental fact in the theory of Eisenstein series is the meromorphic
continuation of $\alpha\mapsto E_{\min}^{(3)}(g;\alpha)$ to
$\mathfrak{a}_{\mathbb{C}}^{(3)}$. This is accomplished explicitly in [Bump84]
and [Bu18] for the special case (2.21).

We will have to keep track of the degenerate terms in the Fourier expansion of
$E_{\min}^{(3)}(g;\alpha)$ for our intended applications. It is preferrrable
to apply the simpler form of such expansion (Lemma 2.15 below) as much as
possible because it detects:

  1. (1)

the equivariance of the Fourier-Whittaker periods (under unipotent
translations); and

  2. (2)

the annihilation of a number of degenerate terms in the Fourier expansion (due
to the unipotent integral in (4.1))

better than the fully explicit form used by [Li14] in her derivation of the
Voronoi formula for the triple divisor function. These two features will
result in much simpler formulae and clearer presentation.

######  Lemma 2.15.

Let $\Phi:\mathfrak{h}^{3}\rightarrow\mathbb{C}$ be an automorphic function of
$\Gamma_{3}$ (not necessarily cuspidal). Then

$\displaystyle\Phi(g)\ =\ (\widehat{\Phi})_{(0,0)}(g)\ +\ \sum_{n_{2}\neq 0}\
(\widehat{\Phi})_{(0,n_{2})}(g)\ +\ \sum_{\gamma\in
U_{2}(\mathbb{Z})\setminus\Gamma_{2}}\ \sum_{n_{1}=1}^{\infty}\
(\widehat{\Phi})_{(n_{1},0)}\left[\begin{pmatrix}\gamma&\\\
&1\end{pmatrix}g\right]\ +\ \Phi^{ND}(g)$ (2.22)

for $g\in\mathfrak{h}^{3}$, where

$\displaystyle\Phi^{ND}(g)\ :=\ \sum_{\gamma\in
U_{2}(\mathbb{Z})\setminus\Gamma_{2}}\ \sum_{m_{1}=1}^{\infty}\
\sum_{m_{2}\neq
0}(\widehat{\Phi})_{(m_{1},m_{2})}\left[\begin{pmatrix}\gamma&\\\
&1\end{pmatrix}g\right]$ (2.23)

is said to be the non-degenerate part of $\Phi$.

###### Proof.

See Theorem 8.3 of Goldfeld-Li [GL18] or Proposition 4.2 of [IY15]. Notice
that the sum of the first two terms on the right side of (2.22) were expressed
as a single integral

$\displaystyle\int_{0}^{1}\int_{0}^{1}\ \Phi\left[\begin{pmatrix}1&&u_{1,3}\\\
&1&u_{2,3}\\\ &&1\end{pmatrix}g\right]\ du_{1,3}\ du_{2,3}$

in [GL18]. The proof can be obtained easily by following the standard argument
(see Theorem 5.3.2 of [Gold]) with a bit of extra book-keeping. ∎

Following [Bu18], we introduce the Weyl group $W_{3}$ of $GL_{3}(\mathbb{R})$
consisting of the elements

$\displaystyle I\ $ $\displaystyle=\ \begin{pmatrix}1&&\\\ &1&\\\
&&1\end{pmatrix},\hskip 15.0ptw_{2}\ =\ -\begin{pmatrix}&1&\\\ 1&&\\\
&&1\end{pmatrix},\hskip 15.0ptw_{3}\ =\ -\begin{pmatrix}1&&\\\ &&1\\\
&1&\end{pmatrix}$ $\displaystyle w_{4}\ $ $\displaystyle=\
\begin{pmatrix}&1&\\\ &&1\\\ 1&&\end{pmatrix},\hskip 15.0ptw_{5}\ =\
-\begin{pmatrix}&&1\\\ 1&&\\\ &1&\end{pmatrix},\hskip 15.0ptw_{\ell}\ =\
-\begin{pmatrix}&&1\\\ &1&\\\ 1&&\end{pmatrix}$ (2.24)

and the corresponding Weyl actions on the Langlands parameters

$\displaystyle(\alpha_{1},\alpha_{2},\alpha_{3})^{I}\ $ $\displaystyle=\
(\alpha_{1},\alpha_{2},\alpha_{3}),\hskip
15.0pt(\alpha_{1},\alpha_{2},\alpha_{3})^{w_{2}}\ =\
(\alpha_{2},\alpha_{1},\alpha_{3}),\hskip
15.0pt(\alpha_{1},\alpha_{2},\alpha_{3})^{w_{3}}\ =\
(\alpha_{1},\alpha_{3},\alpha_{2}),$
$\displaystyle(\alpha_{1},\alpha_{2},\alpha_{3})^{w_{4}}\ $ $\displaystyle=\
(\alpha_{3},\alpha_{1},\alpha_{2}),\hskip
15.0pt(\alpha_{1},\alpha_{2},\alpha_{3})^{w_{5}}\ =\
(\alpha_{2},\alpha_{3},\alpha_{1}),\hskip
15.0pt(\alpha_{1},\alpha_{2},\alpha_{3})^{w_{\ell}}\ =\
(\alpha_{3},\alpha_{2},\alpha_{1}).$ (2.25)

Converting equations (3.40) & (3.45) of [Bump84] into the notations of this
article, we have

######  Definition 2.16.

Let $W_{\mu}^{(2)}(\cdot)$ be the Whittaker function of $GL_{2}(\mathbb{R})$
as defined in (2.1). For each $w\in W_{3}$, we define the degenerate Whittaker
functions of $GL_{3}(\mathbb{R})$ as

$\displaystyle W_{\alpha}^{{}^{(0,1)}}\left[(y_{0},y_{1}),w\right]\ $
$\displaystyle:=\
(y_{0}^{2}y_{1})^{\frac{1-\alpha_{3}^{w}}{2}}\cdot\Gamma_{\mathbb{R}}\left(1+\alpha_{2}^{w}-\alpha_{3}^{w}\right)\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\cdot
W_{(\alpha_{1}^{w}-\alpha_{2}^{w})/2}^{{}^{(2)}}\left(y_{1}\right)$ (2.26)

and

$\displaystyle W_{\alpha}^{{}^{(1,0)}}\left[(y_{0},y_{1}),w\right]\ :=\
\left(y_{0}y_{1}^{2}\right)^{\frac{1+\alpha_{1}^{w}}{2}}\cdot\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{2}^{w}\right)\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)W_{(\alpha_{2}^{w}-\alpha_{3}^{w})/2}^{{}^{(2)}}(y_{0})$
(2.27)

for any $\alpha\in\mathfrak{a}_{\mathbb{C}}^{(3)}$ and $y_{0},y_{1}>0$.

######  Remark 2.17.

Polynomial growths are present in (2.26) and (2.27) when one of $y_{i}$’s
tends to $\infty$.

It will be more convenient to consider the complete minimal parabolic
Eisenstein series of $\Gamma_{3}$, i.e.,

$\displaystyle(E_{\min}^{(3)})^{*}(\ *\ ;\ \alpha)\ :=\
\Lambda(1+\alpha_{1}-\alpha_{2})\Lambda(1+\alpha_{2}-\alpha_{3})\Lambda(1+\alpha_{1}-\alpha_{3})E_{\min}^{(3)}(\
*\ ;\ \alpha).$ (2.28)

The presence of the completion factors in (2.28) simplifies many formulae and
ensures the Hecke combinatorics of the non-degenerate Fourier coefficients
coincide in the cuspidal and Eisenstein cases. Also, it eliminates the need of
invoking the zero-free region for the $\zeta$-function in our argument.
(Notice that the formulae stated in [Li14] and [Bu18] are for incomplete
Eisenstein series.)

We record two results regarding the explicit evaluations of the degenerate
Fourier-Whittaker periods when $\Phi$ is equal to (2.28).

######  Lemma 2.18.

Suppose $\Phi$ is given by (2.28). For any $y_{0},y_{1}>0$ and
$\alpha\in\mathfrak{a}_{\mathbb{C}}^{(3)}$, we have

$\displaystyle\left(\widehat{\Phi}\right)_{(0,1)}(y_{0},y_{1})\ =\
\sum_{w=w_{2},w_{4},w_{\ell}}\
\zeta(w,\alpha,\psi_{(0,1)})W^{{}^{(0,1)}}_{\alpha}[(y_{0},y_{1}),w],$ (2.29)

where

$\displaystyle\zeta(w,\alpha,\psi_{(0,1)})\ :=\
\zeta\left(1+\alpha_{2}^{w}-\alpha_{3}^{w}\right)\zeta\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right).$
(2.30)

###### Proof.

See equation (7.8) of [Bump84] (or (4.6) of [Bu18]). ∎

######  Lemma 2.19.

Suppose $\Phi$ is given by (2.28). For any $y_{0},y_{1}>0$,
$\alpha\in\mathfrak{a}_{\mathbb{C}}^{(3)}$ and $n_{1}\in\mathbb{Z}-\\{0\\}$,
we have

$\displaystyle\left(\widehat{\Phi}\right)_{(n_{1},0)}(y_{0},y_{1})\ =\
\sum_{w=w_{3},w_{5},w_{\ell}}\
\frac{\zeta(w,\alpha,\psi_{(n_{1},0)})}{|n_{1}|^{1-\alpha_{3}^{w}}}\
W^{{}^{(1,0)}}_{\alpha}\left[(n_{1}y_{0},y_{1}),w\right],$ (2.31)

where

$\displaystyle\zeta\left(w,\alpha,\psi_{(n_{1},0)}\right)\ :=\
\zeta\left(1+\alpha_{1}^{w}-\alpha_{2}^{w}\right)\zeta\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\sigma_{\alpha_{2}^{w}-\alpha_{3}^{w}}(|n_{1}|).$
(2.32)

###### Proof.

See equation (7.7) of [Bump84] (or (4.7) of [Bu18]). ∎

###  2.5. The Key Identity

######  Proposition 2.20.

For any smooth automorphic function $\Phi$ of $\Gamma_{3}$, we have, for any
$y_{0},y_{1}>0$,

$\displaystyle\int_{0}^{1}\Phi\left[\begin{pmatrix}1&u_{1,2}&\\\ &1&\\\
&&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]\ $ $\displaystyle e(-u_{1,2})\ du_{1,2}$
$\displaystyle=\
\sum_{a_{0},a_{1}=-\infty}^{\infty}(\widehat{\Phi})_{(a_{1},1)}\left[\begin{pmatrix}1&&\\\
&1&\\\ &-a_{0}&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right].$ (2.33)

###### Proof.

See Proposition 6.1 of [Kw23]. ∎

###  2.6. Automorphic $L$-functions

If $\Phi$ and $\phi$ are Maass cusp forms of $\Gamma_{3}$ and $\Gamma_{2}$,
they are assumed to be Hecke-normalized as part of the conventions for this
paper. We denote their Langlands parameters by
$\alpha=\alpha(\Phi)\in\mathfrak{a}_{\mathbb{C}}^{(3)}$ and
$\mu=\mu(\phi)\in\mathfrak{a}_{\mathbb{C}}^{(2)}$ respectively. Let
$\widetilde{\Phi}(g):=\Phi\left({}^{t}g^{-1}\right)$ be the dual form of
$\Phi$. It is not hard to show that the Langlands parameters of
$\widetilde{\Phi}$ are given by $-\alpha$.

######  Definition 2.21.

Suppose $\Phi$ and $\phi$ are Maass cusp forms (or Eisenstein series) of
$\Gamma_{3}$ and $\Gamma_{2}$ respectively. For $\operatorname{Re}s\gg 1$, the
Rankin-Selberg $L$-function of $\Phi$ and $\phi$ is defined by

$\displaystyle L\left(s,\phi\otimes\Phi\right)\ $ $\displaystyle:=\
\sum_{m_{1}=1}^{\infty}\
\sum_{m_{2}=1}^{\infty}\frac{\mathcal{B}_{\phi}(m_{2})\mathcal{B}_{\Phi}(m_{1},m_{2})}{\left(m_{1}^{2}m_{2}\right)^{s}}.$
(2.34)

######  Proposition 2.22.

Suppose $\Phi$ and $\phi$ are Maass cusp forms (or Eisenstein series) of
$\Gamma_{3}$ and $\Gamma_{2}$ respectively. In addition, assume that $\phi$ is
even. Then for any $\operatorname{Re}s\gg 1$, we have

$\displaystyle\int_{\Gamma_{2}\setminus GL_{2}(\mathbb{R})}\
\phi(g)\left(\widetilde{\Phi}\right)^{ND}\begin{pmatrix}g&\\\
&1\end{pmatrix}|\det g|^{s-\frac{1}{2}}\ dg\ =\
\frac{1}{2}\cdot\Lambda\left(s,\phi\otimes\widetilde{\Phi}\right),$ (2.35)

where

$\displaystyle\Lambda\left(s,\phi\otimes\widetilde{\Phi}\right)\ $
$\displaystyle:=\ L_{\infty}\left(s,\phi\otimes\widetilde{\Phi}\right)\cdot
L\left(s,\phi\otimes\widetilde{\Phi}\right)$ (2.36)

and

$\displaystyle L_{\infty}\left(s,\phi\otimes\widetilde{\Phi}\right)\ $
$\displaystyle:=\ \prod_{k=1}^{3}\
\Gamma_{\mathbb{R}}\left(s\pm\mu-\alpha_{k}\right).$ (2.37)

###### Proof.

Note that the unfolding described in Proposition 5.15 of [Kw23] (or Section
12.2 of [Gold]) remains in place as long as $\operatorname{Re}s\gg 1$ and with
the use of (2.23). ∎

In the rest of this article, we make use of the shorthands
$(\mathbb{P}_{2}^{3}F)(g):=F\begin{pmatrix}g&\\\ &1\end{pmatrix}$ and the
pairing

$\displaystyle\left(\phi,\
\mathbb{P}_{2}^{3}(\Phi^{ND})\cdot|\det*|^{\overline{s}-\frac{1}{2}}\right)_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}$

for the integral on the left side of (2.35).

######  Definition 2.23.

Let $\Phi:\mathfrak{h}^{3}\rightarrow\mathbb{C}$ be a Maass cusp form or
Eisenstein series of $\Gamma_{3}$. For $\operatorname{Re}s\gg 1$, the standard
$L$-function of $\Phi$ is defined by

$\displaystyle L\left(s,\Phi\right)\ :=\ \sum_{n=1}^{\infty}\
\frac{\mathcal{B}_{\Phi}(1,n)}{n^{s}}.$ (2.38)

In the rest of this article, we will not make use of the integral
representation of $L(s,\Phi)$. It suffices to know $L(s,\Phi)$ admits an
entire continuation and satisfies the following functional equation:

######  Proposition 2.24.

Let $\Phi:\mathfrak{h}^{3}\rightarrow\mathbb{C}$ be a Maass cusp form of
$\Gamma_{3}$. For any $s\in\mathbb{C}$, we have

$\displaystyle\Lambda\left(s,\Phi\right)\ =\ \Lambda(1-s,\widetilde{\Phi}),$
(2.39)

where

$\displaystyle\Lambda\left(s,\Phi\right)\ :=\
L_{\infty}\left(s,\Phi\right)\cdot L\left(s,\Phi\right)$ (2.40)

and

$\displaystyle L_{\infty}\left(s,\Phi\right)\ :=\ \prod_{k=1}^{3}\
\Gamma_{\mathbb{R}}\left(s+\alpha_{k}\right).$ (2.41)

###### Proof.

See Chapter 6.5 of [Gold] or [JPSS79]. ∎

Furthermore, the standard $L$-functions $L(s,\phi)$ and $L(s,\Phi)$ for
(Hecke-normalized) Maass cusp forms $\phi$ and $\Phi$ admit Euler products of
the shape

$\displaystyle L(s,\phi)\ =\ \prod_{p}\ \prod_{j=1}^{2}\
\left(1-\beta_{\phi,j}(p)p^{-s}\right)^{-1},\hskip 20.0ptL(s,\Phi)\ =\
\prod_{p}\ \prod_{k=1}^{3}\ \left(1-\alpha_{\Phi,k}(p)p^{-s}\right)^{-1}$
(2.42)

respectively for $\operatorname{Re}s\gg 1$. If $\phi:=E^{*}(*;\mu)$ and
$\Phi:=(E_{\min}^{(3)})^{{}^{*}}(\ *\ ;\ \alpha)$, then we have

$\displaystyle\hskip
20.0pt\left\\{\beta_{\phi,1}(p),\beta_{\phi,2}(p)\right\\}\ =\
\\{p^{\mu},p^{-\mu}\\}\hskip 15.0pt\text{ and }\hskip
15.0pt\left\\{\alpha_{\Phi,1}(p),\alpha_{\Phi,2}(p),\alpha_{\Phi,3}(p)\right\\}\
=\ \left\\{p^{-\alpha_{1}},p^{-\alpha_{2}},p^{-\alpha_{3}}\right\\}$

in (2.42). Then by Cauchy’s identity, one can show that

$\displaystyle L(s,\phi\otimes\Phi)\ =\ \prod_{p}\ \prod_{j=1}^{2}\
\prod_{k=1}^{3}\left(1-\beta_{\phi,j}(p)\alpha_{\Phi,k}(p)p^{-s}\right)^{-1},$
(2.43)

cf. the argument of Proposition 7.4.12 of [Gold]. In particular, we have the
following special cases of Proposition 2.22 in terms of products of the
standard $L$-functions of $GL(1)$, $GL(2)$ and $GL(3)$.

######  Corollary 2.25.

  1. (1)

Suppose $\Phi$ is a Hecke-Maass cusp form of $\Gamma_{3}$. For
$\operatorname{Re}(s\pm\mu)\gg 1$, we have

$\displaystyle\left(E(*;\mu),\
\left(\mathbb{P}_{2}^{3}\Phi\right)\cdot|\det*|^{\bar{s}-\frac{1}{2}}\right)_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}\ \ =\ \ \frac{1}{2}\
\frac{\Lambda(s+\mu,\widetilde{\Phi})\Lambda(s-\mu,\widetilde{\Phi})}{\Lambda(1+2\mu)}.$
(2.44)

  2. (2)

Suppose $\Phi$ is given by (2.28). For $\operatorname{Re}s\gg 1$, we have

$\displaystyle\hskip 61.7pt\left(\phi,\
\mathbb{P}_{2}^{3}\left(\Phi^{ND}\right)\cdot|\det*|^{\overline{s}-\frac{1}{2}}\right)_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}\ \ =\ \ \frac{1}{2}\
\Lambda(s-\alpha_{1},\phi)\Lambda(s-\alpha_{2},\phi_{j})\Lambda(s-\alpha_{3},\phi).$
(2.45)

  3. (3)

Suppose $\Phi$ is given by (2.28). For $\operatorname{Re}(s\pm\mu)\gg 1$, we
have

$\displaystyle\left(E(*;\mu),\
\mathbb{P}_{2}^{3}\left(\Phi^{ND}\right)\cdot|\det*|^{\bar{s}-\frac{1}{2}}\right)_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}\ \ =\ \ \frac{1}{2}\
\frac{\prod\limits_{\pm}\prod\limits_{i=1}^{3}\
\Lambda\left(s\pm\mu-\alpha_{i}\right)}{\Lambda(1+2\mu)}.$ (2.46)

######  Remark 2.26.

When $\Phi$ is a Maass cusp form, the rapid decay of $\Phi$ at $\infty$ and
the automorphy of $\Phi$ imply the integrals on the left sides of (2.35) and
(2.44) converge absolutely for any $s\in\mathbb{C}$, uniformly on any compact
subset of $\mathbb{C}$, and away from the potential poles of $E(*;\mu)$.

###  2.7. Calculations on the Spectral Side

######  Definition 2.27.

Let $a\geq 1$ be an integer and $h\in C^{\infty}(0,\infty)$. The Poincaré
series of $\Gamma_{2}$ is defined by

$P^{a}(z;h)\ :=\ \sum_{\gamma\in
U_{2}(\mathbb{Z})\setminus\Gamma_{2}}h(a\operatorname{Im}\gamma z)\cdot
e\left(a\operatorname{Re}\gamma z\right)\hskip 20.0pt(z\ \in\
\mathfrak{h}^{2})$ (2.47)

provided it converges absolutely.

If the bounds

$\displaystyle h(y)\ \ll\ y^{1+\epsilon}\hskip 10.0pt\text{(as \ $y\to
0$)}\hskip 10.0pt\text{ and }\hskip 10.0pth(y)\ \ll\
y^{\frac{1}{2}-\epsilon}\hskip 10.0pt\text{ (as \ $y\to\infty$)}$ (2.48)

are satisfied, then the Poincaré series $P^{a}(z;h)$ converges absolutely and
represents an $L^{2}$-function. In this article, we take $h:=H^{\flat}$ with
$H\in\mathcal{C}_{\eta}$ and $\eta>40$. By Proposition 2.9, conditions (2.48)
clearly holds. We often use the shorthand $P^{a}:=P^{a}(*;h)$. Also, we use
$\langle\ \cdot\ ,\ \cdot\ \rangle$ to denote the Petersson inner product on
$\Gamma_{2}\setminus\mathfrak{h}^{2}$, i.e.,

$\displaystyle\left\langle\ \phi_{1}\ ,\ \phi_{2}\ \right\rangle\ :=\
\int_{\Gamma_{2}\setminus\mathfrak{h}^{2}}\
\phi_{1}(g)\cdot\overline{\phi_{2}(g)\raisebox{8.53581pt}{}}\ dg$

with $dg$ being the invariant measure on $\mathfrak{h}^{2}$.

######  Lemma 2.28.

Let $\phi$ be a Maass cusp form of $\Gamma_{2}$,
$\Delta\phi=\left(\frac{1}{4}-\mu^{2}\right)\phi$, and $\mathcal{B}_{\phi}(a)$
be the $a$-th Fourier coefficient of $\phi$. Then

$\displaystyle\big{\langle}\ P^{a}\ ,\ \phi\ \big{\rangle}\ $ $\displaystyle=\
|a|^{1/2}\cdot\overline{\mathcal{B}_{\phi}(a)\raisebox{11.38109pt}{}}\cdot
h^{\\#}\left(\overline{\mu\raisebox{8.53581pt}{}}\right).$

###### Proof.

See Lemma 5.22 of [Kw23]. ∎

Similarly, the following holds away from the poles of $E(*;\mu)$:

######  Lemma 2.29.

$\displaystyle\bigg{\langle}P^{a},\ E(*;\mu)\bigg{\rangle}\ =\
|a|^{1/2}\cdot\frac{|a|^{\overline{\mu}}\sigma_{-2\overline{\mu}}(|a|)}{\zeta^{*}(1+2\overline{\mu})}\cdot
h^{\\#}\left(\overline{\mu\raisebox{8.53581pt}{}}\right).$ (2.49)

###### Proof.

See Lemma 5.23 of [Kw23]. ∎

######  Proposition 2.30 (Spectral Expansion).

Suppose $f\in L^{2}(\Gamma_{2}\setminus\mathfrak{h}^{2})$ and $\langle
f,1\rangle=0$. Then

$\displaystyle f(z)\ =\ \sum_{j=1}^{\infty}\
\frac{\big{\langle}f,\phi_{j}\big{\rangle}}{\big{\langle}\phi_{j},\phi_{j}\big{\rangle}}\cdot\phi_{j}(z)\
+\ \int_{(0)}\ \bigg{\langle}f,\ E(*;\mu)\bigg{\rangle}\cdot E(z;\mu)\
\frac{d\mu}{4\pi i}\hskip 20.0pt(z\ \in\ \mathfrak{h}^{2})$ (2.50)

where $(\phi_{j})_{j\geq 1}$ is any orthogonal basis of Maass cusp forms for
$\Gamma_{2}$.

###### Proof.

See Theorem 3.16.1 of [Gold]. ∎

######  Proposition 2.31.

Let $\Phi$ be a Hecke-Maass cusp form or an Eisenstein series of $\Gamma_{3}$,
and $P^{a}$ be a Poincaré series of $\Gamma_{2}$. Then for
$\operatorname{Re}s\gg 1$, we have

$\displaystyle 2\cdot|a|^{-1/2}\cdot\left(P^{a},\
\mathbb{P}_{2}^{3}\left(\Phi^{ND}\right)\cdot|\det*|^{\overline{s}-\frac{1}{2}}\right)_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}$ $\displaystyle\ =\
\sideset{}{{}^{\prime}}{\sum}_{j=1}^{\infty}\
h^{\\#}\left(\overline{\mu_{j}\raisebox{8.53581pt}{}}\right)\cdot\frac{\overline{\mathcal{B}_{j}(a)\raisebox{11.38109pt}{}}\cdot\Lambda(s,\phi_{j}\otimes\widetilde{\Phi})}{\langle\phi_{j},\phi_{j}\rangle}$
$\displaystyle\hskip 40.0pt\ +\ \int_{(0)}\
h^{\\#}\left(\mu\right)\frac{\sigma_{-2\mu}(|a|)|a|^{-\mu}\Lambda(s+\mu,\widetilde{\Phi})\Lambda(s-\mu,\widetilde{\Phi})}{\left|\Lambda(1+2\mu)\right|^{2}}\
\frac{d\mu}{4\pi i},$ (2.51)

where the sum is restricted to an orthogonal basis $(\phi_{j})$ of even Hecke-
normalized Maass cusp forms for $\Gamma_{2}$ with
$\Delta\phi_{j}=\left(\frac{1}{4}-\mu_{j}^{2}\right)\phi_{j}$ and
$\mathcal{B}_{j}(a):=\mathcal{B}_{\phi_{j}}(a)$.

###### Proof.

Follows from a simple modification of Proposition 5.25 in [Kw23]. ∎

###  2.8. Analytic Properties of the Integral Transform

The dual integral transform for the present article as well as [Kw23] is given
by

$\displaystyle(\mathcal{F}_{\Phi}^{(\delta)}H)\left(s_{0},\ s;\ \phi\right)\ \
$ $\displaystyle:=\ \ \int_{(15)}\int_{(\epsilon)}\
\widetilde{h}\left(s-s_{1}-\frac{1}{2}\right)\cdot\mathcal{G}_{\Phi}^{(\delta)}\left(s_{1},u;\
s_{0},s;\ \phi\right)\ \frac{du}{2\pi i}\frac{ds_{1}}{2\pi i},$ (2.52)

where $\delta=\pm$, $h:=H^{\flat}$, $G_{\Phi}:=G_{\alpha(\Phi)}$ as defined in
(2.6), and

$\displaystyle\mathcal{G}_{\Phi}^{(\delta)}\left(s_{1},u;\ s_{0},s;\
\phi\right)\ :=\ $ $\displaystyle
G_{\Phi}\left(s_{0}-u,s_{1}\right)\cdot(2\pi)^{-u}e^{i\delta\phi
u}\Gamma(u)\cdot\frac{\Gamma\left(\frac{u+1-2s+s_{1}-s_{0}}{2}\right)\Gamma\left(\frac{2s-s_{0}-u}{2}\right)}{\Gamma\left(\frac{1+s_{1}}{2}-s_{0}\right)}.$
(2.53)

Adding the ‘$+$’ and ‘$-$’ pieces together and taking the limit
$\phi\to\pi/2$, we find the resultant is

$\displaystyle\left(\mathcal{F}_{\Phi}H\right)\left(s_{0},s\right)\ \ =\ \ $
$\displaystyle\sqrt{\pi}\
\int_{(\eta-1/2)}\widetilde{h}\left(s-s_{1}-\frac{1}{2}\right)\pi^{-s_{1}}\cdot\frac{\prod\limits_{i=1}^{3}\
\Gamma\left(\frac{s_{1}-\alpha_{i}}{2}\right)}{\Gamma\left(\frac{1+s_{1}}{2}-s_{0}\right)}$
$\displaystyle\hskip 50.0pt\cdot\int_{(\epsilon)}\
\frac{\Gamma\left(\frac{u}{2}\right)\Gamma\left(\frac{s_{1}-(s_{0}-u)}{2}+\frac{1}{2}-s\right)\prod\limits_{i=1}^{3}\
\Gamma\left(\frac{(s_{0}-u)+\alpha_{i}}{2}\right)\Gamma\left(s-\frac{s_{0}+u}{2}\right)}{\Gamma\left(\frac{1-u}{2}\right)\Gamma\left(\frac{(s_{0}-u)+s_{1}}{2}\right)}\
\frac{du}{2\pi i}\ \frac{ds_{1}}{2\pi i}$ (2.54)

upon making some simplifications. We write

$\displaystyle s\ =\ \sigma+it,\hskip 15.0pts_{0}=\sigma_{0}+it_{0},\hskip
15.0pts_{1}=\sigma_{1}+it_{1},\hskip 15.0pt\text{ and }\hskip 15.0ptu=\
\epsilon+iv,$

with $\epsilon:=1/100$. It is sufficient to consider $s$ inside the
rectangular box $\epsilon<\sigma<4$ and $|t|\leq T$, for any given $T\geq
1000$. Moreover, $\alpha_{k}:=i\gamma_{k}\in i\mathbb{R}$ $(k=1,2,3)$ by our
assumptions on $\Phi$. The following result constitutes one of the main
archimedean inputs and was proved in greater detail in [Kw23] (Section 8):

######  Proposition 2.32.

Suppose $H\in\mathcal{C}_{\eta}$.

  1. (1)

For any $\phi\in(0,\pi/2]$, the transform
$\big{(}\mathcal{F}_{\Phi}^{(\delta)}H\big{)}(s_{0},s;\phi)$ is holomorphic on
the domain

$\displaystyle\sigma_{0}\ >\ \epsilon,\hskip 15.0pt\sigma\ <\ 4,\hskip
15.0pt\text{ and }\hskip 15.0pt2\sigma-\sigma_{0}-\epsilon\ >\ 0.$ (2.55)

  2. (2)

Whenever $(\sigma_{0},\sigma)\in(\ref{newdomain})$, $|t|<T$, and
$\phi\in(0,\pi/2)$, the transform
$(\mathcal{F}_{\Phi}^{(\delta)}H)\left(s_{0},\ s;\ \phi\right)$ has
exponential decay as $|t_{0}|\to\infty$.

  3. (3)

There exists a constant $B=B_{\eta}$ such that whenever
$(\sigma_{0},\sigma)\in(\ref{newdomain})$, $|t|<T$, and $|t_{0}|\gg_{T}1$, we
have the estimate

$\displaystyle\left|(\mathcal{F}_{\Phi}^{(\delta)}H)\left(s_{0},\ s;\
\pi/2\right)\right|\ \ll\ |t_{0}|^{8-\frac{\eta}{2}}\log^{B}|t_{0}|,$ (2.56)

where the implicit constant depends only on $\eta$, $T$, $\Phi$.

######  Remark 2.33.

The domain (2.55) is chosen in a way that the function
$(s_{0},s)\mapsto\mathcal{G}_{\Phi}^{(\delta)}\left(s_{1},u;\ s_{0},s;\
\phi\right)$ is holomorphic on (2.55) when
$\operatorname{Re}s_{1}=\sigma_{1}\geq 15$ and $\operatorname{Re}u=\epsilon$.
Moreover, if we have $15\leq\sigma_{1}\leq\eta-\frac{1}{2}$ and (2.55), then
$s-s_{1}-1/2$ lies inside the region of holomorphy of $\widetilde{h}$.

######  Remark 2.34.

While (2.8) is the most useful for the purposes of this article, it should be
more conceptual to cast our integral transform in the following form:

$\displaystyle\left(\mathcal{F}_{\Phi}H\right)\left(s_{0},s\right)\ =\
\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\
h\left(\frac{y_{1}}{\sqrt{1+y_{0}^{2}}}\right)\frac{y_{0}^{2s-s_{0}}}{(1+y_{0}^{2})^{\frac{s}{2}-s_{0}+\frac{1}{4}}}W_{-\alpha}\left(X,y_{1}\right)e(Xy_{0})X^{s_{0}-1}y_{1}^{s-\frac{1}{2}}\
d^{\times}X\ \frac{dy_{0}dy_{1}}{y_{0}y_{1}^{2}},$ (2.57)

see Section 4.2 of [Kw23]. Also, we wish to point out several other forms of
the integral transforms obtained by Humphries-Khan [HK22+] (eq. (3.5)) and
Biró [Bi22+] (pp. 5). Interested readers should consult their works for the
formulae and the relevant regularity assumptions.

##  3\. Structures for the Main Terms

The authors of [CFKRS05] have offered an elegant heuristic to obtain the full
sets of main terms for a large class of moments of $L$-functions. Such a
heuristic nicely encodes the combinatorics and symmetries behind. In this
section, we quickly summarize the formulations and discussions regarding the
Moment Conjectures which are of interests to us. Useful references include
[CFKRS05, HY10, CIS12, CF22+]. (See the last one for the recent progress and
refinements of the conjectures.)

###  3.1. CFKRS for the Orthogonal Symmetry

We now describe their set-up in the particular case of our interest, i.e., the
shifted spectral cubic moment over an orthogonal basis of even Hecke-Maass
forms of $SL_{2}(\mathbb{Z})$:

$\displaystyle\sum_{j=1}^{\infty}\ H(\mu_{j})\
\frac{\Lambda(s-\alpha_{1},\phi_{j})\Lambda(s-\alpha_{2},\phi_{j})\Lambda(s-\alpha_{3},\phi_{j})}{\langle\phi_{j},\phi_{j}\rangle}$
$\displaystyle\hskip 120.0pt+\frac{1}{4\pi i}\ \int_{(0)}\ H(\mu)\
\frac{\prod\limits_{i=1}^{3}\Lambda(s+\mu-\alpha_{i})\Lambda(s-\mu-\alpha_{i})}{|\Lambda(1+2\mu)|^{2}}\
d\mu.$ (3.1)

In this case, the Moment Conjecture predicts $2^{3}=8$ main terms in total for
(3.1) which can be classified as follows. Firstly, the diagonal term, or the
‘$0$-swap term’ is of the form:

$\displaystyle\zeta(1-\alpha_{1}-\alpha_{2})\zeta(1-\alpha_{1}-\alpha_{3})\zeta(1-\alpha_{2}-\alpha_{3})\times\text{(archimedean
part)}.$ (3.2)

(For the sake of illustration, we have put $s=1/2$ in (3.1).) The archimedean
part is explicitly given by

$\displaystyle\int_{(0)}\
\frac{H(\mu)}{|\Gamma(\mu)|^{2}}\cdot\prod_{i=1}^{3}\
\Gamma\left(\frac{1/2+\mu-\alpha_{i}}{2}\right)\Gamma\left(\frac{1/2-\mu-\alpha_{i}}{2}\right)\
\frac{d\mu}{2\pi i}.$

Next, one alters the sign of exactly one of the components of
$(-\alpha_{1},-\alpha_{2},-\alpha_{3})$ and obtains the three triples

$\displaystyle\left(\alpha_{1},-\alpha_{2},-\alpha_{3}\right),\hskip
15.0pt\left(-\alpha_{1},\alpha_{2},-\alpha_{3}\right),\hskip
15.0pt\left(-\alpha_{1},-\alpha_{2},\alpha_{3}\right).$

Correspondingly, the following three main terms (up to the archimedean parts)
can be written down:

$\displaystyle\zeta(1+\alpha_{1}-\alpha_{2})\zeta(1+\alpha_{1}-\alpha_{3})\zeta(1-\alpha_{2}-\alpha_{3}),$
$\displaystyle\zeta(1-\alpha_{1}+\alpha_{2})\zeta(1-\alpha_{1}-\alpha_{3})\zeta(1+\alpha_{2}-\alpha_{3}),$
$\displaystyle\zeta(1-\alpha_{1}-\alpha_{2})\zeta(1-\alpha_{1}+\alpha_{3})\zeta(1-\alpha_{2}+\alpha_{3}).$

These three terms are known as the ‘$1$-swap terms’. The three $2$-swap terms
and the one $3$-swap term can be written down in a similar manner.

In this article, the use of the Fourier-Whittaker periods in the form of
Proposition 2.20 allows us to pinpoint the sources of the main terms in a
simple and unified way:

######  Proposition 3.1.

All of the eight main terms of the moment (3.1) originate from the right side
of (2.20) (with $a=1$ therein) and admit the following classification:

  1. (1)

$a_{0}=0$ and $a_{1}\neq 0$ give the $0$-swap term;

  2. (2)

$a_{0}\neq 0$ and $a_{1}=0$ give the three $1$-swap terms;

  3. (3)

$a_{0}\neq 0$ and $a_{1}\neq 0$ contain the three $2$-swap terms and the
$3$-swap term.

Furthermore, there is a single Mellin-Barnes identity for each of the class of
$0,1,2,3$-swap terms (up to the permutations of the shifts $\alpha_{i}$’s) and
is given by Proposition 2.6, 4.4, 4.9, 4.10 respectively.

One shall readily observe that the ‘$4=3+1$’ decomposition for the dual fourth
moment of the $\zeta$-function adopted in this work actually fits with the
aforementioned structures of the main terms of (3.1) better than the more
conventional ‘$4=2+2$’ counterpart (see [Iv02, Fr20] and [Mo93, Mo97] for
instance). Also, it facilitates the matching with the Moment Conjecture, say
without the need of performing multiple Taylor or Laurent expansions which
results in many transcendental constants ([Iv02, Fr20, Mo93, Mo97]), and it
treats the corresponding moments of $L$-functions away from the central point
simultaneously.

Moreover, shifted moments admit a natural classification according to their
symmetry types ([CFKRS05]). For example, the moment (3.1) is of
$SO(\text{even})$ type and the counterparts of $O$ and $SO(\text{odd})$ types
over the Maass spectrum are obvious. If one were studying (3.1) with the
Kuznetsov formulae, one would have to handle the ‘$J$-Bessel’ and the
‘$K$-Bessel’ pieces separately coming from the ‘same-sign’ and the ‘opposite-
sign’ Kuznetsov formulae respectively (cf. Theorem 2.2 & Theorem 2.4 of
[Mo97]). However, only a single off-diagonal piece in terms of the $GL(3)$
Whittaker function is present in our period integral approach (see (4.5)).
This results in simpler computations in our case.

###  3.2. CFKRS for Unitary Symmetry

To state the Moment Conjecture for the Riemann $\zeta$-function, we must
introduce the following Dirichlet series

$\displaystyle\mathcal{Z}_{A;B}(s)\ \ :=\ \
\sum_{\begin{subarray}{c}m=n\end{subarray}}\
\frac{\tau_{A}(m)\tau_{B}(n)}{(mn)^{\frac{1}{2}+s}},$ (3.3)

where $A$ and $B$ are finite sets, and $\tau_{A}(m):=\sum_{d_{1}\cdots
d_{k}=m}\ d_{1}^{-\alpha_{1}}\cdots d_{k}^{-\alpha_{k}}$ if
$A=\\{\alpha_{1},\ldots,\alpha_{k}\\}$ (similarly for $\tau_{B}(n)$). Indeed,
this arises quite naturally as only the diagonals should be preserved
according to the heuristic derivation of [CFKRS05]. It is not hard to show
that

$\displaystyle\mathcal{Z}_{A;B}(s)\ \ :=\ \ L(1+2s,E_{A}\otimes
E_{B})\cdot\widetilde{\mathcal{A}}_{AB}(s)$ (3.4)

for $\operatorname{Re}s>0$, where

$\displaystyle L(1+2s,E_{A}\otimes E_{B})\ \ :=\ \ \prod_{\alpha\in A}\
\prod_{\beta\in B}\ \zeta(1+2s+\alpha+\beta)$ (3.5)

and the infinite Euler product $\widetilde{\mathcal{A}}_{AB}(s)$ can be shown
to converge absolutely on $\Re s>-\delta$ for some $\delta>0$. Hence,
$\mathcal{Z}_{AB}(s)$ admits a meromorphic continuation to
$\operatorname{Re}>-\delta$.

In general, the Euler product $\widetilde{\mathcal{A}}_{AB}(s)$ does not admit
a clean formula, see [CIS12], [Ng21]. However, we only consider the specific
case $|A|=|B|=2$ in this article. Then the standard Rankin-Selberg theory
provides us a nice formula:

$\displaystyle\widetilde{\mathcal{A}}_{AB}(s)\ \ =\ \
\zeta\left(2+2s+\sum_{\alpha\in A}\ \alpha+\sum_{\beta\in B}\
\beta\right)^{-1}.$

######  Conjecture 3.2.

The full set of the main terms for the shifted moment

$\displaystyle\int_{\mathbb{R}}\ \eta\left(t\right)\ $
$\displaystyle\prod_{\alpha\in A}\ \zeta\left(\tfrac{1}{2}+it+\alpha\right)\
\prod_{\beta\in B}\zeta\left(\tfrac{1}{2}-it+\beta\right)\ dt$ (3.6)

is given by

$\displaystyle\int_{\mathbb{R}}\ \eta\left(t\right)\
\left\\{\sum_{\ell=0}^{\infty}\ \sum_{\begin{subarray}{c}U\subset A,\ V\subset
B\\\ |U|=|V|=\ell\end{subarray}}\
\left(\frac{t}{2\pi}\right)^{-U-V}\cdot\mathcal{Z}_{A_{U};B_{V}}(0)\right\\}\
dt,$ (3.7)

where $A_{U}:=A-U+V^{-}$, $B_{V}:=B-V+U^{-}$, and $-U-V:=-\sum_{\alpha\in
U}\alpha-\sum_{\beta\in V}\beta$.

The Conjecture 3.2 formulated above is expected to be true for any finite sets
$A$ and $B$ of shifts. When $|A|=|B|=2$, the expression inside $\\{\cdots\\}$
of (3.7) can be spelled out in more explicit terms:

$\displaystyle
Z_{\\{\alpha_{1},\alpha_{2}\\};\left\\{\beta_{1},\beta_{2}\right\\}}(0)\ $
$\displaystyle+\
Z_{\\{-\beta_{1},\alpha_{2}\\};\left\\{-\alpha_{1},\beta_{2}\right\\}}(0)\left(\frac{t}{2\pi}\right)^{-\alpha_{1}-\beta_{1}}\
+\
Z_{\\{-\beta_{2},\alpha_{2}\\};\left\\{\beta_{1},-\alpha_{1}\right\\}}(0)\left(\frac{t}{2\pi}\right)^{-\alpha_{1}-\beta_{2}}$
$\displaystyle+\
Z_{\\{\alpha_{1},-\beta_{1}\\};\left\\{-\alpha_{2},\beta_{2}\right\\}}(0)\left(\frac{t}{2\pi}\right)^{-\alpha_{2}-\beta_{1}}\
+\
Z_{\\{\alpha_{1},-\beta_{2}\\};\left\\{\beta_{1},-\alpha_{2}\right\\}}(0)\left(\frac{t}{2\pi}\right)^{-\alpha_{2}-\beta_{2}}$
$\displaystyle+\
Z_{\\{-\beta_{1},-\beta_{2}\\};\left\\{-\alpha_{1},-\alpha_{2}\right\\}}(0)\left(\frac{t}{2\pi}\right)^{-\alpha_{1}-\alpha_{2}-\beta_{1}-\beta_{2}}.$
(3.8)

In this case, the first and the last term of (3.2) are known as the ‘$0$-swap’
term and ‘$2$-swap’ term respectively, whereas the middle four terms of (3.2)
are known as the ‘$1$-swap’ terms. While (3.7) is slightly cumbersome in terms
of notations, it has the advantage of illustrating the nice combinatorial
patterns that work in general.

##  4\. Proof of Theorem 1.1

In this section, we fix our automorphic form $\Phi$ to be the complete minimal
parabolic Eisenstein series of $\Gamma_{3}$, i.e.,

$\displaystyle(E_{\min}^{(3)})^{*}(\ *\ ;\ \alpha)\ :=\
\Lambda(1+\alpha_{1}-\alpha_{2})\Lambda(1+\alpha_{2}-\alpha_{3})\Lambda(1+\alpha_{1}-\alpha_{3})E_{\min}^{(3)}(\
*\ ;\ \alpha).$ (4.1)

###  4.1. Unfolding & A Useful Decomposition

Recall the function $\Phi^{ND}$ as defined in in Lemma 2.15 (with (2.28)).
Observe that $\Phi^{ND}$ remains invariant under left translations by elements
of the form $\begin{pmatrix}\gamma&\\\ &1\end{pmatrix}$ with
$\gamma\in\Gamma_{2}$ (and more generally by any mirabolic element of
$\Gamma_{3}$). Thus, the pairing

$\displaystyle\left(P\ ,\
\mathbb{P}_{2}^{3}\left(\Phi^{ND}\right)\cdot|\det*|^{\bar{s}-1/2}\right)_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}$ (4.2)

as described in Definition 2.22 remains well-defined and the expression

$\displaystyle\bigg{(}P,\ \mathbb{P}_{2}^{3}\left(\Phi^{ND}\right)$
$\displaystyle\cdot|\det*|^{\overline{s}-\frac{1}{2}}\bigg{)}_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}$ $\displaystyle\ =\ \int_{0}^{\infty}\int_{0}^{\infty}\
h(y_{1})\cdot(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\cdot\int_{0}^{1}\hskip
5.0pt\left(\widetilde{\Phi}\right)^{ND}\left[\begin{pmatrix}1&u_{1,2}&\\\
&1&\\\ &&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]e(u_{1,2})\ du_{1,2}\
\frac{dy_{0}dy_{1}}{y_{0}y_{1}^{2}}$ (4.3)

converges absoluely whenever $1<\sigma<4$ (see Section 6.2 of [Kw23]).
Moreover, as long as $\sigma>3/2$ the above observation allows us to carry out
the Rankin-Selberg unfoldings of $GL(3)\times GL(2)$ on the spectral side as
in Definition 2.21, Corollary 2.25 and Proposition 2.31 (with the use of
(2.23)).

######  Remark 4.1.

Warning (!): To avoid confusion with the computations already done in the
literature (see Section 2.4, especially about the sign conventions for the
Langlands parameters), we will first compute (4.1) but with the innermost
integral being

$\displaystyle\int_{0}^{1}\ \Phi^{ND}\left[\begin{pmatrix}1&u_{1,2}&\\\ &1&\\\
&&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]\ $ $\displaystyle e(-u_{1,2})\ du_{1,2}.$ (4.4)

Then we replace $\Phi$ by $\widetilde{\Phi}$ and $\alpha$ by $-\alpha$
afterwards in Section 5.

######  Remark 4.2.

It is convenient to assume the $\alpha_{i}$’s to be distinct as in [CFKRS05].
All of the essential ideas of this article are present even with the
assumption $\alpha_{i}\in i\mathbb{R}$ (following [Kw23]). To facilitate
cross-referencing with the relevant propositions in [Kw23], we shall maintain
such an assumption throughout the rest of this article. In fact, this can be
removed easily by taking $|\alpha_{i}|<\epsilon/1000$ for $i=1,2$ and replace
domains of the form $a<\operatorname{Re}s<b$ by the smaller ones
$a+\epsilon<\operatorname{Re}s<b-\epsilon$ in Proposition 2.32 and Theorem
4.12.

From (2.22) and Proposition 2.20 with $a=1$, the last expression (4.4) can be
decomposed as

$\displaystyle\hskip 10.0pt\sum_{(a_{0},a_{1})\in\mathbb{Z}^{2}-\\{(0,0)\\}}\
$ $\displaystyle(\widehat{\Phi})_{(a_{1},1)}\left[\begin{pmatrix}1&&\\\ &1&\\\
&-a_{0}&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]$ (4.5) $\displaystyle+\
\left(\widehat{\Phi}\right)_{(0,1)}(y_{0},y_{1})\ -\ \sum_{n_{2}\neq 0}\
\int_{0}^{1}\ \ (\widehat{\Phi})_{(0,n_{2})}\left[\begin{pmatrix}1&u_{1,2}&\\\
&1&\\\ &&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]e(-u_{1,2})\ du_{1,2}$ (4.6) $\displaystyle\ -\
\sum_{\gamma\in U_{2}(\mathbb{Z})\setminus\Gamma_{2}}\
\sum_{n_{1}=1}^{\infty}\ \int_{0}^{1}\
(\widehat{\Phi})_{(n_{1},0)}\left[\begin{pmatrix}\gamma&\\\
&1\end{pmatrix}\begin{pmatrix}1&u_{1,2}&\\\ &1&\\\
&&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]e(-u_{1,2})\ du_{1,2}.$ (4.7)

In the following subsections, we will treat the expressions (4.5), (4.6) and
(4.7) separately.

###  4.2. The $1$-swap Terms and (4.5)

Observe that

$\displaystyle(\ref{usual})\ -\ \text{(Contributions from $a_{1}\neq 0$ in
(\ref{usual}))}\ =\ $ $\displaystyle\sum_{a_{0}\neq 0}\
(\widehat{\Phi})_{(0,1)}\left(\frac{y_{0}}{1+(a_{0}y_{0})^{2}},y_{1}\sqrt{1+(a_{0}y_{0})^{2}}\right).$
(4.8)

The contributions from $a_{1}\neq 0$ have more-or-less been handled in [Kw23].
We shall briefly summarize the relevant treatment in Section 4.6. In this
subsection, we only treat the terms in (4.5) with $a_{1}=0$ which turn out to
be equal to the three $1$-swap terms predicted by the CFKRS heuristics.

######  Proposition 4.3.

For $\frac{1}{2}<\sigma<4$, we have

$\displaystyle(\mathcal{M}_{\alpha}^{{}^{(1)}}h)(s)\ $ $\displaystyle:=\
\int_{0}^{\infty}\int_{0}^{\infty}\
h(y_{1})\cdot(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\cdot\sum_{a_{0}\neq 0}\
(\widehat{\Phi})_{(0,1)}\left[\begin{pmatrix}1&&\\\ &1&\\\
&-a_{0}&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]\ \frac{dy_{0}dy_{1}}{y_{0}y_{1}^{2}}$ $\displaystyle=\
2\cdot\Gamma_{\mathbb{R}}\left(1+\alpha_{2}^{w}-\alpha_{3}^{w}\right)\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)$
$\displaystyle\hskip 100.0pt\cdot\sum_{w=w_{2},w_{4},w_{\ell}}\
\zeta(w,\alpha,\psi_{(0,1)})\zeta(2s+\alpha_{1}^{w}+\alpha_{2}^{w})\cdot(J_{\alpha}h)(s;w),$
(4.9)

where the integral

$\displaystyle(J_{\alpha}h)(s;w)\ :=\ \int_{0}^{\infty}\int_{0}^{\infty}\
h(y_{1})y_{1}^{s-\frac{\alpha_{3}^{w}}{2}}y_{0}^{2s-\alpha_{3}^{w}}(1+y_{0}^{2})^{-\frac{3}{4}(1-\alpha_{3}^{w})}W_{(\alpha_{1}^{w}-\alpha_{2}^{w})/2}^{{}^{(2)}}\left(y_{1}\sqrt{1+y_{0}^{2}}\right)\frac{dy_{0}dy_{1}}{y_{0}y_{1}^{2}}$
(4.10)

converges absolutely on $0<\sigma<4$.

As a result, the expression on the right side of (4.3) provides a holomorphic
continuation of $(\mathcal{M}_{\alpha}^{{}^{(1)}}h)(s)$ to the domain
$0<\sigma<4$ except for a simple pole at $s=(1+\alpha_{3}^{w})/2$.

###### Proof.

As in the cuspidal case, we begin with the matrix identity

$\displaystyle\begin{pmatrix}1&&\\\ &1&\\\
&-a_{0}&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\ \equiv\ \begin{pmatrix}1&&\\\
&1&-\frac{a_{0}y_{0}^{2}}{1+(a_{0}y_{0})^{2}}\\\
&&1\end{pmatrix}\begin{pmatrix}\frac{y_{0}}{1+(a_{0}y_{0})^{2}}\cdot
y_{1}\sqrt{1+(a_{0}y_{0})^{2}}&&\\\ &\frac{y_{0}}{1+(a_{0}y_{0})^{2}}&\\\
&&1\end{pmatrix}$ (4.11)

under the right quotient by $O_{3}(\mathbb{R})\cdot\mathbb{R}^{\times}$. Then
the left side of (4.3) can be written as

$\displaystyle\sum_{a_{0}\neq 0}\ \int_{0}^{\infty}\int_{0}^{\infty}\
h(y_{1})\cdot(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\cdot(\widehat{\Phi})_{(0,1)}\left(\frac{y_{0}}{1+(a_{0}y_{0})^{2}},y_{1}\sqrt{1+(a_{0}y_{0})^{2}}\right)\
\frac{dy_{0}dy_{1}}{y_{0}y_{1}^{2}}.$ (4.12)

Plugging in (2.26) and (2.29) into the expression above, it becomes

$\displaystyle\Gamma_{\mathbb{R}}\left(1+\alpha_{2}^{w}-\alpha_{3}^{w}\right)\
$
$\displaystyle\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\
\ \sum_{w=w_{2},w_{4},w_{\ell}}\ \zeta(w,\alpha,\psi_{(0,1)})$
$\displaystyle\cdot\ \sum_{a_{0}\neq 0}\ \int_{0}^{\infty}\int_{0}^{\infty}\
h(y_{1})(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\left(\frac{y_{0}}{1+(a_{0}y_{0})^{2}}\right)^{1-\alpha_{3}^{w}}\left(y_{1}\sqrt{1+(a_{0}y_{0})^{2}}\right)^{\frac{1-\alpha_{3}^{w}}{2}}$
$\displaystyle\hskip 80.0pt\cdot
W_{(\alpha_{1}^{w}-\alpha_{2}^{w})/2}^{{}^{(2)}}\left(y_{1}\sqrt{1+(a_{0}y_{0})^{2}}\right)\
\frac{dy_{0}dy_{1}}{y_{0}y_{1}^{2}}.$ (4.13)

Upon making a change of variables $y_{0}\to|a_{0}|^{-1}y_{0}$, the expression
(4.12) is equal to

$\displaystyle\Gamma_{\mathbb{R}}\left(1+\alpha_{2}^{w}-\alpha_{3}^{w}\right)\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\cdot\sum_{w=w_{2},w_{4},w_{\ell}}\
\zeta(w,\alpha,\psi_{(0,1)})\left(\sum_{a_{0}\neq 0}\
|a_{0}|^{-(2s-1)-(1-\alpha_{3}^{w})}\right)\ (J_{\alpha}h)(s;w),$ (4.14)

where the $a_{0}$-sum converges absolutely whenever $\sigma>1/2$ and can be
seen to be

$\displaystyle\sum_{a_{0}\neq 0}\ |a_{0}|^{-(2s-1)-(1-\alpha_{3}^{w})}\ \ =\ \
2\cdot\zeta(2s+\alpha_{1}^{w}+\alpha_{2}^{w}).$

Following the argument of Proposition 6.5 of [Kw23] (the one here is indeed
simpler), one can readily observe the absolute convergence for
$(J_{\alpha}h)(s;w)$ whenever $0<\sigma<4$. For example, the bound

$\displaystyle
W_{(\alpha_{1}^{w}-\alpha_{2}^{w})/2}^{{}^{(2)}}\left(y_{1}\sqrt{1+y_{0}^{2}}\right)\
\ll\ \left(y_{1}\sqrt{1+y_{0}^{2}}\right)^{-A}$

with $A>2\sigma-1/2$, and $H:=h^{\\#}\in\mathcal{C}_{\eta}$ with
$1+\eta>|A-\sigma|$ are sufficient for such a purpose. This completes the
proof of this proposition. ∎

We must also compute the double integrals $(J_{\alpha}h)(s;w)$ in terms of the
Kontorovich-Lebedev transform $H(\mu):=h^{\\#}(\mu)$.

######  Proposition 4.4 ($1$-swap identity).

On the region $0<\sigma<1$, we have

$\displaystyle(J_{\alpha}h)(s;w)\ =\
\frac{\pi^{-(s-\frac{\alpha_{3}^{w}}{2})}}{8}\ $
$\displaystyle\Gamma\left(\frac{1+\alpha_{2}^{w}-\alpha_{3}^{w}}{2}\right)^{-1}\Gamma\left(\frac{1+\alpha_{1}^{w}-\alpha_{3}^{w}}{2}\right)^{-1}$
$\displaystyle\ \cdot\ \int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}\ \prod\limits_{\pm}\
\Gamma\left(\frac{s\pm\mu+\alpha_{1}^{w}}{2}\right)\Gamma\left(\frac{s\pm\mu+\alpha_{2}^{w}}{2}\right)\Gamma\left(\frac{1-s\pm\mu-\alpha_{3}^{w}}{2}\right)\
\frac{d\mu}{2\pi i}.$ (4.15)

###### Proof.

We begin by supposing

$\displaystyle\hskip 20.0pt\frac{1}{2}\ <\ \sigma\ <\ 4,\ \hskip
15.0pt\sigma_{u}\ >\ 2\sigma-\frac{3}{2},\hskip 15.0pt40\ >\
|\sigma_{u}-\sigma+1|,\hskip 15.0ptH\ :=\ h^{\\#}\ \in\
\mathcal{C}_{\eta}\hskip 15.0pt(\text{with }\ \eta\ >\ 40).$ (4.16)

From (2.2), the first two conditions of (4.16) and Mellin inversion, we have

$\displaystyle
W_{(\alpha_{1}^{w}-\alpha_{2}^{w})/2}^{{}^{(2)}}\left(y_{1}\sqrt{1+y_{0}^{2}}\right)\
=\ \int_{(\sigma_{u})}\ \frac{\pi^{-u-\frac{1}{2}}}{2}\ $
$\displaystyle\Gamma\left(\frac{u}{2}+\frac{1+(\alpha_{1}^{w}-\alpha_{2}^{w})}{4}\right)\Gamma\left(\frac{u}{2}+\frac{1-(\alpha_{1}^{w}-\alpha_{2}^{w})}{4}\right)$
$\displaystyle\cdot\left(y_{1}\sqrt{1+y_{0}^{2}}\right)^{-u}\ \frac{du}{2\pi
i}.$ (4.17)

Substitute this into (4.10), we have

$\displaystyle(J_{\alpha}h)(s;w)\ =\ \int_{(\sigma_{u})}\
\frac{\pi^{-u-\frac{1}{2}}}{2}\ $
$\displaystyle\Gamma\left(\frac{u}{2}+\frac{1+(\alpha_{1}^{w}-\alpha_{2}^{w})}{4}\right)\Gamma\left(\frac{u}{2}+\frac{1-(\alpha_{1}^{w}-\alpha_{2}^{w})}{4}\right)$
$\displaystyle\cdot\left(\int_{0}^{\infty}\
y_{0}^{2s-\alpha_{3}^{w}}(1+y_{0}^{2})^{-\frac{3}{4}(1-\alpha_{3}^{w})-\frac{u}{2}}\
d^{\times}y_{0}\right)\cdot\left(\int_{0}^{\infty}\
h(y_{1})y_{1}^{s-\frac{\alpha_{3}^{w}}{2}-u}\ \frac{dy_{1}}{y_{1}^{2}}\right)\
\frac{du}{2\pi i}.$ (4.18)

The $y_{0}$-integral converges absolutely because of the first two conditions
of (4.16) and can be computed by the Euler beta integral formula. We get

$\displaystyle\int_{0}^{\infty}\
y_{0}^{2s-\alpha_{3}^{w}}(1+y_{0}^{2})^{-\frac{3}{4}(1-\alpha_{3}^{w})-\frac{u}{2}}\
d^{\times}y_{0}\ =\ \frac{1}{2}\
\Gamma\left(s-\frac{\alpha_{3}^{w}}{2}\right)\
\frac{\Gamma\left(\frac{3-\alpha_{3}^{w}}{4}-s+\frac{u}{2}\right)}{\Gamma\left(\frac{3}{4}(1-\alpha_{3}^{w})+\frac{u}{2}\right)}.$
(4.19)

The fact that $\eta>40>|\sigma_{u}-\sigma+1|$ guarantees the $y_{1}$-integral
converges absoluetly. As a result, we have the following expression

$\displaystyle(J_{\alpha}h)(s;w)\ =\ \frac{1}{2}\
\Gamma\left(s-\frac{\alpha_{3}^{w}}{2}\right)\cdot\int_{(\sigma_{u})}\
\frac{\pi^{-u-\frac{1}{2}}}{2}\ $
$\displaystyle\Gamma\left(\frac{u}{2}+\frac{1+(\alpha_{1}^{w}-\alpha_{2}^{w})}{4}\right)\Gamma\left(\frac{u}{2}+\frac{1-(\alpha_{1}^{w}-\alpha_{2}^{w})}{4}\right)$
$\displaystyle\cdot\
\frac{\Gamma\left(\frac{3-\alpha_{3}^{w}}{4}-s+\frac{u}{2}\right)}{\Gamma\left(\frac{3}{4}(1-\alpha_{3}^{w})+\frac{u}{2}\right)}\cdot\widetilde{h}\left(s-1-\frac{\alpha_{3}^{w}}{2}-u\right)\
\frac{du}{2\pi i}$ (4.20)

Upon restricting to $\frac{1}{2}<\sigma<1$, we may pick a contour
$\operatorname{Re}u=\sigma_{u}$ satisfying

$\displaystyle 2\sigma-\frac{3}{2}<\ \sigma_{u}\ <\ \sigma-\frac{1}{2}.$
(4.21)

Notice that the conditions (4.16) and the one for (2.15) are satisfied. We may
apply (4.2) with

$\displaystyle\hskip
20.0pt\widetilde{h}\left(s-1-\frac{\alpha_{3}^{w}}{2}-u\right)\ =\
\frac{\pi^{-(s-\frac{\alpha_{3}^{w}}{2}-u)+\frac{1}{2}}}{4}\int_{(0)}\
H(\mu)\cdot\frac{\Gamma\left(\frac{s-\frac{1+\alpha_{3}^{w}}{2}-u+\mu}{2}\right)\Gamma\left(\frac{s-\frac{1+\alpha_{3}^{w}}{2}-u-\mu}{2}\right)}{\left|\Gamma(\mu)\right|^{2}}\
\frac{d\mu}{2\pi i}.$ (4.22)

As a result, we have

$\displaystyle(J_{\alpha}h)(s;w)\ =\ $
$\displaystyle\frac{\pi^{-(s-\frac{\alpha_{3}^{w}}{2})}}{16}\
\Gamma\left(s-\frac{\alpha_{3}^{w}}{2}\right)\cdot\int_{(0)}\ \frac{d\mu}{2\pi
i}\ \frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}$
$\displaystyle\cdot\int_{-i\infty}^{i\infty}\
\frac{\Gamma\left(\frac{u}{2}+\frac{1+(\alpha_{1}^{w}-\alpha_{2}^{w})}{4}\right)\Gamma\left(\frac{u}{2}+\frac{1-(\alpha_{1}^{w}-\alpha_{2}^{w})}{4}\right)\Gamma\left(\frac{3-\alpha_{3}^{w}}{4}-s+\frac{u}{2}\right)\Gamma\left(\frac{s-\frac{1+\alpha_{3}^{w}}{2}-u+\mu}{2}\right)\Gamma\left(\frac{s-\frac{1+\alpha_{3}^{w}}{2}-u-\mu}{2}\right)}{\Gamma\left(\frac{3}{4}(1-\alpha_{3}^{w})+\frac{u}{2}\right)}\
\frac{du}{2\pi i}.$ (4.23)

One can readily check that the contour $\operatorname{Re}u=\sigma_{u}$ with
(4.21) satisfies the Barnes convention (see Section 2.1). Make a change of
variables $u\to 2u$, take

$\displaystyle(a,b,c;d,e)\ =\
\left(\frac{1+(\alpha_{1}^{w}-\alpha_{2}^{w})}{4},\
\frac{1-(\alpha_{1}^{w}-\alpha_{2}^{w})}{4},\ \frac{3-\alpha_{3}^{w}}{4}-s;\
\frac{s+\mu}{2}-\frac{1+\alpha_{3}^{w}}{4},\
\frac{s-\mu}{2}-\frac{1+\alpha_{3}^{w}}{4}\right)$ (4.24)

in the Second Barnes Lemma (Lemma 2.5), and verify that

$\displaystyle(a+b)+c+(d+e)\ \ =\ \ \frac{1}{2}\ +\
\left(\frac{3-\alpha_{3}^{w}}{4}-s\right)\ +\
\left(s-\frac{1+\alpha_{3}^{w}}{2}\right)\ \ =\ \ \frac{3}{4}\
(1-\alpha_{3}^{w})\ \ (:=)\ \ f,$

we conclude that

$\displaystyle(J_{\alpha}h)(s;w)\ $ $\displaystyle=\
\frac{\pi^{-(s-\frac{\alpha_{3}^{w}}{2})}}{8}\
\Gamma\left(s-\frac{\alpha_{3}^{w}}{2}\right)\cdot\int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}\cdot\frac{\prod\limits_{\pm}\
\Gamma\left(\frac{s\pm\mu+\alpha_{1}^{w}}{2}\right)\Gamma\left(\frac{s\pm\mu+\alpha_{2}^{w}}{2}\right)\Gamma\left(\frac{1-s\pm\mu-\alpha_{3}^{w}}{2}\right)}{\Gamma\left(\frac{1+\alpha_{2}^{w}-\alpha_{3}^{w}}{2}\right)\Gamma\left(\frac{1+\alpha_{1}^{w}-\alpha_{3}^{w}}{2}\right)\Gamma\left(s-\frac{\alpha_{3}^{w}}{2}\right)}\
\frac{d\mu}{2\pi i}.$ (4.25)

Note that the relation $\alpha_{1}^{w}+\alpha_{2}^{w}+\alpha_{3}^{w}=0$ was
used multiple times above. The validity of (4.4) on the larger domain
$0<\sigma<1$ follows from analytic continuation. This completes the proof of
Proposition 4.4. ∎

From Proposition 4.3 and 4.4, it immediately follows that

######  Corollary 4.5.

On $0<\sigma<1$, we have

$\displaystyle(\mathcal{M}_{\alpha}^{{}^{(1)}}h)(s)\ =\ $
$\displaystyle\frac{\pi^{-s-1}}{4}\ \sum_{w=w_{2},w_{4},w_{\ell}}\
\pi^{\alpha_{1}^{w}+\alpha_{2}^{w}-\alpha_{3}^{w}}\zeta(1-\alpha_{2}^{w}+\alpha_{3}^{w})\zeta(1-\alpha_{1}^{w}+\alpha_{3}^{w})\zeta(2s-\alpha_{1}^{w}-\alpha_{2}^{w})$
$\displaystyle\hskip 50.0pt\cdot\int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}\ \prod\limits_{\pm}\
\Gamma\left(\frac{s\pm\mu-\alpha_{1}^{w}}{2}\right)\Gamma\left(\frac{s\pm\mu-\alpha_{2}^{w}}{2}\right)\Gamma\left(\frac{1-s\pm\mu+\alpha_{3}^{w}}{2}\right)\
\frac{d\mu}{2\pi i}.$ (4.26)

###  4.3. Cancellation and (4.6)

The following simple lemma is important to the convergence of the dual side of
(4.1) (even when $\operatorname{Re}s\gg 1$).

######  Lemma 4.6.

The expression (4.6) is identical to zero.

###### Proof.

Since

$\displaystyle\left(\widehat{\Phi}\right)_{(0,n_{2})}\left[\begin{pmatrix}1&u_{1,2}&\\\
&1&\\\ &&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]\ =\
e\left(n_{2}u_{1,2}\right)\cdot\left(\widehat{\Phi}\right)_{(0,n_{2})}(y_{0},y_{1}),$

we have

$\displaystyle\sum_{n_{2}\neq 0}\ \int_{0}^{1}\ $ $\displaystyle\
(\widehat{\Phi})_{(0,n_{2})}\left[\begin{pmatrix}1&u_{1,2}&\\\ &1&\\\
&&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]e(-u_{1,2})\ du_{1,2}$ $\displaystyle\hskip 40.0pt\ =\
\sum_{n_{2}\neq 0}\ \left(\widehat{\Phi}\right)_{(0,n_{2})}(y_{0},y_{1})\
\int_{0}^{1}\ e\left((n_{2}-1)u_{1,2}\right)\ du_{1,2}\ =\
\left(\widehat{\Phi}\right)_{(0,1)}(y_{0},y_{1}).$

The result follows.

∎

###  4.4. Degenerate Terms and (4.7)

Recall that (4.7) is given by

$\displaystyle\Phi^{(*,0)}(y_{0},y_{1})\ :=\ -\ \sum_{\gamma\in
U_{2}(\mathbb{Z})\setminus\Gamma_{2}}\ \sum_{n_{1}=1}^{\infty}\ \int_{0}^{1}\
(\widehat{\Phi})_{(n_{1},0)}\left[\begin{pmatrix}\gamma&\\\
&1\end{pmatrix}\begin{pmatrix}1&u_{1,2}&\\\ &1&\\\
&&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]e(-u_{1,2})\ du_{1,2}.$ (4.27)

The main task is once again an exact evaluation of

$\displaystyle(\mathcal{D}_{\alpha}H)(s)\ \ :=\ \ -\ 2\
\int_{0}^{\infty}\int_{0}^{\infty}\
h(y_{1})\cdot(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\cdot\Phi^{(*,0)}(y_{0},y_{1})\ \
\frac{dy_{0}dy_{1}}{y_{0}y_{1}^{2}}$ (4.28)

in terms of the Kontorovich-Lebedev transform $H(\mu):=h^{\\#}(\mu)$. This
results in three degenerate terms that are crucial for the analytic
continuation of (4.1). Interestingly, the $\gamma$-sum of (4.27), which
originates from the $\Gamma_{3}$-Fourier expansion of $\Phi$, will be turned
into the Eisenstein series of $\Gamma_{2}$ during the evaluation of the
expression (4.28). These features are not present in the first two moments of
$GL(2)$ (see [Mo92]).

######  Proposition 4.7.

For $\frac{1}{2}<\sigma<4$, we have

$\displaystyle(\mathcal{D}_{\alpha}H)(s)\ =\ $ $\displaystyle
2\sum_{w=w_{3},w_{5},w_{\ell}}\
H(s+\alpha_{1}^{w})\cdot\frac{\Lambda\left(1+\alpha_{1}^{w}-\alpha_{2}^{w}\right)\Lambda\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\Lambda\left(2s+\alpha_{1}^{w}+\alpha_{2}^{w}\right)\Lambda\left(2s+\alpha_{1}^{w}+\alpha_{3}^{w}\right)}{\Lambda(1+2(s+\alpha_{1}^{w}))}.$
(4.29)

As a result, the expression on the right side of (4.29) provides a holomorphic
continuation of $(\mathcal{D}_{\alpha}h)(s)$ to the domain $0<\sigma<4$ except
for the simple poles at $s=(1+\alpha_{3}^{w})/2,\ (1+\alpha_{2}^{w})/2$
($w=w_{3},w_{5},w_{\ell}$).

###### Proof.

For any $\gamma\in\Gamma_{2}$, the following Iwasawa decomposition:

$\displaystyle\hskip 15.0pt\gamma\begin{pmatrix}1&u_{1,2}\\\
&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&\\\ &y_{0}\end{pmatrix}\ \equiv\
y_{0}y_{1}^{1/2}\begin{pmatrix}1&\operatorname{Re}\gamma z\\\
&1\end{pmatrix}\begin{pmatrix}(\operatorname{Im}\gamma z)^{1/2}&\\\
&(\operatorname{Im}\gamma z)^{-1/2}\end{pmatrix}\hskip
15.0pt\left(\bmod\,SO(2)\right)$ (4.30)

holds, where $z:=u_{1,2}+iy_{1}$. Using this together with the equivariance of
$(\widehat{\Phi})_{(n_{1},0)}$, we have

$\displaystyle(\widehat{\Phi})_{(n_{1},0)}\left[\begin{pmatrix}\gamma&\\\
&1\end{pmatrix}\begin{pmatrix}1&u_{1,2}&\\\ &1&\\\
&&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]$ $\displaystyle\hskip 50.0pt\ =\
(\widehat{\Phi})_{(n_{1},0)}\left[\begin{pmatrix}1&\operatorname{Re}\gamma
z&\\\ &1&\\\
&&1\end{pmatrix}\begin{pmatrix}y_{0}\left(\frac{y_{1}}{\operatorname{Im}\gamma
z}\right)^{1/2}\cdot\operatorname{Im}\gamma z&&\\\
&y_{0}\left(\frac{y_{1}}{\operatorname{Im}\gamma z}\right)^{1/2}&\\\
&&1\end{pmatrix}\right]$ $\displaystyle\hskip 50.0pt\ =\
(\widehat{\Phi})_{(n_{1},0)}\left(y_{0}\left(\frac{y_{1}}{\operatorname{Im}\gamma
z}\right)^{1/2},\ \operatorname{Im}\gamma z\right)$ (4.31)

and

$\displaystyle\Phi^{(*,0)}(y_{0},y_{1})\ =\ -\sum_{\gamma\in
U_{2}(\mathbb{Z})\setminus\Gamma_{2}}\ \sum_{n_{1}=1}^{\infty}\ \int_{0}^{1}\
(\widehat{\Phi})_{(n_{1},0)}\left(y_{0}\left(\frac{y_{1}}{\operatorname{Im}\gamma
z}\right)^{1/2},\ \operatorname{Im}\gamma z\right)e(-u_{1,2})\ du_{1,2}.$
(4.32)

This can be written in a more explicit fashion via Lemma 2.19:

$\displaystyle\Phi^{(*,0)}(y_{0},y_{1})\ =\ -\sum_{w=w_{3},w_{5},w_{\ell}}\
\sum_{\gamma\in U_{2}(\mathbb{Z})\setminus\Gamma_{2}}\
\sum_{n_{1}=1}^{\infty}\ $
$\displaystyle\frac{\zeta(w,\alpha,\psi_{(n_{1},0)})}{|n_{1}|^{1-\alpha_{3}^{w}}}$
$\displaystyle\cdot\int_{0}^{1}\
W^{(1,0)}_{\alpha}\left[\left(n_{1}y_{0}\left(\frac{y_{1}}{\operatorname{Im}\gamma
z}\right)^{1/2},\ \operatorname{Im}\gamma z\right),w\right]e(-u_{1,2})\
du_{1,2}.$ (4.33)

Upon rearrangement, the expression (4.28) becomes

$\displaystyle(\mathcal{D}_{\alpha}H)(s)$ $\displaystyle\ =\
2\sum_{w=w_{3},w_{5},w_{\ell}}\ \sum_{\gamma\in
U_{2}(\mathbb{Z})\setminus\Gamma_{2}}\ \sum_{n_{1}=1}^{\infty}\
\frac{\zeta(w,\alpha,\psi_{(n_{1},0)})}{|n_{1}|^{1-\alpha_{3}^{w}}}$
$\displaystyle\hskip
25.0pt\cdot\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{1}\
h(y_{1})(y_{0}^{2}y_{1})^{s-\frac{1}{2}}W^{(1,0)}_{\alpha}\left[\left(n_{1}y_{0}\left(\frac{y_{1}}{\operatorname{Im}\gamma
z}\right)^{1/2},\ \operatorname{Im}\gamma z\right),w\right]e(-u_{1,2})\
du_{1,2}\ \frac{dy_{0}dy_{1}}{y_{0}y_{1}^{2}}.$ (4.34)

It follows from the change of variables $y_{0}\to
n_{1}^{-1}(y_{1}/\operatorname{Im}\gamma z)^{-1/2}\cdot y_{0}$ that

$\displaystyle(\mathcal{D}_{\alpha}H)(s)\ =\ 2\sum_{w=w_{3},w_{5},w_{\ell}}\ $
$\displaystyle\ \left(\sum_{n_{1}=1}^{\infty}\
\frac{\zeta(w,\alpha,\psi_{(n_{1},0)})}{|n_{1}|^{1-\alpha_{3}^{w}+(2s-1)}}\right)\sum_{\gamma\in
U_{2}(\mathbb{Z})\setminus\Gamma_{2}}$
$\displaystyle\cdot\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{1}\
h(y_{1})\cdot\left(y_{0}^{2}\operatorname{Im}\gamma
z\right)^{s-\frac{1}{2}}\cdot\ W^{(1,0)}_{\alpha}\left[\left(y_{0},\
\operatorname{Im}\gamma z\right),w\right]e(-u_{1,2})\ du_{1,2}\
\frac{dy_{0}dy_{1}}{y_{0}y_{1}^{2}}.$ (4.35)

(Note that $z$ does not depend on $y_{0}$!)

Now, replace $W^{(1,0)}_{\alpha}(\cdots)$ above by its definition (2.27), we
find that $(\mathcal{D}_{\alpha}H)(s)$ is equal to

$\displaystyle\hskip 30.0pt2\sum_{w=w_{3},w_{5},w_{\ell}}\
\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{2}^{w}\right)\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\
$ $\displaystyle\hskip 80.0pt\ \cdot\ \left(\sum_{n_{1}=1}^{\infty}\
\frac{\zeta(w,\alpha,\psi_{(n_{1},0)})}{|n_{1}|^{2s-\alpha_{3}^{w}}}\right)\left(\int_{0}^{\infty}\
W_{\frac{\alpha_{2}^{w}-\alpha_{3}^{w}}{2}}^{{}^{(2)}}(y_{0})y_{0}^{2s-1+\frac{1+\alpha_{1}^{w}}{2}}\
\frac{dy_{0}}{y_{0}}\right)\ $ $\displaystyle\hskip 130.0pt\ \cdot\
\sum_{\gamma\in U_{2}(\mathbb{Z})\setminus\Gamma_{2}}\
\int_{0}^{\infty}\int_{0}^{1}\ h(y_{1})\ \left(\operatorname{Im}\gamma
z\right)^{s-\frac{1}{2}+(1+\alpha_{1}^{w})}\ e(-u_{1,2})\ du_{1,2}\
\frac{dy_{1}}{y_{1}^{2}}.$ (4.36)

In the last line of (4.4), the $\gamma$-sum can be moved inside the double
integrals. Thus, $(\mathcal{D}_{\alpha}H)(s)$ becomes

$\displaystyle 2\ \sum_{w=w_{3},w_{5},w_{\ell}}\
\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{2}^{w}\right)\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)$
$\displaystyle\hskip 50.0pt\ \cdot\ \ \left(\sum_{n_{1}=1}^{\infty}\
\frac{\zeta(w,\alpha,\psi_{(n_{1},0)})}{|n_{1}|^{2s-\alpha_{3}^{w}}}\right)\left(\int_{0}^{\infty}\
W_{\frac{\alpha_{2}^{w}-\alpha_{3}^{w}}{2}}^{{}^{(2)}}(y_{0})y_{0}^{2s-1+\frac{1+\alpha_{1}^{w}}{2}}\
\frac{dy_{0}}{y_{0}}\right)$ $\displaystyle\hskip 160.0pt\ \cdot\ \left(2\
\int_{0}^{\infty}\ h(y_{1})\ \int_{0}^{1}\ E\left(z;s+\alpha_{1}^{w}\right)\
e(-u_{1,2})\ du_{1,2}\ \frac{dy_{1}}{y_{1}^{2}}\right),$ (4.37)

where

$\displaystyle E(z;\mu)\ :=\ \frac{1}{2}\ \ \sum_{\gamma\in
U_{2}(\mathbb{Z})\setminus\Gamma_{2}}(\operatorname{Im}\gamma
z)^{\mu+\frac{1}{2}}\hskip 30.0pt(z\ :=\ u_{1,2}+iy_{1}\ \in\
\mathfrak{h}^{2})$

was defined in (2.17) and it converges absolutely whenever
$\operatorname{Re}\mu>1/2$.

In the following, we evaluate the three expressions in $(\cdots)$ of (4.4).
Firstly, recall from (2.32) that

$\displaystyle\zeta\left(w,\alpha,\psi_{(n_{1},0)}\right)\ =\
\zeta\left(1+\alpha_{1}^{w}-\alpha_{2}^{w}\right)\zeta\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\sigma_{\alpha_{2}^{w}-\alpha_{3}^{w}}(|n_{1}|).$
(4.38)

Hence for $\sigma>1/2$, we have

$\displaystyle\sum_{n_{1}=1}^{\infty}\
\frac{\zeta(w,\alpha,\psi_{(n_{1},0)})}{|n_{1}|^{2s-\alpha_{3}^{w}}}\ \ $
$\displaystyle=\ \
\zeta\left(1+\alpha_{1}^{w}-\alpha_{2}^{w}\right)\zeta\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\sum_{n_{1}=1}^{\infty}\
\frac{\sigma_{\alpha_{2}^{w}-\alpha_{3}^{w}}(|n_{1}|)}{|n_{1}|^{2s-\alpha_{3}^{w}}}$
$\displaystyle=\ \
\zeta\left(1+\alpha_{1}^{w}-\alpha_{2}^{w}\right)\zeta\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\zeta\left(2s-\alpha_{3}^{w}\right)\zeta(2s-\alpha_{2}^{w}).$
(4.39)

Secondly, we have from (2.2) that

$\displaystyle\int_{0}^{\infty}\
W_{\frac{\alpha_{2}^{w}-\alpha_{3}^{w}}{2}}^{{}^{(2)}}(y_{0})y_{0}^{2s-1+\frac{1+\alpha_{1}^{w}}{2}}\
\frac{dy_{0}}{y_{0}}$ $\displaystyle\ \ =\ \ $
$\displaystyle\frac{\pi^{-(2s-1+\frac{1+\alpha_{1}^{w}}{2})-\frac{1}{2}}}{2}\
\Gamma\left(\frac{\left(2s-1+\frac{1+\alpha_{1}^{w}}{2}\right)+\frac{1}{2}+\frac{\alpha_{2}^{w}-\alpha_{3}^{w}}{2}}{2}\right)\Gamma\left(\frac{\left(2s-1+\frac{1+\alpha_{1}^{w}}{2}\right)+\frac{1}{2}-\frac{\alpha_{2}^{w}-\alpha_{3}^{w}}{2}}{2}\right),$
$\displaystyle\ \ =\ \ $
$\displaystyle\frac{\pi^{-2s-\frac{\alpha_{1}^{w}}{2}}}{2}\
\Gamma\left(s-\frac{\alpha_{3}^{w}}{2}\right)\Gamma\left(s-\frac{\alpha_{2}^{w}}{2}\right)$
(4.40)

for $\sigma>0$, where the last line follows from the relation
$\alpha_{1}^{w}+\alpha_{2}^{w}+\alpha_{3}^{w}=0$.

Thirdly, it follows from (2.18) and (2.10), i.e.,

$\displaystyle\int_{0}^{1}\ E\left(z;\mu\right)e(-u_{1,2})\ du_{1,2}\ =\
\frac{W^{(2)}_{\mu}(y)}{\Lambda(1+2\mu)}\hskip 10.0pt\text{ and }\hskip
10.0pth^{\\#}(\mu)\ :=\ \int_{0}^{\infty}h(y_{1})\ W_{\mu}(y_{1})\
\frac{dy_{1}}{y_{1}^{2}},$ (4.41)

that the last line of (4.4) is equal to

$\displaystyle 2\cdot\int_{0}^{\infty}\ h(y_{1})\ \int_{0}^{1}\
E\left(z;s+\alpha_{1}^{w}\right)\ e(-u_{1,2})\ du_{1,2}\
\frac{dy_{1}}{y_{1}^{2}}\ \ =\ \ $
$\displaystyle\frac{2}{\Lambda(1+2(s+\alpha_{1}^{w}))}\ \int_{0}^{\infty}\
h(y_{1})\ W_{s+\alpha_{1}^{w}}^{{}^{(2)}}(y_{1})\ \frac{dy_{1}}{y_{1}^{2}}$
$\displaystyle\ \ =\ \ $
$\displaystyle\frac{2}{\Lambda(1+2(s+\alpha_{1}^{w}))}\cdot
h^{\\#}(s+\alpha_{1}^{w}).$ (4.42)

Altogether, we conclude from (4.4), (4.4), (4.4) and (4.4) that

$\displaystyle\hskip 5.0pt(\mathcal{D}_{\alpha}H)(s)\ =\ $ $\displaystyle
2\sum_{w=w_{3},w_{5},w_{\ell}}\
\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{2}^{w}\right)\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)$
$\displaystyle\hskip 60.0pt\ \cdot\
\zeta\left(1+\alpha_{1}^{w}-\alpha_{2}^{w}\right)\zeta\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\zeta\left(2s-\alpha_{3}^{w}\right)\zeta(2s-\alpha_{2}^{w})$
$\displaystyle\hskip 90.0pt\ \cdot\
\frac{\pi^{-2s-\frac{\alpha_{1}^{w}}{2}}}{2}\
\Gamma\left(s-\frac{\alpha_{3}^{w}}{2}\right)\Gamma\left(s-\frac{\alpha_{2}^{w}}{2}\right)\cdot\frac{2}{\Lambda(1+2(s+\alpha_{1}^{w}))}\cdot
h^{\\#}(s+\alpha_{1}^{w})$ $\displaystyle\ =\ $ $\displaystyle
2\sum_{w=w_{3},w_{5},w_{\ell}}\ h^{\\#}(s+\alpha_{1}^{w})\
\frac{\Lambda\left(1+\alpha_{1}^{w}-\alpha_{2}^{w}\right)\Lambda\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\Lambda\left(2s+\alpha_{1}^{w}+\alpha_{2}^{w}\right)\Lambda\left(2s+\alpha_{1}^{w}+\alpha_{3}^{w}\right)}{\Lambda(1+2(s+\alpha_{1}^{w}))}.$
(4.43)

This completes the proof.

∎

###  4.5. Interlude: A Summary for Section 4.1-4.4

######  Theorem 4.8.

Let

$\displaystyle\Phi\ :=\
\Lambda(1+\alpha_{1}-\alpha_{2})\Lambda(1+\alpha_{2}-\alpha_{3})\Lambda(1+\alpha_{1}-\alpha_{3})E_{\min}^{(3)}(\
*\ ;\ \alpha).$ (4.44)

Then on the vertical strip $1<\sigma<4$ we have

$\displaystyle\bigg{(}P,\
\mathbb{P}_{2}^{3}\left(\Phi^{ND}\right)\cdot|\det*|^{\overline{s}-\frac{1}{2}}\bigg{)}_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}$ $\displaystyle\hskip 5.0pt\ =\
\frac{\pi^{-3s}}{4}\zeta(2s+\alpha_{1})\zeta(2s+\alpha_{2})\zeta(2s+\alpha_{3})\int_{(0)}\frac{H(\mu)}{|\Gamma(\mu)|^{2}}\cdot\prod_{i=1}^{3}\
\prod_{\pm}\Gamma\left(\frac{s\pm\mu-\alpha_{i}}{2}\right)\ \frac{d\mu}{2\pi
i}\ +\ OD_{\alpha}(s)$ (4.45) $\displaystyle\hskip 30.0pt\ +\
\frac{\pi^{-s-1}}{4}\ \sum_{w=w_{2},w_{4},w_{\ell}}\
\pi^{\alpha_{1}^{w}+\alpha_{2}^{w}-\alpha_{3}^{w}}\zeta(1-\alpha_{2}^{w}+\alpha_{3}^{w})\zeta(1-\alpha_{1}^{w}+\alpha_{3}^{w})\zeta(2s-\alpha_{1}^{w}-\alpha_{2}^{w})$
$\displaystyle\hskip 90.0pt\cdot\int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}\ \prod\limits_{\pm}\
\Gamma\left(\frac{s\pm\mu-\alpha_{1}^{w}}{2}\right)\Gamma\left(\frac{s\pm\mu-\alpha_{2}^{w}}{2}\right)\Gamma\left(\frac{1-s\pm\mu+\alpha_{3}^{w}}{2}\right)\
\frac{d\mu}{2\pi i}$ (4.46) $\displaystyle\hskip 35.0pt\ -\ \frac{1}{2}\
\sum_{w=w_{3},w_{5},w_{\ell}}\ H(s-\alpha_{1}^{w})\
\frac{\Lambda\left(1-\alpha_{1}^{w}+\alpha_{2}^{w}\right)\Lambda\left(1-\alpha_{1}^{w}+\alpha_{3}^{w}\right)\Lambda\left(2s-\alpha_{1}^{w}-\alpha_{2}^{w}\right)\Lambda\left(2s-\alpha_{1}^{w}-\alpha_{3}^{w}\right)}{\Lambda(1+2(s-\alpha_{1}^{w}))},$
(4.47)

where $\Phi^{ND}$ and $L(2s,\Phi)$ were defined in (2.23) and (2.38)
respectively, whereas $OD_{\alpha}(s)$ will be defined in (4.5) below.

Every term on the right side of the equation above except $OD_{\Phi}(s)$ have
been shown to admit an analytic continuation to the vertical strip
$0<\sigma<4$.

###### Proof.

Recall the expression (4.1) for the pairing and the decomposition of the
unipotent period (4.4) into the pieces (4.5), (4.6) and (4.7). In view of
Remark 4.1, we also make the replacements $\Phi\rightarrow\widetilde{\Phi}$
and $\alpha\rightarrow-\alpha$. Here is a list of the origins for the summands
in Theorem 4.8:

  1. (1)

(4.45) comes from the contributions from $a_{1}\neq 0$ in (4.5). From equation
(6.7) of [Kw23], it is given by

$\displaystyle 2\cdot
L(2s,\Phi)\cdot\int_{0}^{\infty}\int_{0}^{\infty}h(y_{1})\cdot(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\cdot
W_{-\alpha(\Phi)}(y_{0},y_{1})\ \frac{dy_{0}dy_{1}}{y_{0}y_{1}^{2}}\ +\
OD_{\Phi}(s).$ (4.48)

Since $\Phi$ is given by (4.44), we have

$\displaystyle L(s_{0},\Phi)\ =\ \prod_{p}\ \prod_{k=1}^{3}\
\left(1-p^{-\alpha_{k}}\cdot p^{-s_{0}}\right)^{-1}\ =\
\zeta(s_{0}+\alpha_{1})\zeta(s_{0}+\alpha_{2})\zeta(s_{0}+\alpha_{3})$ (4.49)

for any $\operatorname{Re}s_{0}>1$, see (2.42) (or Theorem 10.8.6 of [Gold]).
The double integral in (4.48) can be evaluated easily (2.11) and Proposition
2.6.

  2. (2)

(4.46) is the contribution from $a_{1}=0$ in (4.5) which was computed in
Proposition 4.3-4.4.

  3. (3)

(4.47) is obtained from Proposition 4.7.

This completes the proof. ∎

By off-diagonal, we mean the contributions from the terms with $a_{0}a_{1}\neq
0$ in (4.5) and was denoted by $OD_{\Phi}(s)$ above. The analytic continuation
argument performed in [Kw23] is robust and carries over to the present case
with minor modifications (e.g., with extra polar and Euler products
considerations). Recall from equation (6.8) of [Kw23] and (4.49) that

$\displaystyle OD_{\alpha}(s)\ $ $\displaystyle=\ OD_{\Phi}(s)\ :=\
\sum_{a_{0}\neq 0}\sum_{a_{1}\neq 0}\
\frac{\mathcal{B}_{\Phi}(1,a_{1})}{|a_{0}|^{2s-1}|a_{1}|}\cdot\int_{0}^{\infty}\int_{0}^{\infty}h(y_{1})\cdot(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\cdot
e\left(\frac{a_{1}}{a_{0}}\cdot\frac{y_{0}^{2}}{1+y_{0}^{2}}\right)$
$\displaystyle\hskip 230.0pt\cdot
W_{-\alpha}\left(\left|\frac{a_{1}}{a_{0}}\right|\cdot\frac{y_{0}}{1+y_{0}^{2}},\
y_{1}\sqrt{1+y_{0}^{2}}\right)\frac{dy_{0}dy_{1}}{y_{0}y_{1}^{2}}$
$\displaystyle=\ \lim_{\phi\to\pi/2}\ \frac{1}{4}\ \int_{(1+2\epsilon)}\
\zeta\left(2s-s_{0}\right)\zeta(s_{0}+\alpha_{1})\zeta(s_{0}+\alpha_{2})\zeta(s_{0}+\alpha_{3})\cdot\sum_{\delta=\pm}\left(\mathcal{F}_{\alpha}^{(\delta)}H\right)\left(s_{0},\
s;\ \phi\right)\ \frac{ds_{0}}{2\pi i},$ (4.50)

where absolute convergence are in place whenever $1<\sigma<4$ and
$H\in\mathcal{C}_{\eta}$ ($\eta>40$).

###  4.6. Analysis of the Off-Diagonal

As before, we take $\epsilon:=1/100$.

####  4.6.1. The identities for $2$-swap and $3$-swap terms

We now investigate the extra symmetries for our integral transform
$\left(\mathcal{F}_{\alpha}H\right)\left(s_{0},s\right)$ which turn out to be
important in matching the $2$-swap terms and $3$-swap term of the Moment
Conjecture. Also, they will take part in our analytic continuation argument.

######  Proposition 4.9 ($2$-swap identity).

The function
$s\mapsto\left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{1},s\right)$ admits
a holomorphic continuation to $0<\sigma<4$ except for a simple pole
$s=(1-\alpha_{1})/2$. In particular, we have

$\displaystyle\left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{1},s\right)\
=\ $ $\displaystyle\pi^{\frac{1}{2}-s}\
\frac{\Gamma\left(s-\frac{1}{2}+\frac{\alpha_{1}}{2}\right)}{\Gamma\left(1-s-\frac{\alpha_{1}}{2}\right)}$
$\displaystyle\hskip 15.0pt\cdot\int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}\ \prod_{\pm}\
\Gamma\left(\frac{s\pm\mu-\alpha_{1}}{2}\right)\Gamma\left(\frac{1+\alpha_{2}\pm\mu-s}{2}\right)\Gamma\left(\frac{1+\alpha_{3}\pm\mu-s}{2}\right)\frac{d\mu}{2\pi
i}$ (4.51)

on $0<\sigma<1$. The results for
$\left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{i},s\right)$ ($i=2,3$)
follow by symmetry.

###### Proof.

The holomorphy of
$s\mapsto\left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{1},s\right)$ on
$\frac{1}{2}+\epsilon<\sigma<4$ follows immediately from Proposition 2.32. We
now restrict ourselves to the smaller domain $\frac{1}{2}+\epsilon<\sigma<1$.

Upon putting $s_{0}=1-\alpha_{1}$, notice that the factor
$\Gamma\left(\frac{1-u}{2}\right)$ in the denominator of (2.8) cancels with
the factor $\Gamma\left(\frac{s_{0}+\alpha_{1}-u}{2}\right)$ in the numerator
of (2.8) which results in

$\displaystyle\left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{1},s\right)\
=\ $ $\displaystyle\sqrt{\pi}\ \int_{(\eta-1/2)}\
\widetilde{h}\left(s-s_{1}-\frac{1}{2}\right)\frac{\pi^{-s_{1}}\prod\limits_{i=1}^{3}\
\Gamma\left(\frac{s_{1}-\alpha_{i}}{2}\right)}{\Gamma\left(\frac{s_{1}-1}{2}+\alpha_{1}\right)}$
$\displaystyle\hskip 5.0pt\cdot\int_{(\epsilon)}\
\frac{\Gamma(\frac{u}{2})\Gamma\left(\frac{s_{1}+\alpha_{1}+u}{2}-s\right)\Gamma\left(\frac{1-\alpha_{1}+\alpha_{2}-u}{2}\right)\Gamma\left(\frac{1-\alpha_{1}+\alpha_{3}-u}{2}\right)\Gamma\left(s+\frac{-1+\alpha_{1}-u}{2}\right)}{\Gamma\left(\frac{1-\alpha_{1}+s_{1}-u}{2}\right)}\
\frac{du}{2\pi i}\ \frac{ds_{1}}{2\pi i}.$ (4.52)

We make a change of variable $u\to-2u$ and take

$\displaystyle\left(a,b,c;d,e\right)\ =\
\left(\frac{1-\alpha_{1}+\alpha_{2}}{2},\ \frac{1-\alpha_{1}+\alpha_{3}}{2},\
s-\frac{1-\alpha_{1}}{2};\ 0,\ \frac{s_{1}+\alpha_{1}}{2}-s\right)$ (4.53)

in (2.5). Observe that

$\displaystyle(a+b)+c+d+e\ =\
\left(1-\alpha_{1}-\frac{\alpha_{1}}{2}\right)+s-\frac{1-\alpha_{1}}{2}+\frac{s_{1}+\alpha_{1}}{2}-s\
=\ \frac{s_{1}-\alpha_{1}+1}{2}\ (:=f).$

The $u$-integral is thus equal to

$\displaystyle
2\cdot\frac{\Gamma\left(\frac{1-\alpha_{1}+\alpha_{2}}{2}\right)\Gamma\left(\frac{1-\alpha_{1}+\alpha_{3}}{2}\right)\Gamma\left(s-\frac{1}{2}+\frac{\alpha_{1}}{2}\right)\Gamma\left(\frac{s_{1}+1+\alpha_{2}}{2}-s\right)\Gamma\left(\frac{s_{1}+1+\alpha_{3}}{2}-s\right)\Gamma\left(\frac{s_{1}-1}{2}+\alpha_{1}\right)}{\Gamma\left(\frac{s_{1}-\alpha_{2}}{2}\right)\Gamma\left(\frac{s_{1}-\alpha_{3}}{2}\right)\Gamma\left(1-\alpha_{1}-s+\frac{s_{1}}{2}\right)}.$

Note that the factors
$\Gamma\left(\frac{s_{1}-\alpha_{2}}{2}\right)\Gamma\left(\frac{s_{1}-\alpha_{3}}{2}\right)$
occurred in both the denominator of the last expression and the numerator of
the first line of (4.6.1). Upon making this cancellation, we have

$\displaystyle\left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{1},s\right)\
=\ $ $\displaystyle\
\Gamma\left(\frac{1-\alpha_{1}+\alpha_{2}}{2}\right)\Gamma\left(\frac{1-\alpha_{1}+\alpha_{3}}{2}\right)\Gamma\left(s-\frac{1}{2}+\frac{\alpha_{1}}{2}\right)$
$\displaystyle\cdot 2\sqrt{\pi}\ \int_{(\eta-1/2)}\
\widetilde{h}\left(s-s_{1}-\frac{1}{2}\right)\
\frac{\pi^{-s_{1}}\Gamma\left(\frac{s_{1}-\alpha_{1}}{2}\right)\Gamma\left(\frac{s_{1}+1+\alpha_{2}}{2}-s\right)\Gamma\left(\frac{s_{1}+1+\alpha_{3}}{2}-s\right)}{\Gamma\left(1-\alpha_{1}-s+\frac{s_{1}}{2}\right)}\
\frac{ds_{1}}{2\pi i}.$ (4.54)

Then Lemma 2.11 implies

$\displaystyle\left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{1},s\right)\
=\ $ $\displaystyle\frac{\pi^{-s}}{2}\
\Gamma\left(\frac{1-\alpha_{1}+\alpha_{2}}{2}\right)\Gamma\left(\frac{1-\alpha_{1}+\alpha_{3}}{2}\right)\Gamma\left(s-\frac{1}{2}+\frac{\alpha_{1}}{2}\right)\int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}$ $\displaystyle\cdot\
\int_{(\sigma_{1})}\ \
\frac{\Gamma\left(\frac{s_{1}-\alpha_{1}}{2}\right)\Gamma\left(\frac{s_{1}+1+\alpha_{2}}{2}-s\right)\Gamma\left(\frac{s_{1}+1+\alpha_{3}}{2}-s\right)\Gamma\left(\frac{s-s_{1}+\mu}{2}\right)\Gamma\left(\frac{s-s_{1}-\mu}{2}\right)}{\Gamma\left(1-\alpha_{1}-s+\frac{s_{1}}{2}\right)}\
\frac{ds_{1}}{2\pi i}\ \frac{d\mu}{2\pi i}.$ (4.55)

For the $s_{1}$-integral right above, apply the change of variable $s_{1}\to
2s_{1}$ and (2.5) the second time but with

$\displaystyle(a,b,c;d,e)\ :=\ \left(-\frac{\alpha_{1}}{2},\
\frac{1+\alpha_{2}}{2}-s,\ \frac{1+\alpha_{3}}{2}-s;\ \frac{s+\mu}{2},\
\frac{s-\mu}{2}\right).$ (4.56)

Observe that

$\displaystyle a+(b+c)+d+e\ =\
-\frac{\alpha_{1}}{2}+\left(1-\frac{\alpha_{1}}{2}-2s\right)\ +s\ =\
1-\alpha_{1}-s\ (:=f).$

The $s_{1}$-integral is now equal to

$\displaystyle
2\cdot\frac{\Gamma\left(\frac{s+\mu-\alpha_{1}}{2}\right)\Gamma\left(\frac{s-\mu-\alpha_{1}}{2}\right)\Gamma\left(\frac{1+\alpha_{2}+\mu-s}{2}\right)\Gamma\left(\frac{1+\alpha_{2}-\mu-s}{2}\right)\Gamma\left(\frac{1+\alpha_{3}+\mu-s}{2}\right)\Gamma\left(\frac{1+\alpha_{3}-\mu-s}{2}\right)}{\Gamma\left(1-s-\frac{\alpha_{1}}{2}\right)\Gamma\left(\frac{1-\alpha_{2}}{2}-\alpha_{1}\right)\Gamma\left(\frac{1-\alpha_{3}}{2}-\alpha_{1}\right)}.$

The conclusion (4.9) follows from the fact that the factors
$\Gamma\left(\frac{1-\alpha_{2}}{2}-\alpha_{1}\right)\Gamma\left(\frac{1-\alpha_{3}}{2}-\alpha_{1}\right)$
in the denominator of the last expression coincide with the factors
$\Gamma\left(\frac{1-\alpha_{1}+\alpha_{2}}{2}\right)\Gamma\left(\frac{1-\alpha_{1}+\alpha_{3}}{2}\right)$
in the first line of (4.6.1) because of $\alpha_{1}+\alpha_{2}+\alpha_{3}=0$.

Finally, observe that (4.9) is holomorphic on $0<\sigma<1$ except at
$s=(1-\alpha_{1})/2$. Combining with the initial region of holomorphy, the
meromorphic continuation of
$s\mapsto\left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{1},s\right)$ to
$0<\sigma<4$ is now established.

∎

######  Proposition 4.10 ($3$-swap identity).

The function $s\mapsto\left(\mathcal{F}_{\alpha}H\right)\left(2s-1,s\right)$
admits a holomorphic continuation to $0<\sigma<4$ except for three simple
poles $s=(1-\alpha_{i})/2$, $i=1,2,3$. On the domain $0<\sigma<1$, we have

$\displaystyle\left(\mathcal{F}_{\alpha}H\right)\left(2s-1,s\right)\ $
$\displaystyle=\ \pi^{\frac{1}{2}-s}\cdot\prod_{i=1}^{3}\
\frac{\Gamma\left(s-\frac{1}{2}+\frac{\alpha_{i}}{2}\right)}{\Gamma\left(1-s-\frac{\alpha_{i}}{2}\right)}\cdot\int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}\cdot\prod\limits_{i=1}^{3}\
\prod\limits_{\pm}\ \Gamma\left(\frac{1-s+\alpha_{i}\pm\mu}{2}\right)\
\frac{d\mu}{2\pi i}.$ (4.57)

###### Proof.

This was first proved in [Kw23]. For completeness, we record its proof in
Appendix B. ∎

######  Remark 4.11.

A salient feature in many explicit archimedean computations is the complete
reduction of the Mellin-Barnes integrals into ratios of products of
$\Gamma$-factors — provided the arguments are aligned in certain ‘nice’
configurations. This phenomenon is usually caused by deeper underlying
structures of the integrals. Of course, showing these to be indeed the case
requires effort. For example,

  * •

Theorem 3.3 in Stade [St01] was predicted by the theory of exterior square
lifting of $GL(n)$;

  * •

The simplifications in our case are actually predicted by the Moment
Conjecture. The off-diagonal main terms predicted by the Moment Conjecture for
the shifted cubic moment (3.1) suggest that
$\left(\mathcal{F}_{\alpha}H\right)\left(s_{0},s\right)$ as well as the
closely-related $(J_{\alpha}h)(s;w)$ to be defined in (4.10) must simplify in
a very specific and exact fashion as stated in Theorem 1.2.

####  4.6.2. Analytic Continuation

######  Theorem 4.12.

The expression $OD_{\alpha}(s)$ admits a meromorphic continuation to the
domain $1/4+\epsilon/2\ <\ \sigma\ <\ 4$. On the domain $1/4+\epsilon/2\ <\
\sigma\ <\ 3/4$ we have

$\displaystyle 4\cdot OD_{\alpha}(s)\ =\ $ $\displaystyle\ \ \sum_{i=1}^{3}\
\zeta(2s-1+\alpha_{i})\prod_{\begin{subarray}{c}1\leq j\leq 3\\\ j\neq
i\end{subarray}}\zeta(1-\alpha_{i}+\alpha_{j})\cdot\left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{i},s\right)$
(4.58) $\displaystyle\hskip 20.0pt\ +\ \
\zeta(2s-1+\alpha_{1})\zeta(2s-1+\alpha_{2})\zeta(2s-1+\alpha_{3})\cdot\left(\mathcal{F}_{\alpha}H\right)\left(2s-1,s\right)$
(4.59) $\displaystyle\hskip 40.0pt\ +\ \
\int_{(1/2)}\zeta\left(2s-s_{0}\right)\zeta(s_{0}+\alpha_{1})\zeta(s_{0}+\alpha_{2})\zeta(s_{0}+\alpha_{3})\cdot\left(\mathcal{F}_{\alpha}H\right)\left(s_{0},s\right)\
\frac{ds_{0}}{2\pi i}.$ (4.60)

###### Proof.

Readers are invited to consult Section 9 of [Kw23] for fuller details. Also,
recall the conventions imposed in Remark 4.2. Take $\epsilon=1/100$ (say).
Parallel to ‘Step $1$’ of [Kw23] (Section 9.1 therein), we shift the line of
integration in (4.5) to $\operatorname{Re}s_{0}=2\epsilon$. This time,
however, we pick up the residues of three extra simple poles at
$s_{0}=1-\alpha_{i}$ ($i=1,2,3$), which results in

$\displaystyle 4\cdot OD_{\alpha}(s)\ =\ \lim_{\phi\to\pi/2}\ \bigg{\\{}$
$\displaystyle\sum_{i=1}^{3}\
\zeta(2s-1+\alpha_{i})\prod_{\begin{subarray}{c}1\leq j\leq 3\\\ j\neq
i\end{subarray}}\zeta(1-\alpha_{i}+\alpha_{j})\sum_{\delta=\pm}\left(\mathcal{F}_{\alpha}^{(\delta)}H\right)\left(1-\alpha_{i},\
s;\ \phi\right)$ $\displaystyle\hskip
10.0pt+\int_{(2\epsilon)}\zeta\left(2s-s_{0}\right)\zeta(s_{0}+\alpha_{1})\zeta(s_{0}+\alpha_{2})\zeta(s_{0}+\alpha_{3})\sum_{\delta=\pm}\left(\mathcal{F}_{\alpha}^{(\delta)}H\right)\left(s_{0},\
s;\ \phi\right)\ \frac{ds_{0}}{2\pi i}\bigg{\\}},$ (4.61)

which provides an analytic continuation to $\frac{1}{2}+\epsilon<\sigma<4$
except at three simple poles $s=1-\frac{\alpha_{i}}{2}$ ($i=1,2,3$). Parallel
to ‘Step $2$’ of [Kw23] (Section 9.2 therein), we first restrict to the
vertical strip $\frac{1}{2}+\epsilon<\sigma<\frac{3}{4}$ and then shift the
line of integration from $\operatorname{Re}s_{0}=2\epsilon$ to
$\operatorname{Re}s_{0}=1/2$, crossing the simple pole of $\zeta(2s-s_{0})$
with residue equal to $-1$. This essentially arrives at the desired conclusion
of Theorem 4.12.

‘Step 3’ of [Kw23] (Section 9.3 therein) carries over to the present context
without change as it merely takes care of the necessary regularity (using
Proposition 2.32 and the imposed assumptions on the class of test functions we
adopted). ‘Step 4’ of [Kw23] (Section 9.4 therein) concerns the analytic
continuation of (4.58) and (4.59). Here, we instead adopt an explicit approach
and the desired conclusion follows directly from Proposition 4.9-4.10. ‘Step
5’ of [Kw23] (Section 9.5 therein) is not necessary for this proof because
here we restrict our attention to $OD_{\alpha}(s)$ only. ∎

###  4.7. Contributions from Continuous Spectrum

Recall that the spectral expansion of

$\displaystyle\left(P,\
\mathbb{P}_{2}^{3}\left(\Phi^{ND}\right)\cdot|\det*|^{\overline{s}-\frac{1}{2}}\right)_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}$

is given by (3.1) for $\operatorname{Re}s>1$ using Proposition 2.31. The
cuspidal part clearly admits an entire continuation, whereas the analytic
continuation of the continuous part, i.e.,

$\displaystyle(\mathcal{C}_{\alpha}H)(s)\ \ :=\ \ \int_{(0)}\
H\left(\mu\right)\cdot\frac{\prod\limits_{i=1}^{3}\
\Lambda\left(s+\mu-\alpha_{i}\right)\Lambda\left(s-\mu-\alpha_{i}\right)}{\Lambda(1+2\mu)\Lambda(1-2\mu)}\
\frac{d\mu}{2\pi i},$ (4.62)

will take a small amount of work. Since $H\in\mathcal{C}_{\eta}$, the
integrand of (4.62) is holomorphic on $|\operatorname{Re}\mu|<2\eta$ except
for the poles at

$\displaystyle\mu\ \in\ \big{\\{}-s+\alpha_{i},\
1-s+\alpha_{i}\big{\\}}_{i=1}^{3}\hskip 20.0pt\text{ and }\hskip 20.0pt\mu\
\in\ \big{\\{}s-\alpha_{i},\ s-1-\alpha_{i}\big{\\}}_{i=1}^{3},$ (4.63)

as well as the zeros of $\Lambda(1\pm 2\mu)$.

######  Lemma 4.13.

The function $s\mapsto(\mathcal{C}_{\alpha}H)(s)$ admits a meromorphic
continuation to the domain $\epsilon<\operatorname{Re}s<1-\epsilon$ and it is
explicitly given by $(\mathcal{R}_{\alpha}H)(s)+(\mathcal{C}_{\alpha}H)(s)$,
where

$\displaystyle(\mathcal{R}_{\alpha}H)(s)\ \ :=\ \ 2\ \sum_{i=1}^{3}\
H(1-s+\alpha_{i})\ $
$\displaystyle\frac{\prod\limits_{\begin{subarray}{c}1\leq j\leq 3\\\ j\neq
i\end{subarray}}\
\Lambda\left(1+\alpha_{i}-\alpha_{j}\right)\Lambda(2s-1-\alpha_{i}-\alpha_{j})}{\Lambda\left(3-2s+2\alpha_{i}\right)}$
(4.64)

and $(\mathcal{C}_{\alpha}H)(s)$ is defined as in (4.62).

###### Proof.

This is an adaptation of an argument from the book [Mo97] (pp. 118). We
provide a sketch for our case as follows.

Firstly, suppose $1<\operatorname{Re}s<1+\epsilon$. The regularity of
$\mathcal{C}_{\eta}$ allows us to shift the line of integration to
$\operatorname{Re}\mu=\eta>40$. Such a shift only picks up the second set of
poles in (4.63) together with the zeros of $\Lambda(1-2\mu)$. In other words,

$\displaystyle(\mathcal{C}_{\alpha}H)(s)\ \ =\ \ $ $\displaystyle-\
\sum_{i=1}^{3}\ \operatorname{Res}_{\mu=s-1-\alpha_{i}}\ -\ \sum_{i=1}^{3}\
\operatorname{Res}_{\mu=s-\alpha_{i}}\ -\ \sum_{\rho}\
\operatorname{Res}_{\mu=(1-\rho)/2}$ $\displaystyle\hskip 100.0pt\ \ +\ \
\int_{(\eta)}\ H\left(\mu\right)\cdot\frac{\prod\limits_{i=1}^{3}\
\Lambda\left(s+\mu-\alpha_{i}\right)\Lambda\left(s-\mu-\alpha_{i}\right)}{\Lambda(1+2\mu)\Lambda(1-2\mu)}\
\frac{d\mu}{2\pi i},$ (4.65)

where $\rho$ runs over all nontrivial zeros of $\zeta(s)$.

Secondly, observe that the function on the second line of (4.7) is holomorphic
on

$\displaystyle 1-\eta+\epsilon\ <\ \operatorname{Re}s\ <\ 1+\epsilon$ (4.66)

using the facts that $\Lambda(s)$ is holomorphic on $\operatorname{Re}s<0$ or
$\operatorname{Re}s>1$, as well as $\Lambda(1\pm 2\mu)\neq 0$ on
$\operatorname{Re}\mu=\eta$. Together with the residual terms in the first
line of (4.7), a meromorphic continuation of $(\mathcal{C}_{\alpha}H)(s)$ to
the domain (4.66) is obtained.

Thirdly, we restrict the right side of (4.7) to the region
$\epsilon<\operatorname{Re}s<1-\epsilon$. It is easy to check that only the
poles $\mu=1-s+\alpha_{i}$, $\mu=s-\alpha_{i}$ and the zeros of
$\Lambda(1-2\mu)$ are between the contour $\operatorname{Re}\mu=0$ and
$\operatorname{Re}\mu=\eta$. We shift the line of integration in the last
integral of (4.7) back to $\operatorname{Re}\mu=0$. The residual contributions
of $\mu=s-\alpha_{i}$ and $\mu=(1-\rho)/2$ in (4.7) are now cancelled, leaving

$\displaystyle\hskip 20.0pt-\ \sum_{i=1}^{3}\
\operatorname{Res}_{\mu=s-1-\alpha_{i}}\ \ +\ \ \sum_{i=1}^{3}\
\operatorname{Res}_{\mu=1-s+\alpha_{i}}\ \ +\ \ \int_{(0)}\
H\left(\mu\right)\cdot\frac{\prod\limits_{i=1}^{3}\
\Lambda\left(s+\mu-\alpha_{i}\right)\Lambda\left(s-\mu-\alpha_{i}\right)}{\Lambda(1+2\mu)\Lambda(1-2\mu)}\
\frac{d\mu}{2\pi i}$ (4.67)

as the meromorphic continuation of $(\mathcal{C}_{\alpha}H)(s)$ to the domain
$\epsilon<\operatorname{Re}s<1-\epsilon$. A little calculation shows that

$\displaystyle(\mathcal{R}_{\alpha}H)(s)\ \ =\ \ -\ \sum_{i=1}^{3}\
\operatorname{Res}_{\mu=s-1-\alpha_{i}}\ \ +\ \ \sum_{i=1}^{3}\
\operatorname{Res}_{\mu=1-s+\alpha_{i}}$

and this completes our sketch.

∎

###  4.8. Proof of Theorem 1.1 and Concluding Remarks

###### Proof of Theorem 1.1.

Putting Theorem 4.8, Theorem 4.12, Proposition 2.31 and Lemma 4.13 together,
Theorem 1.1 follows from the uniqueness of analytic continuation.

∎

In the literature, it is more common to study moments of $L$-functions
starting with the approximate functional equations. A clear advantage is that
one would be able to take $s=1/2$ right at the beginning of such an argument.
However, the situation we are in is quite unusual — Theorem 1.1 connects two
very different-looking moments, and in principle, one should be able to
transition between them for two distinct applications, much like the case for
the Kuznetsov formulae.

For example, if one traces through the arguments of [Pe15] for the cubic
moments of $GL(2)$, which adopted the ‘approximate’ approach and can be
adjusted easily to the Maass case (closely related to [Y17]), the main terms
for the fourth moment of the $GL(1)$ $L$-functions are nowhere to be seen (or
hidden in the error terms in the analysis of [Pe15]) and this is rather
different from the treatment of this work (see (1.13)-(1.14) of Theorem 1.1).
In other words, the ‘approximate’ treatment eliminates the possibility of
performing spectral inversion by changing the test vectors (see the end of
Section 1.5). Also, this might explain the significance of the parts within
[Ne20+] and [Wu22] in handling the degenerate functionals.

Towards the end of Section 1.4, we have briefly compared the sources of the
main terms in the fourth moment of the $\zeta$-function with earlier methods
involving shifted divisor sums. In the context of cubic moments of $GL(2)$,
previous works such as [Iv02], [Fr20] leveraged on the shifted divisor sums to
understand the main terms with the latter one (in the weight aspect) being
more in-depth and closer to the spirit of [CFKRS05]. The work [Fr20] begins
with an exact moment identity for the twisted second moment of $GL(2)$, which
is a consequence of the Petersson formula, and its dual side consists of three
pieces of shifted divisor sums. In total, $12$ different residues are obtained
from these sums. Similar to the scenarios of [HY10, Y11], delicate
combinations of terms are required and [Fr20] managed to carry out this task.
Somewhat interestingly, some combinations yield main term contributions, some
yield error contributions, and some yield a mixture of both! This should be
the primary reason why [Fr20] arrived at an ‘approximate’ cubic moment
identity, distinct in nature from the moment identities for the first two
moments of $GL(2)$ (see [Mo92]), as well as those appeared in [Mo93, BM05,
Ne20+, Wu22, BFW21+] and this work. This interesting phenomenon deserves
futher investigations.

##  5\. Assembling à la CFKRS

We are now ready to match the results obtained in Theorem 1.1 with the Moment
Conjecture prediction (see Section 3).

###  5.1. The Orthogonal Symmetry

The terms in (4.46), (4.58), (4.59) will be shown to be the $1$, $2$, $3$-swap
terms respectively and the first term in line (4.45) will be the $0$-swap
term. Following Section 3.1, we put $s=1/2$ and the desired result follows
from an analytic continuation in $\alpha_{i}$’s. (Alternatively, we can
perform a reparametrization of the variables as in the next subsection.) Now,
observe that:

  1. (1)

The first term of (4.45) is equal to $\frac{\pi^{-3/2}}{4}$ times

$\displaystyle\hskip
30.0pt\zeta(1-\alpha_{1}-\alpha_{2})\zeta(1-\alpha_{1}-\alpha_{3})\zeta(1-\alpha_{2}-\alpha_{3})\int_{(0)}\
\frac{H(\mu)}{|\Gamma(\mu)|^{2}}\ \prod_{i=1}^{3}\ \prod_{\pm}\
\Gamma\left(\frac{1/2\pm\mu-\alpha_{i}}{2}\right)\ \frac{d\mu}{2\pi i}$

which follows from (4.49) and the relation
$\alpha_{1}+\alpha_{2}+\alpha_{3}=0$. This matches up with the $0$-swap
prediction.

  2. (2)

The terms in (4.46) is already in a form that matches up with the $1$-swap
prediction.

  3. (3)

Consider the term with $i=1$ in (4.58). It follows from Proposition 4.9 and
the functional equation of $\zeta$ in the form

$\displaystyle\hskip 30.0pt\zeta(2s-1+\alpha_{1})\cdot\pi^{\frac{1}{2}-s}\
\frac{\Gamma\left(s-\frac{1}{2}+\frac{\alpha_{1}}{2}\right)}{\Gamma\left(1-s-\frac{\alpha_{1}}{2}\right)}\
=\ \pi^{s-1+\alpha_{1}}\zeta(2-2s-\alpha_{1})\ =\
\pi^{s-1+\alpha_{1}}\zeta(2-2s+\alpha_{2}+\alpha_{3}).$

that such a term at $s=1/2$ is equal to $\frac{\pi^{-1/2}}{4}$ times

$\displaystyle\hskip 30.0pt\pi^{\alpha_{1}}$
$\displaystyle\cdot\zeta(1+\alpha_{2}+\alpha_{3})\zeta(1-\alpha_{1}+\alpha_{2})\zeta(1-\alpha_{1}+\alpha_{3})$
$\displaystyle\hskip 20.0pt\cdot\int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}\cdot\prod_{\pm}\
\Gamma\left(\frac{1/2\pm\mu-\alpha_{1}}{2}\right)\Gamma\left(\frac{1/2+\alpha_{2}\pm\mu}{2}\right)\Gamma\left(\frac{1/2+\alpha_{3}\pm\mu}{2}\right)\
\frac{d\mu}{2\pi i}.$ (5.1)

This matches up with the 2-swap prediction and the matching for the terms with
$i=2,3$ follows by symmetry.

  4. (4)

The term in (4.59) at $s=1/2$ can be written as $\frac{\pi^{-3/2}}{4}$ times

$\displaystyle\hskip
30.0pt\pi^{\alpha_{1}+\alpha_{2}+\alpha_{3}}\cdot\zeta(1+\alpha_{1}+\alpha_{2})$
$\displaystyle\zeta(1+\alpha_{1}+\alpha_{3})\zeta(1+\alpha_{2}+\alpha_{3})$
$\displaystyle\hskip 20.0pt\ \cdot\ \int_{(0)}\
\frac{H(\mu)}{|\Gamma(\mu)|^{2}}\ \prod_{i=1}^{3}\ \prod_{\pm}\
\Gamma\left(\frac{1/2+\mu+\alpha_{i}}{2}\right)\ \frac{d\mu}{2\pi i}$ (5.2)

by applying Proposition 4.10 and the functional equation of $\zeta$ three
times. This matches with the 3-swap prediction.

###  5.2. The Unitary Symmetry

In order to align well with the set-up and notations of Section 3.2, we re-
label the dual moment of Theorem 1.1 as

$\displaystyle\int_{(\frac{1}{2}+\gamma)}\
\zeta(2s-s_{0})\zeta(s_{0}+\nu_{1})\zeta(s_{0}+\nu_{2})\zeta(s_{0}+\nu_{3})\left(\mathcal{F}_{\nu}H\right)\left(s_{0},s\right)\
\frac{ds_{0}}{2\pi i},$ (5.3)

where the line of integration has been shifted slightly by $\gamma$. We also