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

Chung-Hang Kwan<EMAIL_ADDRESS>University College London

###### Abstract.

This is a sequel to our previous articles [Kw23, Kw23a+]. In this work, we
apply recent techniques that fall under the banner of ‘Period Reciprocity’ to
study moments of $GL(3)\times GL(2)$ $L$-functions in the non-archimedean
aspects, with a view towards the ‘Twisted Moment Conjectures’ formulated by
CFKRS.

###### Key words and phrases:

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

###### 2010 Mathematics Subject Classification:

11F55 (Primary) 11F72 (Secondary)

##  1\. Introduction

###  1.1. Reciprocity

This article is motivated by the important works of Young [Y11] and Blomer-
Humphries-Khan-Milinovic [BHKM20] regarding the spectral theory for the fourth
moment of the Dirichlet $L$-functions in the prime conductor aspect. Their
works featured a very curious reciprocity relation of the form

$\displaystyle\mathcal{M}_{4}(p)\ :=\ \sum_{\begin{subarray}{c}\chi\
(\bmod\,p)\\\ \chi\neq\chi_{0}\end{subarray}}\ \int_{|t|\ll 1}\
\left|L\left(\frac{1}{2}+it,\chi\right)\right|^{4}\ dt\ =\ p^{1/2}\
\sum_{\begin{subarray}{c}f:\text{ level $1$ }\\\ \text{cusp
forms}\end{subarray}}\ L\left(\frac{1}{2},f\right)^{3}\lambda_{f}(p)\ +\
(***),$ (1.1)

where $p$ is a prime number.

The formula above can be recognized as a non-archimedean generalization of the
celebrated spectral reciprocity formula due to Motohashi [Mo93, Mo97]:

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

Indeed, (1.1) incorporates an additional duality between the $GL(1)$ twists of
Dirichlet characters and the $GL(2)$ twists of Hecke eigenvalues when compared
to (1.2). Furthermore, the reciprocity (1.1) succinctly captures the essential
inputs that lead to a sharp asymptotic formula for $\mathcal{M}_{4}(p)$ — it
suffices to apply the spectral large sieve and the best approximation towards
the Ramanujan-Petersson conjecture for $GL(2)$ cuspidal Hecke eigenforms to
the spectral side of (1.1).

The arguments culminating in (1.1), as presented by [Y11] and [BHKM20], have
rendered the elegant-looking non-archimedean reciprocity even more surprising.
On one hand, Blomer et. al. [BHKM20] devised a strategic compositum of
spectral and harmonic summation formulae which can be summarized as
‘Poisson-(Arithmetic) Reciprocity-Poisson-Voronoi-Kuznetsov’. One arrived at
the reciprocity formula (1.1) upon carefully tracing through the delicate
transformations of exponential/character sums in each step. On the other hand,
Young [Y11] applied the spectral method in conjunction with the delta method
to a generalized binary additive problem. His strategy was favourable in
establishing the conjecture of [CFKRS05] for the moment $\mathcal{M}_{4}(p)$
and turned out to be useful in many other instances, see [HY10, BBLR20, Wu22,
Za19, Be19, Ng21, HN22]. An entirely different sequence of transformations led
to the same formula (1.1) once again. Regardless, both [Y11] and [BHKM20]
greatly enriched the Kuznetsov-based approach introduced by [Mo93].

The first goal of this article is to understand the source of the non-
archimedean reciprocity discovered by [Y11, BHKM20] using period integrals. In
our previous work [Kw23], we focused on the archimedean aspects of our period
identity and a number of technical features of this article were thus not
present in [Kw23]. For the sake of elucidating the key ideas and the other
applications (our second goal, see Section 1.3), we establish a higher-rank
analogue of the reciprocity (1.1) involving $GL(3)\times GL(2)$ Rankin-Selberg
$L$-functions. In this case, it is more natural to describe the result in the
spectral direction. Nevertheless, it is important to note that the reciprocity
of our interest stays the same for both the spectral and the arithmetic
directions.

###  1.2. Main Results

In this article, we work with the same admissible class of test functions
$\mathcal{C}_{\eta}$ ($\eta>40$) as in [Kw23]. The class $\mathcal{C}_{\eta}$
consists of holomorphic functions $H$ defined on the vertical strip
$|\operatorname{Re}\mu|<2\eta$ satisfying $H(\mu)=H(-\mu)$ and $H(\mu)\ll
e^{-2\pi|\mu|}$ on such a strip.

Let $\Phi$ be a fixed Hecke-normalized Maass cusp form of $SL_{3}(\mathbb{Z})$
with Langlands parameters
$(\alpha_{1},\alpha_{2},\alpha_{3})\in(i\mathbb{R})^{3}$, Hecke eigenvalues
$\lambda_{\Phi}(m)$ ($m\geq 1$), and $\widetilde{\Phi}$ being the dual form of
$\Phi$. We pick an orthogonal basis $(\phi_{j})_{j=1}^{\infty}$ of even Hecke-
normalized Maass cusp forms of $SL_{2}(\mathbb{Z})$ with
$\Delta\phi_{j}=(1/4-\mu_{j}^{2})\,\phi_{j}$ and $\lambda_{j}(a)$ ($a\geq 1$)
being the Hecke eigenvalues of $\phi_{j}$.

Let $L\left(s,\phi_{j}\otimes\Phi\right)$ and $L\left(s,\Phi\right)$ be the
Rankin-Selberg $L$-function of the pair $(\phi_{j},\Phi)$ and the standard
$L$-function of $\Phi$ respectively. We use the notation $\Lambda(\cdots)$ to
denote the complete $L$-function of $L(\cdots)$. The Hurwitz $\zeta$-function
and the additively-twisted $L$-function of $\Phi$ are denoted by $\zeta(s,a)$
and $L(s,a/c;\Phi)$ respectively, and define

$\displaystyle\mathcal{L}^{(a)}_{\pm}\left(s_{0},s;\Phi\right)\ \ :=\ \
\sum_{dr\mid a}\ \frac{d^{2s_{0}-1}\mu(r)}{r^{s_{0}}}\
\overline{\lambda_{\Phi}}\left(\frac{a}{dr}\right)\
\sideset{}{{}^{*}}{\sum}_{\ell\ (\bmod\,d)}\
\zeta\left(2s-s_{0},\frac{\ell}{d}\right)L\left(s_{0};\mp\frac{r\overline{\ell}}{d};\Phi\right).$
(1.3)

See Section 2.5 for details.

######  Theorem 1.1.

For $\frac{1}{4}+\frac{1}{200}<\sigma<\frac{3}{4}$ and
$H\in\mathcal{C}_{\eta}$, the following reciprocity formula holds:

$\displaystyle\hskip 15.0pt\sum_{j=1}^{\infty}\ H\left(\mu_{j}\right)$
$\displaystyle\
\frac{\lambda_{j}(a)\Lambda(s,\phi_{j}\otimes\widetilde{\Phi})}{\langle\phi_{j},\phi_{j}\rangle}\
+\ \int_{(0)}\
H\left(\mu\right)\frac{\sigma_{-2\mu}(a)a^{-\mu}\Lambda(s+\mu,\widetilde{\Phi})\Lambda\left(1-s+\mu,\Phi\right)}{\left|\Lambda(1+2\mu)\right|^{2}}\
\frac{d\mu}{4\pi i}$ $\displaystyle\ =\ \frac{a^{-s}}{2}\ L(2s,\Phi)\
\prod_{p\mid a}\ \left\\{\overline{\lambda_{\Phi}}\left(p^{o_{p}(a)}\right)\
-\
\frac{\overline{\lambda_{\Phi}}\left(p^{o_{p}(a)-1}\right)}{p^{2s}}\right\\}$
$\displaystyle\hskip 40.0pt\ \cdot\
\int_{(0)}\frac{H(\mu)}{|\Gamma(\mu)|^{2}}\ \prod_{\pm}\ \prod_{i=1}^{3}\
\Gamma_{\mathbb{R}}\left(s\pm\mu-\alpha_{i}\right)\ \frac{d\mu}{2\pi i}$
$\displaystyle\hskip 80.0pt\ +\ \frac{a^{s-1}}{2}\ L(2(1-s),\widetilde{\Phi})\
\prod_{p\mid a}\ \left\\{\lambda_{\Phi}\left(p^{o_{p}(a)}\right)\ -\
\frac{\lambda_{\Phi}\left(p^{o_{p}(a)-1}\right)}{p^{2(1-s)}}\right\\}$
$\displaystyle\hskip 120.0pt\ \cdot\ \int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}\ \prod_{\pm}\ \prod_{i=1}^{3}\
\Gamma_{\mathbb{R}}\left(1-s\pm\mu+\alpha_{i}\right)\ \frac{d\mu}{2\pi i}$
$\displaystyle\hskip 150.0pt\ +\ \frac{a^{-s}}{2}\ \sum_{\pm}\ \int_{(1/2)}\
\mathcal{L}^{(a)}_{\pm}\left(s_{0},s;\Phi\right)(\mathcal{F}_{\Phi}^{(\pm)}H)\left(s_{0},s\right)\
\frac{ds_{0}}{2\pi i},$ (1.4)

where $\sigma_{-2\mu}(a):=\sum_{d\mid a}d^{-2\mu}$, $o_{p}(a)$ denotes the
power of $p$ in the prime factorization of $a$, and
$(\mathcal{F}_{\Phi}^{(\pm)}H)\left(s_{0},s\right)$ are certain explicit
integral transforms to be defined in Section 2.7.

Our theorem holds for general twists of $GL(2)$ Hecke eigenvalues (i.e., $a$
is not necessarily prime). In this instance, we find it more convenient to
describe the dual moment of (1.1) in terms of additively-twisted $L$-functions
and this actually explains the origin of our reciprocity better (see Remark
5.1). In a number of previous works on twisted moments, e.g., [Liu10, C07,
Be16, Y11b, AK18], one had to restrict to Hecke eigenvalue twists at primes
for various technical reasons.

When $a=p$ is prime, the dual moment can be expressed in terms of primitive
multiplicatively-twisted $L$-functions as in (1.1) (see Proposition 4.7).
Furthermore, if $s=1/2$ and $\Phi$ is replaced by the (complete) minimal
parabolic Eisenstein series $(E_{\min}^{(3)})^{{}^{*}}(\,*\,;\,(0,0,0))$ of
$SL_{3}(\mathbb{Z})$, then the dual moments of (1.1) is precisely the fourth
moments of the Dirichlet $L$-functions:

$\displaystyle\hskip 10.0pt\frac{p^{\frac{1}{2}}}{\phi(p)}\ \ \
\sideset{}{{}^{\pm}}{\sum}_{\chi\ (\bmod\,p)}\ \int_{(1/2)}\ \
\left|L(s_{0},\chi)\right|^{4}\,(\mathcal{F}_{\alpha}^{(+)}H\pm\mathcal{F}_{\alpha}^{(-)}H)\left(s_{0},\,1/2\right)\frac{\Gamma_{\mathbb{R}}\left(1+\frac{1\mp
1}{2}-s_{0}\right)}{\Gamma_{\mathbb{R}}\left(s_{0}+\frac{1\mp 1}{2}\right)}\
\frac{ds_{0}}{2\pi i},$ (1.5)

which we will show in Section 5.1. When $a$ is composite, the relevant
conversion remains valid but this time (1.5) would average over all Dirichlet
characters $(\bmod\,a)$ — both primitive and imprimitive ones. Judging from
the experiences of [BHKM20] and [Wu22], this phenomenon should not be
surprising even to period integral approaches.

###  1.3. Twisting and the ‘Recipe’

Theorem 1.1 exhibits a number of interesting technical similarities with the
asymptotic formulae for the twisted moments of the Riemann $\zeta$-function
and the Dirichlet $L$-functions. There is an extensive literature on this
classical topic: Levinson [Le74], Deshouillers-Iwaniec [DI84],
Balasubramanian-Conrey-Heath-Brown [BCHB85], Conrey [C89], Watt [W95] and
Iwaniec-Sarnak [IS99] — just to name a few.

For more basic applications (say [Le74], [Y10], [BCHB85], [IS99]) involving
twisted moments of lower degrees such as

$\displaystyle\int_{T}^{2T}\
\left(\frac{h}{k}\right)^{it}\zeta(1/2+\alpha+it)\zeta(1/2+\beta-it)\ dt,$
(1.6)

or

$\displaystyle\frac{1}{\phi(q)}\ \ \sideset{}{{}^{*}}{\sum}_{\chi\
(\bmod\,q)}\
\chi(h)\overline{\chi}(k)L\left(1/2+\alpha,\chi\right)L\left(1/2+\beta,\overline{\chi}\right),$
(1.7)

for any $h,k\geq 1$ and $\alpha,\beta\ll(\log qT)^{-1}$, the diagonals usually
dominate the asymptotic evaluations upon inserting the approximate functional
equations (AFEs). However, the twisted variant of the Moment Conjecture (or
‘twisted recipe’) of [CFKRS05] does predict the existence of off-diagonal
contributions which are non-negligible for more refined applications, see
Bettin-Chandee-Radziwiłł [BCR17], Conrey-Iwaniec-Soundararajan [CIS19], Bui-
Pratt-Robles-Zaharescu [BPRZ20] and Conrey [C07]. In these works, the search
of the essential off-diagonal terms turned out to be extremely subtle. The
‘shifts’ (i.e., $\alpha,\beta$ in (1.6)-(1.7)) in the twisted moments are
crucial in unraveling the combinatorics behind the assembling of various
distinct-looking terms.

A similar situation is encountered in the context of Theorem 1.1. In view of
the applications of Liu [Liu10] and Khan [Kh15] regarding the $GL(3)\times
GL(2)$ $L$-values, it was sufficient to extract only the diagonal as the main
term. As in Li [Li11], their moments were expanded as sums of geometric terms
via the AFEs and Petersson/ Kuznetsov/ $GL(3)$ Voronoi formulae.
Unfortunately, it is not clear from their works the locations of the off-
diagonal main terms. A recurring theme in the area, as exemplified by [CIS12,
CIS19, C07, BCR17], involves the identification of specific polar divisors
subsequent to the re-packaging of the off-diagonal pieces. This procedure is
usually delicate and depends heavily on the arithmetic of the moment problem.
In Section 4.4, we shall observe the Motohashi-type structure depicted in
Theorem 1.1 is indeed very suitable for a simple extraction of the off-
diagonal main term. In particular, the twisted sums of $L$-functions (1.3)
admit a clear polar divisor $2s-s_{0}=1$. Overall speaking, we seek the most
canonical and straightforward mean to reveal the desired structure. It appears
to us that the use of period integrals is favourable in this regard.

In Theorem 1.1, we examine the case when $\Phi$ is a cusp form of
$SL_{3}(\mathbb{Z})$. This is nevertheless adequate to illustrate the
intricacies in understanding the main terms for twisted moments while keeping
our exposition to a reasonable length. In fact, our twisted moment is of
degree $6$ and its off-diagonal main term corresponds precisely to the
‘$3$-swap term’, according to the set-up and terminology of Conrey-Keating
[CK19, CK16, CK15a, CK15b, CK15c] and Conrey-Fazzari [CF23], of the twisted
cubic moment of $GL(2)$ $L$-functions:

$\displaystyle\sum_{j=1}^{\infty}\ H(\mu_{j})\
\frac{\lambda_{j}(a)\Lambda(1/2-\alpha_{1},\phi_{j})\Lambda(1/2-\alpha_{2},\phi_{j})\Lambda(1/2-\alpha_{3},\phi_{j})}{\langle\phi_{j},\phi_{j}\rangle}\
+\ (\text{cont.}),$ (1.8)

which is simply the Eisenstein case of Theorem 1.1, i.e., taking $\Phi$ to be
$(E_{\min}^{(3)})^{{}^{*}}(\ *\ ;\,(\alpha_{1},\alpha_{2},\alpha_{3}))$.

Our current knowledge on the Dirichlet polynomial approximations of
$L$-functions enables us to extract the ‘$0$-swap’ and ‘$1$-swap’ terms across
various moments and families, see [HN22, Ng21, BTB22+, CK15c], [CF23] and
[CR22+] for, respectively, the unitary, orthogonal and symplectic examples. To
go beyond the $0$-swap and $1$-swap terms, one would require extra arithmetic
structures, say the Motohashi-type structure present in our case. Furthermore,
since our method is based on period integrals, we are able to sidestep the
technical complications stem from the use of AFEs and treating the $J$\- and
$K$-Bessel pieces of the spectral Kuznetsov formula separately (see [Kw23] for
further discussions).

Given a twisted moment of $L$-functions, showing agreement with the relevant
twisted recipe often requires additional symmetries besides the functional
equations of the $L$-functions. For instance, certain finite Euler products,
which account for the ramifications coming from twisting, satisfy unexpected
functional relations. Within the unitary families, readers are referred to
[HY10] (Section 6.2-6.3), [CIS19] (Section 8) and [BTB22+] (Section 9.3) (cf.
[CK15c] and [HN22] (Section 5-6)) for the examples. Established in a prime-by-
prime (or local) fashion with leverage on the delicate combinatorics of
divisors, these relations result in remarkable combinations and cancellations
of terms, ultimately leading to the desired prediction up to an acceptable
error. In the case of Theorem 1.1, one has to unravel the complicated
expression (4.4) for the hidden functional relations if one follows the
standard approach in the literature.

In Section 4.4 of this article, we proceed globally instead by studying the
triple Dirichlet series:

$\displaystyle\sum_{a=1}^{\infty}\ a^{-w}\ \sum_{d\mid a}\ d^{4s-3}\
\sum_{a_{1}=1}^{\infty}\
\frac{\mathcal{B}_{\Phi}\left(a/d,a_{1}\right)}{(a_{1})^{2s-1}}\cdot
S(0,a_{1};d),$ (1.9)

where $\mathcal{B}_{\Phi}(\,\cdot\,,\,\cdot\,)$’s are the $GL(3)$ Fourier
coefficients of $\Phi$ and $S(0,a_{1};d)$ is the Ramanujan sum. We will
explain how (1.9) arises from our period integral method as well as how it
takes up the role of extra functional relations as in [HY10, CIS19, HN22,
BTB22+]. The (multiple) Dirichlet series of $GL(3)$ will provide convenient
packagings of the underlying combinatorics. From which, the recipe prediction
is confirmed with an exact reciprocity identity and for arbitrary twists of
Hecke eigenvalues of $GL(2)$. In many ways, the content of Section 4.4 serves
as a non-archimedean analogue of the Barnes-type integral identities found in
[Kw23a+]. This kind of analogies was also investigated by Bump [Bu84] but in a
different situation. Bump constructed an interesting double Dirichlet series
(see equation (9.2) of [Bu84]) that mirrors the Vinogradov-Takhtadzhyan
formula (see equation (10.1) of [Bu84]).

In a separate article [Kw23a+], we tackled the analytic issues for the
untwisted Eisenstein case of Theorem 1.1, including explication of the
integral transforms, regularization and analytic continuation (with the
Fourier expansion for the $GL(3)$ Eisenstein series). By combining the
techniques of [Kw23a+] with the non-archimedean ingredients of this article,
the twisted recipe for (1.8) (see Conjecture 5.2) should follow with some
extra book-keeping. It is noteworthy that the majority of our arguments work
equally well for the Eisenstein case.

Both Theorem 1.1 and Conjecture 5.2 express their main terms in a local
fashion. This better displays the symmetries — the finite part of each term
now consists of a product of $L$-values and a ‘modified’ Hecke eigenvalue of
$GL(3)$. As a matter of fact, this is a somewhat intriguing phenomenon for the
off-diagonal main terms because the $GL(3)$ Hecke combinatorics are exploited
in rather non-trivial ways. In a broader context, it might be of interest to
study the twisted recipes for different families (perhaps also revisiting some
of the previous works) by searching for suitable candidates of multiple
Dirichlet series and their combinatorial relations.

###  1.4. Related Works

In contrast to the Motohashi-type reciprocities which relate two different-
looking moments of $L$-functions, there are also the recent ‘level
reciprocities’ of Blomer-Khan [BK19a, BK19b] roughly taking the shape

$\displaystyle\sum_{\begin{subarray}{c}f:\text{ level $p$ }\\\ \text{cusp
forms}\end{subarray}}\ \lambda_{f}(q)L(1/2,\Phi\otimes f)L(1/2,f)\ \
\longleftrightarrow\ \ \sum_{\begin{subarray}{c}f:\text{ level $q$ }\\\
\text{cusp forms}\end{subarray}}\ \lambda_{f}(p)L(1/2,\widetilde{\Phi}\otimes
f)L(1/2,f)$ (1.10)

for $p,q$ being primes. In the latter case, two similar-looking moments are
connected, akin to an earlier example due to Conrey [C07] (see also [Y11b,
Be16, DN22+]):

$\displaystyle\sideset{}{{}^{*}}{\sum}_{\chi\,(\bmod\,p)}\
\chi(q)|L(1/2,\chi)|^{2}\ \ \longleftrightarrow\ \
\sideset{}{{}^{*}}{\sum}_{\chi\,(\bmod\,q)}\ \chi(-p)|L(1/2,\chi)|^{2}.$
(1.11)

The level reciprocity (1.10) features the switching of two distinct twists of
$GL(2)$ Hecke eigenvalues appeared on both sides of the formula. It is
conceivable that a certain invariant automorphic linear functional which
incorporates two distinct $GL(2)$ Hecke operators is responsible for such kind
of reciprocities. This was nicely confirmed by two different constructions of
such functionals, see Zacharias [Za21, Za20+] (for the ‘$4=2+2$’ case) and
Nunes [Nu23] (for the ‘$4=3+1$’ case), and for Hecke twists at two coprime
square-free integral ideals.

While the interchange of harmonics
$\left(\lambda_{f}(p)\right)\leftrightarrow\left(\chi\bmod\,p\right)$
described in (1.1)-(1.5) and (1.1) is also of non-archimedean nature (and to
some extent a hybrid of the ones in (1.10) and (1.11)), its primary arithmetic
cause has less to do with the actions of the Hecke operators and is actually
rather different from [Za20+, Za21, Nu23], see Remark 5.1.

Moreover, Theorem 1.1 is a higher-degree and higher-rank analogue of the
famous result of Iwaniec-Sarnak (Theorem 17 of [IS00]), Motohashi (Equation
(2.33)-(2.36) of [Mo92]) and Bykovskii ([By96]) which has found plenty of
applications in the area, see [Mo92], [Iv01, Iv02], [J99], [Fr20], [IS00],
[Liu18], [BF21].

###  1.5. Acknowledgement

The reseach is supported by the EPSRC grant: EP/W009838/1. Part of the work
was completed during the author’s stay at American Institute of Mathematics,
Texas A&M University, the Chinese University of Hong Kong and Queen’s
University. The author would like to express his gratitude for their generous
hospitality.

##  2\. Preliminary

In this section, we collect results and notions that are essential to our
arguments. Some of which have appeared in [Kw23] (Section 2.2-2.3).

###  2.1. Notations and Conventions

Throughout this article, $\Gamma_{\mathbb{R}}(s):=\pi^{-s/2}\,\Gamma(s/2)$
$(s\in\mathbb{C})$, $e(x):=e^{2\pi ix}$ $(x\in\mathbb{R})$, and
$\Gamma_{n}:=SL_{n}(\mathbb{Z})$ $(n\geq 2)$. Our test function $H$ lies in
the class $\mathcal{C}_{\eta}$ and $H:=h^{\\#}$ (see Definition 2.2) without
otherwise specified. 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 [Go15]. We often
use the same symbol to denote a function (in $s$) and its analytic
continuation. Also, we adopt the following sets 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 [Go15].

  2. (2)

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

###  2.2. Whittaker Functions & Transforms

All of the Whittaker functions in this article are spherical. 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 $y>0$. Denote by
$W_{\alpha}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\ &&1\end{pmatrix}$ the
Whittaker function of $GL_{3}(\mathbb{R})$, where

$\displaystyle\hskip 15.0pty_{0},\,y_{1}>0\hskip 15.0pt\text{ and }\hskip
15.0pt\alpha\in\mathfrak{a}_{\mathbb{C}}^{(3)}\ :=\
\left\\{(\alpha_{1},\alpha_{2},\alpha_{3})\in\mathbb{C}^{3}:\alpha_{1}+\alpha_{2}+\alpha_{3}=0\right\\}.$
(2.2)

To define $W_{\alpha}$, we first introduce the function

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

whenever (2.2) holds. Then we define

$\displaystyle W_{\alpha}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\ :=\ $ $\displaystyle\prod_{1\leq j<k\leq
3}\,\Gamma_{\mathbb{R}}(1+\alpha_{j}-\alpha_{k})$ $\displaystyle\hskip
20.0pt\cdot\,\int_{\mathbb{R}^{3}}\ I_{\alpha}\left[\begin{pmatrix}&&1\\\
&-1&\\\ 1&&\end{pmatrix}\begin{pmatrix}1&u_{1,2}&u_{1,3}\\\ &1&u_{2,3}\\\
&&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]\,e(-u_{1,2}-u_{2,3})\,\prod_{1\leq j<k\leq
3}\,du_{j,k},$ (2.4)

once again for (2.2). The last integral is known as the Jacquet integral. See
Chapter 5.5 of [Go15] for details. We recall the explicit evaluation of the
Rankin-Selberg integral of $GL_{3}(\mathbb{R})\times GL_{2}(\mathbb{R})$.

######  Proposition 2.1.

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

$\int_{0}^{\infty}\int_{0}^{\infty}\
W_{\mu}(y_{1})\,W_{-\alpha}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\,(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\
\frac{dy_{0}dy_{1}}{y_{0}y_{1}^{2}}\ =\
\frac{1}{4}\,\prod_{\pm}\prod_{k=1}^{3}\
\Gamma_{\mathbb{R}}\left(s\pm\mu-\alpha_{k}\right).$ (2.5)

###### Proof.

See [Bu88]. ∎

As in [Kw23, Kw23a+], the following pair of integral transforms plays an
important role in the description of the archimedean component of our main
theorem.

######  Definition 2.2.

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

$h^{\\#}(\mu)\ :=\ \int_{0}^{\infty}h(y)\,W_{\mu}(y)\ \frac{dy}{y^{2}},$ (2.6)

whereas its inverse transform is defined by

$\displaystyle H^{\flat}(y)\ :=\ \frac{1}{4\pi i}\int_{(0)}\
H(\mu)\,W_{\mu}(y)\ \frac{d\mu}{\left|\Gamma(\mu)\right|^{2}},$ (2.7)

provided the integrals converge absolutely.

######  Definition 2.3.

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

$\displaystyle H(\mu)\ \ll\ e^{-2\pi|\mu|}\hskip
50.0pt(|\operatorname{Re}\mu|\ <\ 2\eta).$ (2.8)

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

######  Proposition 2.4.

For any $H\in\mathcal{C}_{\eta}$, the integral (2.7) defining $H^{\flat}$
converges absolutely. Moreover, we have

$\displaystyle H^{\flat}(y)\ \ll\ \min\\{y,y^{-1}\\}^{\eta}\hskip 20.0pt(y\ >\
0).$ (2.9)

###### Proof.

See Lemma 2.10 of [Mo97]. ∎

######  Proposition 2.5.

Under the same assumptions of Proposition 2.4, we have

$\displaystyle(h^{\\#})^{\flat}(g)\ =\ h(g)\hskip 20.0pt\text{ and }\hskip
20.0pt(H^{\flat})^{\\#}(\mu)\ =\ H(\mu).$ (2.10)

###### Proof.

See Lemma 2.10 of [Mo97]. ∎

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

Let

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

with its invariant measure given by $dg:=y^{-2}du\,dy$. We use $\langle\
\cdot\ ,\ \cdot\ \rangle$ to denote the Petersson inner product on
$\Gamma_{2}\setminus\mathfrak{h}^{2}$, i.e.,

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

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

$(\widehat{\phi})_{a}(y)\ :=\ \int_{0}^{1}\ \phi\left[\begin{pmatrix}1&u\\\
&1\end{pmatrix}\begin{pmatrix}y&\\\ &1\end{pmatrix}\right]e(-au)\ du\ =\
\frac{\mathcal{B}_{\phi}(a)}{\sqrt{|a|}}\,W_{\mu(\phi)}(|a|y)\hskip 15.0pt(y\
>\ 0).$ (2.11)

Let $I_{\mu}(y):=y^{\mu+\frac{1}{2}}$. The Eisenstein series of $\Gamma_{2}$
is given by

$\displaystyle E(z;\mu)\ :=\ \frac{1}{2}\ \sum_{\gamma\in
U_{2}(\mathbb{Z})\setminus\Gamma_{2}}I_{\mu}(\operatorname{Im}\gamma z)\hskip
20.0pt(z\ \in\ \mathfrak{h}^{2}).$ (2.12)

It is well-known that the series (2.12) converges absolutely for
$\operatorname{Re}\mu>1/2$ and admits a meromorphic continuation to the whole
complex plane. Also, we have $\Delta
E(\,*\,;\mu)\,=\,\left(\frac{1}{4}-\mu^{2}\right)E(\,*\,;\mu)$ and the Fourier
coefficients of $E(*;\mu)$ are given by

$\displaystyle\mathcal{B}(a;\mu)\ =\ \mathcal{B}_{E(\,*\,;\,\mu)}(a)\ =\
\frac{|a|^{\mu}\sigma_{-2\mu}(|a|)}{\Lambda(1+2\mu)},$ (2.13)

where

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

We also write $E^{*}(*;\mu):=\Lambda(1+2\mu)\,E(*;\mu)$ for the complete
Eisenstein series of $\Gamma_{2}$ and
$\mathcal{B}_{E^{{}^{*}}(\,*\,;\,\mu)}(a)$ for its Fourier coefficients.

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

Next, let

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

Let $\Phi:\mathfrak{h}^{3}\rightarrow\mathbb{C}$ be a Maass cusp form of
$\Gamma_{3}$ as defined in Definition 5.1.3 of [Go15]. In particular, there
exists $\alpha=\alpha(\Phi)\in\mathfrak{a}_{\mathbb{C}}^{(3)}$ such that for
any $D\in Z(U\mathfrak{gl}_{3}(\mathbb{C}))$ (i.e., 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 2.6.

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
$SL_{3}(\mathbb{Z})$. The integral defined by

$\displaystyle\hskip 10.0pt(\widehat{\Phi})_{(m_{1},m_{2})}(g)\ $
$\displaystyle:=\
\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\Phi\left[\begin{pmatrix}1&u_{1,2}&u_{1,3}\\\
&1&u_{2,3}\\\
&&1\end{pmatrix}g\right]e\left(-m_{1}u_{2,3}-m_{2}u_{1,2}\right)\ du_{1,2}\
du_{1,3}\ du_{2,3},$ (2.14)

for $g\in GL_{3}(\mathbb{R})$, 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

$\displaystyle(\widehat{\Phi})_{(m_{1},m_{2})}\begin{pmatrix}y_{0}y_{1}&&\\\
&y_{0}&\\\ &&1\end{pmatrix}\ =\
\frac{\mathcal{B}_{\Phi}(m_{1},m_{2})}{|m_{1}m_{2}|}\
W_{\alpha(\Phi)}^{{}^{\operatorname{sgn}(m_{2})}}\begin{pmatrix}(|m_{1}|y_{0})(|m_{2}|y_{1})&&\\\
&|m_{1}|y_{0}&\\\ &&1\end{pmatrix}$ (2.15)

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

######  Remark 2.7.

  1. (1)

The archimedean multiplicity-one theorem of Shalika guarantees the well-
definedness of the Fourier coefficients for $\Phi$. For a proof, see Theorem
6.1.6 of [Go15].

  2. (2)

We shall make use of the following simple observation frequently:

$\displaystyle(\widehat{\Phi})_{(m_{1},m_{2})}\left[\begin{pmatrix}1&u_{1,2}&u_{1,3}\\\
&1&u_{2,3}\\\ &&1\end{pmatrix}g\right]\ \ =\ \
e\left(m_{1}u_{2,3}+m_{2}u_{1,2}\right)\cdot(\widehat{\Phi})_{(m_{1},m_{2})}(g)$
(2.16)

for any $g\in GL_{3}(\mathbb{R})$ and $u_{i,j}\in\mathbb{R}$ ($1\leq i<j\leq
3$).

If $\Phi$ is Hecke-normalized (part of the convention of this article, see
Section 2.1.(1)), then $\mathcal{B}_{\Phi}(1,n)$ can be shown to be a Hecke
eigenvalue of $\Phi$ and thus is a multiplicative function. In this case, we
write $\lambda_{\Phi}(n):=\mathcal{B}_{\Phi}(1,n)$. Furthermore, we have

$\displaystyle\mathcal{B}_{\Phi}(a,a_{1})\ =\ \sum\limits_{r\mid(a,a_{1})}\
\mu(r)\overline{\lambda_{\Phi}}(a/r)\lambda_{\Phi}(a_{1}/r)$ (2.17)

for any $a,a_{1}\in\mathbb{Z}-\\{0\\}$, where $\mu(\cdot)$ denotes the Möbius
$\mu$-function. See Section 6.4 of [Go15] for details.

###  2.5. Automorphic $L$-functions

The Maass cusp forms $\Phi$ and $\phi$ of $\Gamma_{3}$ and $\Gamma_{2}$ are
assumed to be Hecke-normalized. 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. If
$\widetilde{\Phi}(g):=\Phi\left({}^{t}g^{-1}\right)$ is the dual form of
$\Phi$, then the Langlands parameters of $\widetilde{\Phi}$ are given by
$-\alpha$.

######  Definition 2.8.

Suppose $\Phi$ (resp. $\phi$) is a Maass cusp form or an Eisenstein series of
$\Gamma_{3}$ (resp. $\Gamma_{2}$). 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)\ $ $\displaystyle:=\
\sum_{m_{1}=1}^{\infty}\
\sum_{m_{2}=1}^{\infty}\frac{\mathcal{B}_{\phi}(m_{2})\mathcal{B}_{\Phi}(m_{1},m_{2})}{\left(m_{1}^{2}m_{2}\right)^{s}}.$
(2.18)

######  Proposition 2.9.

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

$\displaystyle\left(\phi,\
\mathbb{P}_{2}^{3}\Phi\cdot|\det*|^{\overline{s}-\frac{1}{2}}\right)_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}\ \ :=\ \ \int_{\Gamma_{2}\setminus GL_{2}(\mathbb{R})}\
\phi(g)\widetilde{\Phi}\begin{pmatrix}g&\\\ &1\end{pmatrix}|\det
g|^{s-\frac{1}{2}}\ dg\ =\ \frac{1}{2}\
\Lambda(s,\phi\otimes\widetilde{\Phi}),$ (2.19)

where

$\displaystyle\Lambda(s,\phi\otimes\widetilde{\Phi})\ $ $\displaystyle:=\
L_{\infty}(s,\phi\otimes\widetilde{\Phi})\,L(s,\phi\otimes\widetilde{\Phi})$
(2.20)

and

$\displaystyle L_{\infty}(s,\phi\otimes\widetilde{\Phi})\ $ $\displaystyle:=\
\prod_{k=1}^{3}\ \Gamma_{\mathbb{R}}\left(s\pm\mu-\alpha_{k}\right).$ (2.21)

###### Proof.

See [Go15] and the relevant remarks in [Kw23]. ∎

######  Definition 2.10.

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

$\displaystyle L\left(s,\Phi\right)\ :=\ \sum_{n=1}^{\infty}\
\frac{\mathcal{B}_{\Phi}(1,n)}{n^{s}}.$ (2.22)

The standard $L$-function admits an entire continuation and satisfies a
functional equation:

######  Proposition 2.11.

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

$\displaystyle\Lambda\left(s,\Phi\right)\ =\ \Lambda(1-s,\widetilde{\Phi}),$
(2.23)

where

$\displaystyle\Lambda\left(s,\Phi\right)\ :=\
L_{\infty}\left(s,\Phi\right)\,L\left(s,\Phi\right)$ (2.24)

and

$\displaystyle L_{\infty}\left(s,\Phi\right)\ :=\ \prod_{k=1}^{3}\
\Gamma_{\mathbb{R}}\left(s+\alpha_{k}\right).$ (2.25)

###### Proof.

See Chapter 6.5 of [Go15]. ∎

Furthermore, the standard $L$-function $L(s,\phi)$ (resp. $L(s,\Phi)$) for
$\phi$ (resp. $\Phi$), which is either a Hecke-normalized Maass cusp form or a
complete Eisenstein series of $\Gamma_{2}$ (resp. $\Gamma_{3}$), admits an
Euler product of the form

$\displaystyle L(s,\phi)\ =\ \prod_{p}\ \prod_{j=1}^{2}\
\left(1-\beta_{\phi,j}(p)p^{-s}\right)^{-1},$ (2.26)

resp.

$\displaystyle L(s,\Phi)\ =\ \prod_{p}\ \prod_{k=1}^{3}\
\left(1-\alpha_{\Phi,k}(p)p^{-s}\right)^{-1}$ (2.27)

for $\operatorname{Re}s\gg 1$. One can show that

$\displaystyle L(s,\phi\otimes\Phi)\ \ =\ \ \prod_{p}\ \prod_{j=1}^{2}\
\prod_{k=1}^{3}\
\left(1-\beta_{\phi,j}(p)\alpha_{\Phi,k}(p)p^{-s}\right)^{-1},$ (2.28)

see Proposition 7.4.12 of [Go15]. If $\phi=E^{*}(*;\mu)$, then (2.26) holds
for $\left\\{\beta_{\phi,1}(p),\beta_{\phi,2}(p)\right\\}$
$=\\{p^{\mu},p^{-\mu}\\}$ and (2.26)-(2.28) imply the Dirichlet series
$L(s,\phi\otimes\Phi)$ is given by the product of $L$-functions
$L(s+\mu,\Phi)L(s-\mu,\Phi)$.

######  Proposition 2.12.

Suppose $\Phi$ is a Hecke-Maass cusp form of $\Gamma_{3}$. For any
$s,\mu\in\mathbb{C}$, we have

$\displaystyle\left(\,E^{*}(\,*\,;\,\mu)\,,\,\left(\mathbb{P}_{2}^{3}\Phi\right)\cdot|\det*|^{\bar{s}-\frac{1}{2}}\right)_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}\ \ =\ \ \frac{1}{2}\
\Lambda(s+\mu,\widetilde{\Phi})\Lambda(s-\mu,\widetilde{\Phi}).$ (2.29)

###### Proof.

Parallel to Proposition 2.9. ∎

We will also need the basic analytic properties of certain ‘generalized’ zeta-
functions and automorphic $L$-functions which are stated as follow.

######  Proposition 2.13.

  1. (1)

For $\operatorname{Re}s>1$ and $a\neq 0,-1,-2,\ldots$, the Hurwitz
$\zeta$-function, which is defined by

$\displaystyle\zeta(s,a)\ :=\ \sum_{n=0}^{\infty}\,(n+a)^{-s},$ (2.30)

admits a holomorphic continuation to $\mathbb{C}$ except at $s=1$. It has a
simple pole at $s=1$ and the residue is equal to $1$. Moreover, it has
polynomial growth in every vertical strip.

  2. (2)

Let $\Phi$ be a Maass cusp form of $\Gamma_{3}$. For
$\operatorname{Re}s>1+\theta$ and $a/c\in\mathbb{Q}$, the additively-twisted
$L$-series of $\Phi$ by $a/c$, which is defined by

$\displaystyle L\left(s,\frac{a}{c};\Phi\right)\ :=\
\sum_{\begin{subarray}{c}n=1\end{subarray}}^{\infty}\ \
\frac{\mathcal{B}_{\Phi}(1,n)}{n^{s}}\,e\left(\frac{na}{c}\right),$ (2.31)

admits an entire continuation and it has polynomial growth in every vertical
strip.

###### Proof.

See [Ap76] and [GL06]. ∎

It is well-known that both (2.30) and (2.31) satisfy some forms of functional
equations, see Theorem 12.6 of [Ap76] and Theorem 3.1 of [GL06] respectively.
However, we will make no use of such formulae in the proof of Theorem 1.1. The
polynomial growth of (2.30) and (2.31) in vertical strips can be deduced from
that of the Dirichlet $L$-functions and the multiplicatively-twisted
$L$-functions of $\Phi$. Moreover, the regularity of our class of test
functions $\mathcal{C}_{\eta}$ is more than sufficient for all of our analytic
purposes provided $\eta>40$.

###  2.6. Calculations on the Spectral Side

######  Definition 2.14.

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

$P^{a}(z;h)\ :=\ \sum_{\gamma\in
U_{2}(\mathbb{Z})\setminus\Gamma_{2}}h(a\operatorname{Im}\gamma z)\cdot
e\left(a\operatorname{Re}\gamma z\right)\hskip 20.0pt(z\ \in\
\mathfrak{h}^{2})$ (2.32)

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 \ $y\to\infty$)}$ (2.33)

are satisfied, then the Poincaré series $P^{a}(z;h)$ converges absolutely and
is an $L^{2}$-function on $\Gamma_{2}\setminus\mathfrak{h}^{2}$. In this
article, we take $h:=H^{\flat}$ with $H\in\mathcal{C}_{\eta}$ and $\eta>40$.
The conditions of (2.33) clearly holds because of Proposition 2.4. We often
use the shorthand $P^{a}:=P^{a}(*;h)$.

######  Proposition 2.15.

Let $\Phi$ be a Hecke-Maass cusp form of $\Gamma_{3}$ and $P^{a}$ be a
Poincaré series of $\Gamma_{2}$. Then for $s\in\mathbb{C}$, we have

$\displaystyle 2a^{-1/2}\,\left(P^{a},\
\mathbb{P}_{2}^{3}\Phi\cdot|\det*|^{\overline{s}-\frac{1}{2}}\right)_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}$ $\displaystyle\ =\ \sum_{j=1}^{\infty}\
h^{\\#}\left(\mu_{j}\right)\,\frac{\overline{\mathcal{B}_{j}(a)\raisebox{11.38109pt}{}}\,\Lambda(s,\phi_{j}\otimes\widetilde{\Phi})}{\langle\phi_{j},\phi_{j}\rangle}$
$\displaystyle\hskip 40.0pt\ +\ \int_{(0)}\
h^{\\#}\left(\mu\right)\,\frac{\sigma_{-2\mu}(a)a^{-\mu}\Lambda(s+\mu,\widetilde{\Phi})\Lambda(s-\mu,\widetilde{\Phi})}{\left|\Lambda(1+2\mu)\right|^{2}}\
\frac{d\mu}{4\pi i},$ (2.34)

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.

See [Kw23]. ∎

###  2.7. Archimedean Analysis: Properties of the Integral Transforms
$\mathcal{F}^{(\pm)}_{\Phi}$

Let $H=h^{\\#}$. The integral transforms for this article as well as [Kw23,
Kw23a+] are given by

$\displaystyle\hskip 10.0pt(\mathcal{F}^{(\pm)}_{\Phi}H)\left(s_{0},s\right)\
=\ \int_{0}^{\infty}\int_{0}^{\infty}\
h\left(\frac{y_{1}}{\sqrt{1+y_{0}^{2}}}\right)\ $
$\displaystyle\frac{y_{0}^{2s-s_{0}}}{(1+y_{0}^{2})^{\frac{s}{2}-s_{0}+\frac{1}{4}}}$
$\displaystyle\cdot\,\left(\int_{0}^{\infty}\
W_{-\alpha(\Phi)}\begin{pmatrix}Xy_{1}&&\\\ &X&\\\ &&1\end{pmatrix}e(\pm
Xy_{0})X^{s_{0}-1}\ d^{\times}X\right)y_{1}^{s-\frac{1}{2}}\ \
\frac{dy_{0}\,dy_{1}}{y_{0}y_{1}^{2}}.$

Readers should consult [Kw23] for various formulations of (2.7) in terms of
Mellin-Barnes integrals. For technical reasons, it is often convenient to work
with the ‘perturbed’ version of
$(\mathcal{F}^{(\pm)}_{\Phi}H)\left(s_{0},s\right)$, denoted by
$(\mathcal{F}^{(\pm)}_{\Phi}H)\left(s_{0},s;\,\phi\right)$ (for
$\phi\in(0,\pi/2]$), see equation (7.6)-(7.7) of [Kw23]. We have
$(\mathcal{F}^{(\pm)}_{\Phi}H)\left(s_{0},s\right)=(\mathcal{F}^{(\pm)}_{\Phi}H)\left(s_{0},s;\,\pi/2\right)$.
As in [Kw23, Kw23a+], we take $\epsilon:=1/100$ throughout this article.

######  Proposition 2.16.

Suppose $H\in\mathcal{C}_{\eta}$, $s:=\sigma+it$ and
$s_{0}:=\sigma_{0}+it_{0}$. Then

  1. (1)

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

$\displaystyle\sigma_{0}\ >\ \epsilon,\hskip 15.0pt\sigma\ <\ 4,\hskip
15.0pt\text{ and }\hskip 15.0pt2\sigma-\sigma_{0}-\epsilon\ >\ 0.$ (2.36)

  2. (2)

Whenever $(\sigma_{0},\sigma)\in(\ref{newdomain})$, $|t|<T$ and
$\phi\in(0,\pi/2)$, the transform
$(\mathcal{F}_{\Phi}^{(\pm)}H)\left(s_{0},\,s;\,\phi\right)$ 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

$\displaystyle\left|(\mathcal{F}_{\Phi}^{(\pm)}H)\left(s_{0},\,s\right)\right|\
\ll\ |t_{0}|^{8-\frac{\eta}{2}}\log^{B}|t_{0}|,$ (2.37)

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

###### Proof.

See Proposition 8.1 and Proposition 9.1 of [Kw23]. ∎

######  Proposition 2.17.

Suppose $1/2<\sigma<1$. Then

$\displaystyle\hskip 10.0pt\sum_{\pm}\
(\mathcal{F}^{(\pm)}_{\Phi}H)\left(2s-1,s\right)\ =\ \pi^{\frac{1}{2}-s}$
$\displaystyle\cdot\,\prod_{i=1}^{3}\
\frac{\Gamma\left(s-\frac{1}{2}+\frac{\alpha_{i}}{2}\right)}{\Gamma\left(1-s-\frac{\alpha_{i}}{2}\right)}$
$\displaystyle\hskip 25.0pt\cdot\,\int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}\cdot\prod\limits_{i=1}^{3}\
\prod\limits_{\pm}\ \Gamma\left(\frac{1-s+\alpha_{i}\pm\mu}{2}\right)\
\frac{d\mu}{2\pi i}.$ (2.38)

###### Proof.

See Theorem 1.2 of [Kw23]. ∎

##  3\. Twisted Key Identity

In this section, we illustrate how the arithmetic (or non-archimedean) twists
allude to Section 1 enter the picture of our period integral approach. We must
include a couple of technical adjustments to our argument in [Kw23] in order
to incorporate the desired new features.

######  Proposition 3.1.

Let $a\in\mathbb{Z}-\\{0\\}$ and $\Phi$ be a smooth automorphic function of
$\Gamma_{3}$. Then for any $g\in GL_{3}(\mathbb{R})$, we have

$\displaystyle\hskip 10.0pt\int_{0}^{1}\Phi\left[\begin{pmatrix}1&u_{1,2}&\\\
&1&\\\ &&1\end{pmatrix}g\right]\,$ $\displaystyle e(-au_{1,2})\ du_{1,2}$
$\displaystyle=\ \ \sum_{d\mid a}\
\sum_{\begin{subarray}{c}a_{0}\in\mathbb{Z}\\\ \gcd(a_{0},d)=1\end{subarray}}\
\sum_{a_{1}=-\infty}^{\infty}\
\left(\widehat{\Phi}\right)_{(a_{1},\,a/d)}\left[\begin{pmatrix}1&&\\\
&\alpha&\beta\\\ &-a_{0}&d\end{pmatrix}g\right],$ (3.1)

where $(\alpha,\beta)$ is any pair of integers satisfying

$\displaystyle d\alpha+a_{0}\beta\ =\ 1.$ (3.2)

###### Proof.

Firstly, we perform a Fourier expansion along the abelian unipotent subgroup
$\left\\{\begin{pmatrix}1&&*\\\ &1&\\\ &&1\end{pmatrix}\right\\}$, which leads
to

$\displaystyle\hskip 20.0pt\int_{0}^{1}\
\Phi\left[\begin{pmatrix}1&u_{1,2}&\\\ &1&\\\ &&1\end{pmatrix}g\right]\,$
$\displaystyle e(-au_{1,2})\ du_{1,2}$ $\displaystyle\ =\ $
$\displaystyle\sum_{a_{0}=-\infty}^{\infty}\ \int_{0}^{1}\ \int_{0}^{1}\
\Phi\left[\begin{pmatrix}1&u_{1,2}&u_{1,3}\\\ &1&\\\
&&1\end{pmatrix}g\right]e(-au_{1,2}-a_{0}u_{1,3})\ du_{1,2}\ du_{1,3}.$ (3.3)

Secondly, we consider a change of variables of the form $(u_{1,2},u_{1,3})\ =\
(u_{1,2}^{\prime},u_{1,3}^{\prime})\cdot\begin{pmatrix}\alpha&\beta\\\
\gamma&\delta\end{pmatrix}$ for each $a_{0}\in\mathbb{Z}$, where the matrix
$\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix}$ satisfies

  * •

$\gamma\ =\ \gamma(a_{0})\ :=\ -a_{0}/\gcd\left(a_{0},a\right)$ and $\delta\
=\ \delta(a_{0})\ :=\ a/\gcd\left(a_{0},a\right)$;

  * •

$\alpha=\alpha(a_{0}),\ \beta=\beta(a_{0})$ be any pair of integers that
satisfy $\delta\alpha-\gamma\beta\ =\ 1$.

Then $au_{1,2}+a_{0}u_{1,3}=a^{\prime}u_{1,2}^{\prime}$ with
$a^{\prime}=a^{\prime}(a_{0}):=a\alpha+a_{0}\beta$, and one can easily verify
that

$\displaystyle\hskip 20.0pt\begin{pmatrix}1&u_{1,2}&u_{1,3}\\\ &1&\\\
&&1\end{pmatrix}\ =\ \begin{pmatrix}1&&\\\ &\delta&-\beta\\\
&-\gamma&\alpha\end{pmatrix}\begin{pmatrix}1&u_{1,2}^{\prime}&u_{1,3}^{\prime}\\\
&1&\\\ &&1\end{pmatrix}\begin{pmatrix}1&&\\\ &\alpha&\beta\\\
&\gamma&\delta\end{pmatrix}.$

The automorphy of $\Phi$ with respect to $\Gamma_{3}$ gives

$\displaystyle\hskip 20.0pt\int_{0}^{1}\
\Phi\left[\begin{pmatrix}1&u_{1,2}&\\\ &1&\\\ &&1\end{pmatrix}g\right]\,$
$\displaystyle e(-au_{1,2})\ du_{1,2}$ $\displaystyle\ =\
\sum_{a_{0}=-\infty}^{\infty}\ \int_{0}^{1}\ \int_{0}^{1}\
\Phi\left[\begin{pmatrix}1&u_{1,2}^{\prime}&u_{1,3}^{\prime}\\\ &1&\\\
&&1\end{pmatrix}\begin{pmatrix}1&&\\\ &\alpha&\beta\\\
&\gamma&\delta\end{pmatrix}g\right]e(-a^{\prime}u_{1,2}^{\prime})\
du_{1,2}^{\prime}\ du_{1,3}^{\prime}.$ (3.4)

Thirdly, we perform a Fourier expansion along another abelian unipotent
subgroup $\left\\{\begin{pmatrix}1&&\\\ &1&*\\\ &&1\end{pmatrix}\right\\}$.
From this, we obtain

$\displaystyle\hskip 25.0pt\int_{0}^{1}\
\Phi\left[\begin{pmatrix}1&u_{1,2}&\\\ &1&\\\
&&1\end{pmatrix}g\right]\,e(-au_{1,2})\ du_{1,2}\ \ =\ \
\sum_{a_{0},a_{1}=-\infty}^{\infty}\
\left(\widehat{\Phi}\right)_{(a_{1},a^{\prime})}\left[\begin{pmatrix}1&&\\\
&\alpha&\beta\\\ &\gamma&\delta\end{pmatrix}g\right].$ (3.5)

Upon breaking up the $a_{0}$-sum according to the value of $d:=(a_{0},a)$ and
noticing that $a^{\prime}=d$, we have

$\displaystyle\hskip 10.0pt\int_{0}^{1}\
\Phi\left[\begin{pmatrix}1&u_{1,2}&\\\ &1&\\\
&&1\end{pmatrix}g\right]\,e(-au_{1,2})\ du_{1,2}\ =\ \sum_{d\mid a}\
\sum_{\begin{subarray}{c}a_{0}\in\mathbb{Z}\\\ (a_{0},a)=d\end{subarray}}\
\sum_{a_{1}\in\mathbb{Z}}\
\left(\widehat{\Phi}\right)_{(a_{1},d)}\left[\begin{pmatrix}1&&\\\
&\alpha&\beta\\\ &-a_{0}/d&a/d\end{pmatrix}g\right],$ (3.6)

where $(\alpha,\beta)$ is any pair of integers satisfying
$a\alpha+a_{0}\beta=d$. The conclusion (3.1) now follows from the successive
replacements $a_{0}\to da_{0}$ and $d\to a/d$.

∎

The next step concerns the explication of the pair $(\alpha,\beta)$ in (3.1)
in which condition (3.2) will naturally come into play. To some extent, this
may be compared to the step of applying ‘additive reciprocity’ in [BHKM20],
where its usage (and actually the full compositum of the transformations) was
guided by the lengths of summations instead, see the sketch 111 Justifying (or
motivating) the insertion of various auxiliary additive factors in the main
argument of [BHKM20] in which the method of continuation was used.  in Section
1.4 therein.

######  Corollary 3.2.

Suppose $\Phi:\mathfrak{h}^{3}\rightarrow\mathbb{C}$ is an automorphic form of
$\Gamma_{3}$. Then for any $y_{0},y_{1}>0$, we have

$\displaystyle\hskip 10.0pt\int_{0}^{1}\
\Phi\left[\begin{pmatrix}1&u_{1,2}&\\\ &1&\\\
&&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]$ $\displaystyle\ e(-au_{1,2})\ du_{1,2}\ $
$\displaystyle=\ \ \sum_{d\mid a}\ \ \sum_{a_{1}\in\mathbb{Z}}\
\left(\widehat{\Phi}\right)_{(a_{1},\,a/d)}\begin{pmatrix}y_{0}y_{1}&&\\\
&y_{0}&\\\ &&1\end{pmatrix}$ $\displaystyle\ +\ \sum_{d\mid a}\
\sum_{\begin{subarray}{c}a_{0}\in\mathbb{Z}-\\{0\\}\\\
\gcd(a_{0},d)=1\end{subarray}}\
\left(\widehat{\Phi}\right)_{(0,\,a/d)}\begin{pmatrix}\frac{y_{0}y_{1}}{\sqrt{(a_{0}y_{0})^{2}+d^{2}}}&&\\\
&\frac{y_{0}}{(a_{0}y_{0})^{2}+d^{2}}&\\\ &&1\\\ \end{pmatrix}$
$\displaystyle\ +\ \sum_{d\mid a}\
\sum_{\begin{subarray}{c}a_{0}\in\mathbb{Z}-\\{0\\}\\\
\gcd(a_{0},d)=1\end{subarray}}\ \sum_{a_{1}\in\mathbb{Z}-\\{0\\}}\
e\left(\frac{a_{1}\overline{a_{0}}}{d}\right)\ $
$\displaystyle\cdot\left(\widehat{\Phi}\right)_{(a_{1},\,a/d)}\left[\begin{pmatrix}1&&\\\
&1&-\frac{1}{da_{0}}\frac{(a_{0}y_{0})^{2}}{d^{2}+(a_{0}y_{0})^{2}}\\\
&&1\end{pmatrix}\begin{pmatrix}\frac{y_{0}y_{1}}{\sqrt{(a_{0}y_{0})^{2}+d^{2}}}&&\\\
&\frac{y_{0}}{(a_{0}y_{0})^{2}+d^{2}}&\\\ &&1\\\ \end{pmatrix}\right],$

where $a_{0}\overline{a_{0}}\equiv 1\ (\bmod\,d)$ for $\gcd(a_{0},d)=1$.

######  Remark 3.3.

  1. (1)

Corollary 3.2 can be understood as follows: the exponetial factor
$e\left(a_{1}\overline{a_{0}}/d\right)$ in (3.2) is our desired ‘arithmetic
twist’, whereas the last line of (3.2) will be regarded as the ‘analytic
component’.

  2. (2)

The mentioned ‘analytic component’ is more-or-less the same as the one for the
untwisted case of [Kw23, Kw23a+]. This is anticipated and is a reflection of
the local-global nature of our method.

  3. (3)

The expression (3.2) will be simplified considerably in (4.1) upon plugging
into the period integral
$\left(P^{a},\,\mathbb{P}_{2}^{3}\Phi\cdot|\det*|^{\overline{s}-\frac{1}{2}}\right)_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}$.

  4. (4)

Corollary 3.2 was stated in a slightly more general form as it has further
applications in verifying Conjecture 5.2, i.e., the twisted version of the
main theorem of [Kw23a+], which requires us to take
$\Phi=(E_{\min})^{{}^{*}}(\,*\,;\,(\alpha_{1},\alpha_{2},\alpha_{3}))$. When
$\Phi$ is cuspidal, we have $(\widehat{\Phi})_{(0,\,a/d)}\equiv 0$, so the
second summand on the right side of (3.2) vanishes.

###### Proof of Corollary 3.2.

When $a_{0}=0$, we have $d=\alpha=1$ and we may choose $\beta=0$ in
Proposition 3.1. This results in the first term on the right side of (3.2).

Suppose now $a_{0}\neq 0$ (with $\gcd(a_{0},d)=1$). Let
$w_{\ell}:=\begin{pmatrix}&&1\\\ &1&\\\ 1&&\end{pmatrix}$ and $g^{\iota}\ :=\
^{t}g^{-1}$. To reveal the non-archimedean component behind our calculations,
we must deviate from [Kw23] by first considering the product of matrices:

$\displaystyle\hskip 10.0ptw_{\ell}\ \left[\begin{pmatrix}1&&\\\
&\alpha&\beta\\\ &-a_{0}&d\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\
&y_{0}&\\\ &&1\end{pmatrix}\right]^{\iota}w_{\ell},$

which can be seen to be

$\displaystyle\hskip 70.0pt\begin{pmatrix}y_{1}\begin{pmatrix}\alpha&-\beta\\\
a_{0}&d\end{pmatrix}\begin{pmatrix}y_{0}&\\\ &1\end{pmatrix}&\\\
&1\end{pmatrix}\hskip 15.0pt(\bmod\ \mathbb{R}^{\times}).$

Thus, we are left with a simple $2$-by-$2$ Iwasawa computation:

$\displaystyle\hskip 10.0pt\begin{pmatrix}\alpha&-\beta\\\
a_{0}&d\end{pmatrix}\begin{pmatrix}y_{0}&\\\ &1\end{pmatrix}\ \equiv\
\sqrt{d^{2}+(a_{0}y_{0})^{2}}\
\begin{pmatrix}1&\frac{\alpha}{a_{0}}-\frac{d/a_{0}}{d^{2}+(a_{0}y_{0})^{2}}\\\
&1\end{pmatrix}\begin{pmatrix}\frac{y_{0}}{d^{2}+(a_{0}y_{0})^{2}}&\\\
&1\end{pmatrix}\hskip 10.0pt(\bmod\,SO(2)).$ (3.8)

From (3.8), the Iwasawa decomposition of $\begin{pmatrix}1&&\\\
&\alpha&\beta\\\ &-a_{0}&d\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\
&y_{0}&\\\ &&1\end{pmatrix}\ (\bmod\,O_{3}(\mathbb{R})\,\mathbb{R}^{\times})$
can be readily obtained and it follows that

$\displaystyle\left(\widehat{\Phi}\right)_{(a_{1},\,a/d)}\left[\begin{pmatrix}1&&\\\
&\alpha&\beta\\\ &-a_{0}&d\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\
&y_{0}&\\\ &&1\end{pmatrix}\right]$ $\displaystyle\hskip 70.0pt\ \ =\ \
\left(\widehat{\Phi}\right)_{(a_{1},\,a/d)}\left[\begin{pmatrix}1&&\\\
&1&\frac{d/a_{0}}{d^{2}+(a_{0}y_{0})^{2}}-\frac{\alpha}{a_{0}}\\\
&&1\end{pmatrix}\begin{pmatrix}\frac{y_{0}y_{1}}{\sqrt{(a_{0}y_{0})^{2}+d^{2}}}&&\\\
&\frac{y_{0}}{(a_{0}y_{0})^{2}+d^{2}}&\\\ &&1\\\ \end{pmatrix}\right].$

When $a_{1}=0$, we have

$\displaystyle\hskip
5.0pt\left(\widehat{\Phi}\right)_{(a_{1},\,a/d)}\left[\begin{pmatrix}1&&\\\
&\alpha&\beta\\\ &-a_{0}&d\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\
&y_{0}&\\\ &&1\end{pmatrix}\right]\ \ =\ \ $
$\displaystyle\left(\widehat{\Phi}\right)_{(0,\,a/d)}\begin{pmatrix}\frac{y_{0}y_{1}}{\sqrt{(a_{0}y_{0})^{2}+d^{2}}}&&\\\
&\frac{y_{0}}{(a_{0}y_{0})^{2}+d^{2}}&\\\ &&1\\\ \end{pmatrix}$

from (3) and (2.16). This results in the second term on the right side of
(3.2). When $a_{1}\in\mathbb{Z}-\\{0\\}$, observe from (3.2) that

$\displaystyle\frac{d/a_{0}}{d^{2}+(a_{0}y_{0})^{2}}\ -\ \frac{\alpha}{a_{0}}\
\equiv\ \frac{\overline{a_{0}}}{d}\ +\
\frac{d/a_{0}}{d^{2}+(a_{0}y_{0})^{2}}-\frac{1}{a_{0}d}\ =\
\frac{\overline{a_{0}}}{d}\ -\
\frac{1}{da_{0}}\,\frac{(a_{0}y_{0})^{2}}{d^{2}+(a_{0}y_{0})^{2}}\ \
(\bmod\,1),$

and (3) can be rewritten as

$\displaystyle\left(\widehat{\Phi}\right)_{(a_{1},\,a/d)}\left[\begin{pmatrix}1&&\\\
&1&\overline{a_{0}}/d\\\ &&1\end{pmatrix}\,\cdot\,\begin{pmatrix}1&&\\\
&1&-\frac{1}{da_{0}}\,\frac{(a_{0}y_{0})^{2}}{d^{2}+(a_{0}y_{0})^{2}}\\\
&&1\end{pmatrix}\begin{pmatrix}\frac{y_{0}y_{1}}{\sqrt{(a_{0}y_{0})^{2}+d^{2}}}&&\\\
&\frac{y_{0}}{(a_{0}y_{0})^{2}+d^{2}}&\\\ &&1\\\ \end{pmatrix}\right].$

We thus obatin the third term on the right side of (3.2) by invoking (2.16).
This concludes the proof of Corollary 3.2.

∎

##  4\. Proof of Theorem 1.1

###  4.1. Unfolding and Clean-up

Let $\epsilon:=1/100$. On the vertical strip $1+\theta/2+\epsilon<\sigma<4$,
we begin by replacing the Poincaré series $P^{a}$ by its definition in the
pairing

$\displaystyle 2a^{-1/2}\,\left(P^{a},\
\mathbb{P}_{2}^{3}\Phi\cdot|\det*|^{\overline{s}-\frac{1}{2}}\right)_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}.$ (4.1)

An unfolding shows that (4.1) is equal to

$\displaystyle 2a^{-1/2}\,\int_{0}^{\infty}\int_{0}^{\infty}\
h(y_{1})\,(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\,\int_{0}^{1}\hskip
5.0pt\widetilde{\Phi}\left[\begin{pmatrix}1&u_{1,2}&\\\ &1&\\\
&&1\end{pmatrix}\begin{pmatrix}y_{0}y_{1}&&\\\ &y_{0}&\\\
&&1\end{pmatrix}\right]e(au_{1,2})\ du_{1,2}\
\frac{dy_{0}\,dy_{1}}{y_{0}y_{1}^{2}}.$ (4.2)

From Corollary 3.2 (replacing $\Phi$ therein by $\widetilde{\Phi}$ and notice
that $(\widehat{\Phi})_{(0,a/d)}\equiv 0$ as $\Phi$ is a cusp form here), we
obtain

$\displaystyle 2a^{-1/2}\,$ $\displaystyle\sum_{d\mid a}\ \
\sum_{a_{1}\in\mathbb{Z}-\\{0\\}}\
\int_{0}^{\infty}\int_{0}^{\infty}h(ay_{1})\left(\widehat{\widetilde{\Phi}}\right)_{(a_{1},a/d)}\begin{pmatrix}y_{0}y_{1}&&\\\
&y_{0}&\\\ &&1\end{pmatrix}(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\
\frac{dy_{0}\,dy_{1}}{y_{0}y_{1}^{2}}$ $\displaystyle\ +\
2a^{-1/2}\,\sum_{d\mid a}\
\sum_{\begin{subarray}{c}a_{0}\in\mathbb{Z}-\\{0\\}\\\
\gcd(a_{0},d)=1\end{subarray}}\ \sum_{a_{1}\in\mathbb{Z}-\\{0\\}}\ \
\frac{\mathcal{B}_{\Phi}\left(a/d,a_{1}\right)}{|a_{0}|^{2s-1}|a_{1}|}\
e\left(-\frac{a_{1}\overline{a_{0}}}{d}\right)$ $\displaystyle\hskip
50.0pt\cdot\,\int_{0}^{\infty}\int_{0}^{\infty}h(ay_{1})\,W_{-\alpha(\Phi)}\left(\frac{|a_{1}|y_{0}}{(a_{0}y_{0})^{2}+d^{2}},\
\frac{ay_{1}}{d}\sqrt{(a_{0}y_{0})^{2}+d^{2}}\right)$ $\displaystyle\hskip
200.0pt\cdot\,e\left(\frac{1}{da_{0}}\frac{(a_{0}y_{0})^{2}}{d^{2}+(a_{0}y_{0})^{2}}\right)(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\
\ \frac{dy_{0}\,dy_{1}}{y_{0}y_{1}^{2}}$ (4.3)

and the absolute convergence of (4.1) follows from Proposition 6.5 of [Kw23].

Denote by $D_{\Phi}^{(a)}(s)$ the expression on the first line of (4.1) and
$OD_{\Phi}^{(a)}(s)$ the expression spanning the last three lines of (4.1). In
other words,

$\displaystyle 2a^{-1/2}\,\left(P^{a},\
\mathbb{P}_{2}^{3}\Phi\cdot|\det*|^{\overline{s}-\frac{1}{2}}\right)_{\Gamma_{2}\setminus
GL_{2}(\mathbb{R})}\ \ =\ \ D_{\Phi}^{(a)}(s)\ +\ OD_{\Phi}^{(a)}(s).$ (4.4)

By the changes of variables $y_{0}\to|a_{1}|^{-1}y_{0}$ and $y_{1}\to
a^{-1}y_{1}$, we have

$\displaystyle D_{\Phi}^{(a)}(s)\ =\
2a^{-s}\sum_{a_{1}\in\mathbb{Z}-\\{0\\}}\frac{\mathcal{B}_{\Phi}(a,a_{1})}{|a_{1}|^{2s}}\cdot\int_{0}^{\infty}\int_{0}^{\infty}h(y_{1})\,(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\,W_{-\alpha(\Phi)}\left(y_{0},y_{1}\right)\,\frac{dy_{0}\,dy_{1}}{y_{0}y_{1}^{2}},$
(4.5)

where the double integral can be computed by (2.5) and (2.7). For
$OD_{\Phi}^{(a)}(s)$, we apply another set of changes of variables
$y_{0}\to(d|a_{0}|^{-1})y_{0}$ and $y_{1}\to a^{-1}y_{1}$, it follows at once

$\displaystyle OD_{\Phi}^{(a)}(s)\ \ =\ \ $ $\displaystyle 2a^{-s}\
\sum_{d\mid a}\ d^{2s}\sum_{\begin{subarray}{c}a_{0}\in\mathbb{Z}-\\{0\\}\\\
\gcd(a_{0},d)=1\end{subarray}}\ \sum_{a_{1}\in\mathbb{Z}-\\{0\\}}\
\frac{\mathcal{B}_{\Phi}\left(a/d,a_{1}\right)}{|a_{0}|^{2s-1}|a_{1}|}\,e\left(-\frac{a_{1}\overline{a_{0}\raisebox{8.53581pt}{}}}{d}\right)$
$\displaystyle\hskip 90.0pt\cdot\
\int_{0}^{\infty}\int_{0}^{\infty}h(y_{1})\,(y_{0}^{2}y_{1})^{s-\frac{1}{2}}\,e\left(\frac{a_{1}}{da_{0}}\,\frac{y_{0}^{2}}{1+y_{0}^{2}}\right)\
$ $\displaystyle\hskip 165.0pt\cdot
W_{-\alpha(\Phi)}\left(\left|\frac{a_{1}}{da_{0}}\right|\,\frac{y_{0}}{1+y_{0}^{2}},\
y_{1}\sqrt{1+y_{0}^{2}}\right)\ \frac{dy_{0}\,dy_{1}}{y_{0}y_{1}^{2}}.$

Following the same argument of Proposition 7.2 in [Kw23], i.e., applying the
Mellin inversion formulae for $e(\cdots)$ and $W_{-\alpha(\Phi)}(\cdots)$, we
obtain

$\displaystyle OD_{\Phi}^{(a)}(s;\,\phi)\ \ =\ \
2a^{-s}\cdot\frac{1}{8}\cdot\,\int_{(1+\theta+2\epsilon)}\ \sum_{\pm}\ $
$\displaystyle\sum_{d\mid a}\ d^{2s+s_{0}-1}\
\sum_{\begin{subarray}{c}a_{0},a_{1}\in\mathbb{Z}-\\{0\\}\\\
\gcd(a_{0},d)=1\\\ \operatorname{sgn}(a_{0}a_{1})=\pm\end{subarray}}\ \
\frac{\mathcal{B}_{\Phi}\left(a/d,a_{1}\right)}{|a_{0}|^{2s-s_{0}}|a_{1}|^{s_{0}}}\,e\left(-\frac{a_{1}\overline{a_{0}\raisebox{8.53581pt}{}}}{d}\right)$
$\displaystyle\hskip
30.0pt\cdot\,(\mathcal{F}_{\Phi}^{(\pm)}H)\left(s_{0},\,s;\,\phi\right)\
\frac{ds_{0}}{2\pi i},$ (4.7)

where $\mathcal{F}_{\Phi}^{(\pm)}H$ was defined in Section 2.7, and

$\displaystyle\lim_{\phi\to\pi/2}\ OD_{\Phi}^{(a)}(s;\,\phi)\ \ =\ \
OD_{\Phi}^{(a)}(s)$ (4.8)

holds on the domain $(3+\theta)/2<\sigma<4$ (see Proposition 7.3 of [Kw23]).
Note: the constant $1/8$ in (4.1) comes from the constants $1/4$ and $1/2$
from Vinogradov-Takhtadzhyan’s formula and Euler’s beta integral formula
respectively.

However, one must beware of the signs of $a_{0}a_{1}$ due to the extra factor
$e(-a_{1}\overline{a_{0}}/d)$ present in this article (but not in [Kw23,
Kw23a+]), which will in turn influence the shapes of the dual moments, see
(4.7) and (5.1). Consider the double Dirichlet series

$\displaystyle\mathcal{L}^{(a)}_{\pm}\left(s_{0},s;\Phi\right)\ \ $
$\displaystyle:=\ \ \sum_{d\mid a}\
d^{2s+s_{0}-1}\sum_{\begin{subarray}{c}a_{0}\geq 1\\\
\gcd(a_{0},d)=1\end{subarray}}\ \sum_{a_{1}\geq 1}\
\frac{\mathcal{B}_{\Phi}\left(a/d,a_{1}\right)}{(a_{0})^{2s-s_{0}}(a_{1})^{s_{0}}}\
e\left(\mp\frac{a_{1}\overline{a_{0}\raisebox{8.53581pt}{}}}{d}\right),$ (4.9)

which converge absolutely on the region

$\displaystyle\operatorname{Re}(2s-s_{0})\ >\ 1\hskip 20.0pt\text{ and }\hskip
20.0pt\operatorname{Re}s_{0}\ >\ 1+\theta.$ (4.10)

Using the fact that $\Phi$ is an even form, we readily observe that

$\displaystyle OD_{\Phi}^{(a)}(s;\,\phi)\ \ =\ \ $
$\displaystyle\frac{a^{-s}}{2}\ \sum_{\pm}\ \int_{(1+\theta+2\epsilon)}\
\mathcal{L}^{(a)}_{\pm}\left(s_{0},s;\Phi\right)(\mathcal{F}_{\Phi}^{(\pm)}H)\left(s_{0},\,s;\,\phi\right)\
\frac{ds_{0}}{2\pi i}.$ (4.11)

for $1+\theta/2+\epsilon<\sigma<4$ and $\phi\in(0,\pi/2)$.

###  4.2. Analytic Continuation

We will make use of Proposition 2.13 to obtain the analytic properties of
$\mathcal{L}_{\pm}^{(a)}\left(s_{0},s;\Phi\right)$.

######  Proposition 4.1.

The double Dirichlet series $\mathcal{L}_{\pm}^{(a)}(s_{0},s;\Phi)$ admits a
holomorphic continuation to $\mathbb{C}^{2}$ except on the polar divisor
$2s-s_{0}=1$.

###### Proof.

Suppose $\operatorname{Re}(2s-s_{0})>1$ and $\operatorname{Re}s_{0}>1+\theta$.
The double Dirichlet series (4.9) converge absolutely and can be written as

$\displaystyle\mathcal{L}_{\pm}^{(a)}\left(s_{0},s;\Phi\right)\ $
$\displaystyle=\ \ \sum_{d\mid a}\ d^{2s+s_{0}-1}\
\sideset{}{{}^{*}}{\sum}_{\ell\ (\bmod\,d)}\
\sum_{\begin{subarray}{c}a_{0}\geq 1\\\ a_{0}\ \equiv\ \ell\
(\bmod\,d)\end{subarray}}\ \sum_{a_{1}\geq 1}\
\frac{\mathcal{B}_{\Phi}\left(a/d,a_{1}\right)}{(a_{0})^{2s-s_{0}}(a_{1})^{s_{0}}}\
e\left(\mp\frac{a_{1}\overline{\ell\raisebox{8.53581pt}{}}}{d}\right)$ (4.12)

upon splitting the $a_{0}$-sum into residue classes $(\bmod\,d)$. Since

$\displaystyle\sum_{\begin{subarray}{c}a_{0}\geq 1\\\ a_{0}\ \equiv\ \ell\
(\bmod\,d)\end{subarray}}\frac{1}{a_{0}^{2s-s_{0}}}\ $ $\displaystyle=\
d^{-(2s-s_{0})}\cdot\zeta\left(2s-s_{0},\frac{\ell}{d}\right),$ (4.13)

we have

$\displaystyle\mathcal{L}_{\pm}^{(a)}\left(s_{0},s;\Phi\right)\ \ =\ \
\sum_{d\mid a}\ d^{2s_{0}-1}\ \sideset{}{{}^{*}}{\sum}_{\ell\ (\bmod\,d)}\
\zeta\left(2s-s_{0},\frac{\ell}{d}\right)\left(\sum_{a_{1}=1}^{\infty}\
\frac{\mathcal{B}_{\Phi}\left(a/d,a_{1}\right)}{a_{1}^{s_{0}}}\
e\left(\mp\frac{a_{1}\overline{\ell\raisebox{8.53581pt}{}}}{d}\right)\right).$
(4.14)

The Hecke relation (2.17) implies that

$\displaystyle\mathcal{L}^{(a)}_{\pm}\left(s_{0},s;\Phi\right)\ \ =\ \
\sum_{dr\mid a}\ \frac{d^{2s_{0}-1}\mu(r)}{r^{s_{0}}}\
\overline{\lambda_{\Phi}}\left(\frac{a}{dr}\right)\
\sideset{}{{}^{*}}{\sum}_{\ell\ (\bmod\,d)}\
\zeta\left(2s-s_{0},\frac{\ell}{d}\right)L\left(s_{0};\mp\frac{r\overline{\ell}}{d};\Phi\right)$
(4.15)

and its analytic continuation now follows from Proposition 2.13. Also,

$\displaystyle-\ \operatorname{Res}_{s_{0}=2s-1}\
\mathcal{L}_{\pm}^{(a)}(s_{0},s;\Phi)\ \ =\ \ \sum_{dr\mid a}\
\frac{d^{4s-3}\mu(r)}{r^{2s-1}}\
\overline{\lambda_{\Phi}}\left(\frac{a}{dr}\right)\ \
\sideset{}{{}^{*}}{\sum}_{\ell\ (\bmod\,d)}\
L\left(2s-1;\mp\frac{r\overline{\ell}}{d};\Phi\right)$ (4.16)

which clearly admits an entire continuation. This finishes the proof. ∎

Denote by $\mathcal{R}_{\pm}^{(a)}(s;\Phi)$ the expression on the right side
of (4.16). By Corollary 4.1 and repeating the argument described in Section 9
of [Kw23] (with the exact same analysis at the archimedean place, see Section
2.7), we arrive at

######  Proposition 4.2.

On $1/4+\epsilon/2<\sigma<3/4$, we have

$\displaystyle\hskip 30.0ptOD_{\Phi}^{(a)}(s)\ \ =\ \ $
$\displaystyle\frac{a^{-s}}{2}\ \sum_{\pm}\ \int_{(1/2)}\
\mathcal{L}^{(a)}_{\pm}\left(s_{0},s;\Phi\right)(\mathcal{F}_{\Phi}^{(\pm)}H)\left(s_{0},\,s\right)\
\frac{ds_{0}}{2\pi i}$ $\displaystyle\hskip 120.0pt\ +\ \frac{a^{-s}}{2}\
\sum_{\pm}\ \
\mathcal{R}_{\pm}^{(a)}(s;\Phi)(\mathcal{F}_{\Phi}^{(\pm)}H)\left(2s-1,\,s\right).$
(4.17)

###  4.3. The Diagonal Main Term

######  Lemma 4.3.

For $\operatorname{Re}s\gg 1$, we have

$\displaystyle\frac{1}{2}\ \sum_{a_{1}\in\mathbb{Z}-\\{0\\}}\
\frac{\mathcal{B}_{\Phi}(a,a_{1})}{|a_{1}|^{2s}}\ \ $ $\displaystyle=\
L(2s,\Phi)\ \sum_{r\mid a}\
\frac{\mu(r)\overline{\lambda_{\Phi}}(a/r)}{r^{2s}}$ $\displaystyle=\ \
L(2s,\Phi)\ \prod_{p\mid a}\
\left\\{\overline{\lambda_{\Phi}}\left(p^{o_{p}(a)}\right)\ -\
\frac{\overline{\lambda_{\Phi}}\left(p^{o_{p}(a)-1}\right)}{p^{2s}}\right\\},$
(4.18)

where $o_{p}(a)$ denotes the power of $p$ in the prime factorization of $a$.
Thus, the Dirichlet series above admits an entire continuation.

###### Proof.

The first equality follows from (2.17) and the second equality follows from
the multiplicativity of $\lambda_{\Phi}(\,\cdot\,)$, $\mu(\,\cdot\,)$, and
their multiplicative convolution. ∎

######  Proposition 4.4.

On the domain $1+\theta/2+\epsilon<\sigma<4$, we have

$\displaystyle D_{\Phi}^{(a)}(s)\ =\ \frac{a^{-s}}{2}\ L(2s,\Phi)\
\prod_{p\mid a}\ $
$\displaystyle\left\\{\overline{\lambda_{\Phi}}\left(p^{o_{p}(a)}\right)\ -\
\frac{\overline{\lambda_{\Phi}}\left(p^{o_{p}(a)-1}\right)}{p^{2s}}\right\\}$
$\displaystyle\hskip 40.0pt\ \cdot\
\int_{(0)}\frac{H(\mu)}{|\Gamma(\mu)|^{2}}\cdot\prod_{\pm}\ \prod_{i=1}^{3}\
\Gamma_{\mathbb{R}}\left(s\pm\mu-\alpha_{i}\right)\ \frac{d\mu}{2\pi i}.$
(4.19)

As a result, $D_{\Phi}^{(a)}(s)$ admits a holomorphic continuation to
$0<\sigma<4$.

###### Proof.

Follows from (4.5), (2.7), (2.5) and Lemma 4.3. ∎

###  4.4. The Off-diagonal Main Term

If $\Phi$ is a Maass cusp form of $SL_{3}(\mathbb{Z})$, the twisted recipe of
[CFKRS05] for the spectral moment of (1.1) is implied by the functional
equation

$\displaystyle\Lambda(s,\,\phi\otimes\widetilde{\Phi})\ =\
\Lambda(1-s,\,\phi\otimes\Phi),$ (4.20)

where $\phi$ is any even Maass form of $SL_{2}(\mathbb{Z})$. Indeed, the
spectral side of (1.1) is invariant under the replacements $s\to 1-s$ and
$\Phi\rightarrow\widetilde{\Phi}$. This observation together with the diagonal
evaluation (4.4) predict the existence of an off-diagonal main term:

$\displaystyle\hskip
15.0pt\frac{a^{s-1}}{2}\,L\left(2\,(1-s),\widetilde{\Phi}\right)$
$\displaystyle\,\prod_{p\mid a}\
\left\\{\lambda_{\Phi}\left(p^{o_{p}(a)}\right)\ -\
\frac{\lambda_{\Phi}\left(p^{o_{p}(a)-1}\right)}{p^{2(1-s)}}\right\\}$
$\displaystyle\hskip 60.0pt\cdot\,\int_{(0)}\
\frac{H(\mu)}{\left|\Gamma(\mu)\right|^{2}}\cdot\prod_{\pm}\ \prod_{i=1}^{3}\
\Gamma_{\mathbb{R}}\left(1-s\pm\mu+\alpha_{i}\right)\ \frac{d\mu}{2\pi i}.$
(4.21)

In other words, we must show that

######  Proposition 4.5.

For $1/4+\epsilon/2<\sigma<3/4$, the expression

$\displaystyle\frac{a^{-s}}{2}\ \sum_{\pm}\
\mathcal{R}_{\pm}^{(a)}(s;\Phi)\,(\mathcal{F}_{\Phi}^{(\pm)}H)\left(2s-1,s\right)$
(4.22)

is equal to (4.4).

The archimedean functional relation for $\sum_{\pm}\
(\mathcal{F}_{\Phi}^{(\pm)}H)\left(2s-1,s\right)$ has been established in
[Kw23] before, see Proposition 2.17. Hence, the conclusion for Proposition 4.5
would follow from the functional equation of $GL(3)$ (Proposition 2.11), as
well as:

######  Proposition 4.6.

For any $s\in\mathbb{C}$, we have

$\displaystyle\mathcal{R}_{\pm}^{(a)}(s;\Phi)\ \ =\ \ a^{2s-1}\cdot
L(2s-1,\Phi)\cdot\prod_{p\mid a}\ \left\\{\lambda_{\Phi}(p^{o_{p}(a)})\ -\
\frac{\lambda_{\Phi}(p^{o_{p}(a)-1})}{p^{2(1-s)}}\right\\}.$ (4.23)

This is the non-archimedean analogue of Proposition 2.17. In principle, it
should be a finite checking at the ‘ramified’ places upon expanding
$\mathcal{R}_{\pm}^{(a)}(s;\Phi)$ into an Euler product (as in [HY10, CIS19,
BTB22+, HN22]). Unfortunately, our situation is even more complicated than the
aforementioned works — one has to match the right side of (4.6) with the
expression

$\displaystyle\mathcal{R}_{\pm}^{(a)}(s;\Phi)\ $ $\displaystyle=\
\sum_{der\mid a}\ \
\frac{\mu(r)\mu(d)}{d}\left(\frac{d^{2}e}{\widetilde{r}}\right)^{2s-1}\overline{\lambda_{\Phi}}\left(\frac{a}{der}\right)\
\sum_{\begin{subarray}{c}a_{1}\geq 1\end{subarray}}\
\frac{\lambda_{\Phi}(\widetilde{e}a_{1})}{(a_{1})^{2s-1}}$ $\displaystyle=\
L(2s-1,\Phi)\sum_{der\mid a}\ \
\frac{\mu(r)\mu(d)}{d}\left(\frac{d^{2}e}{\widetilde{r}}\right)^{2s-1}\overline{\lambda_{\Phi}}\left(\frac{a}{der}\right)$
$\displaystyle\hskip 30.0pt\cdot\prod_{p\,\mid\,\widetilde{e}}\ \
\left(1-\lambda_{\Phi}(p)p^{-(2s-1)}+\overline{\lambda_{\Phi}}(p)p^{-2(2s-1)}-p^{-3(2s-1)}\right)\sum\limits_{m=0}^{\infty}\
\frac{\lambda_{\Phi}(p^{m+o_{p}(\widetilde{e})})}{(p^{m})^{2s-1}},$ (4.24)

where $\widetilde{e}:=e/(r,e)$ and $\widetilde{r}:=r/(r,e)$. Equation (4.4)
follows from (4.16), the multiplicativity of the $GL(3)$ Hecke eigenvalues,
and the Euler product identity

$\displaystyle L(s,\Phi)\ $ $\displaystyle=\ \prod_{p}\
\left(1-\lambda_{\Phi}(p)p^{-s}+\overline{\lambda_{\Phi}}(p)p^{-2s}-p^{-3s}\right)^{-1}\hskip
20.0pt(\operatorname{Re}s\ \gg\ 1)$ (4.25)

for $GL(3)$ $L$-functions, see pp. 173-174 of [Go15]. When $a$ is prime, one
can indeed show the desired agreement using (4.4). However, it is already
unclear how to proceed even if we assume $a$ is square-free. Nevertheless, we
have found a simple proof that works for any integer $a\geq 1$.

###### Proof of Proposition 4.6.

Suppose $\operatorname{Re}s\gg 1$. By Corollary 4.1, we may evaluate
$\mathcal{R}_{\pm}^{(a)}(s;\Phi)$ by taking the residue of (4.14) at
$s_{0}=2s-1$. Switching the order of summation, we find that the $\ell$-sum in
(4.14) can be identified with the Ramanujan sum $S(0,\mp a_{1};d)$. In other
words,

$\displaystyle\mathcal{R}_{\pm}^{(a)}(s;\Phi)\ \ =\ \ \ \sum_{d\mid a}\
d^{4s-3}\ \sum_{a_{1}\geq 1}\
\frac{\mathcal{B}_{\Phi}\left(a/d,a_{1}\right)}{(a_{1})^{2s-1}}\cdot S(0,\mp
a_{1};d),$ (4.26)

Note: the last expression is independent of the sign and we shall now drop
‘$\pm$’ sign in our notation $\mathcal{R}_{\pm}^{(a)}(s;\Phi)$ from now on.

Suppose $\operatorname{Re}w\gg 1$. We define the triple Dirichlet series:

$\displaystyle\mathcal{C}(w,s;\Phi)\ \ :=\ \ \sum_{a=1}^{\infty}\
\frac{\mathcal{R}^{(a)}(s;\Phi)}{a^{w}}\ \ =\ \ \sum_{a,d,a_{1}=1}^{\infty}\
\frac{d^{4s-3}\mathcal{B}_{\Phi}(a,a_{1})S(0,a_{1};d)}{(da)^{w}a_{1}^{2s-1}},$
(4.27)

where the last expression follows from the change of variables $a\to da$. The
$d$-sum can be readily computed by the well-known identity:

$\displaystyle\sum_{d=1}^{\infty}\ \frac{S(0,a_{1};d)}{d^{w-4s+3}}\ \ =\ \
\frac{\sigma_{-w+4s-2}(a_{1})}{\zeta(w-4s+3)}.$ (4.28)

From (2.13), observe that
$\sigma_{-w+4s-2}(|a_{1}|)=|a_{1}|^{\mu}\cdot\mathcal{B}_{E^{*}(*;\,\mu)}(a_{1})$
with $\mu:=-w/2+2s-1$. As a result, we recognize

$\displaystyle\mathcal{C}(w,s;\Phi)\ \ $ $\displaystyle=\ \
\frac{1}{\zeta(w-4s+3)}\ \ \sum_{a,a_{1}=1}^{\infty}\
\frac{\mathcal{B}_{E^{*}(*;\,\mu)}(a_{1})\mathcal{B}_{\Phi}(a,a_{1})}{(a^{2}a_{1})^{w/2}}$
(4.29)

as a double Dirichlet series of $GL(3)\times GL(2)$ type, see (2.18). Next, by
considering the Euler products (2.27)-(2.28) of $GL(3)$ and $GL(3)\times
GL(2)$ types, we obtain

$\displaystyle\mathcal{C}(w,s;\Phi)\ =\
\frac{L(\frac{w}{2}+\mu,\Phi)L(\frac{w}{2}-\mu,\Phi)}{\zeta(w-4s+3)}\ =\
\frac{L(2s-1,\Phi)L(w+1-2s,\Phi)}{\zeta(w-4s+3)}.$ (4.30)

Then we may rewrite (4.30) as a Dirichlet series in $w$ using the $GL(3)$
$L$-series (2.22) with Möbius inversion:

$\displaystyle\mathcal{C}(w,s;\Phi)\ $ $\displaystyle=\ L(2s-1,\Phi)\
\sum_{r=1}^{\infty}\ \frac{1}{r^{w}}\sum_{mn=r}\
\frac{\lambda_{\Phi}(m)\mu(n)}{m^{1-2s}n^{3-4s}}.$ (4.31)

Upon comparing coefficients, we readily conclude that

$\displaystyle\mathcal{R}^{(a)}(s;\Phi)\ =\
L(2s-1,\Phi)\,\cdot\,a^{2s-1}\,\sum_{n\mid a}\
\frac{\lambda_{\Phi}(a/n)\mu(n)}{n^{2(1-s)}}$ (4.32)

and this holds for any $s\in\mathbb{C}$ by analytic continuation. The
convolution of multiplicative functions in (4.32) can be computed as

$\displaystyle\sum_{n\mid a}\ \frac{\lambda_{\Phi}(a/n)\mu(n)}{n^{2(1-s)}}\ $
$\displaystyle=\ \prod_{p\mid a}\ \left\\{\lambda_{\Phi}(p^{o_{p}(a)})\ -\
\frac{\lambda_{\Phi}(p^{o_{p}(a)-1})}{p^{2(1-s)}}\right\\}.$ (4.33)

This completes the proof of Proposition 4.6. ∎

###  4.5. Concluding Theorem 1.1

Theorem 1.1 follows from putting Proposition 2.15, equation (4.4), Proposition
4.2, Proposition 4.4, Proposition 4.5 and equation (4.15) together.

###  4.6. Shape of the Dual Moment

In this section, we assume that $a=p$ is a prime as in [Y11, BHKM20]. The
connection between the twisted spectral moment and the moment averages over
Dirichlet characters is now a straight-forward consequence of the
orthogonality relation thanks to the structures casted early on by our period
integral construction (see Corollary 3.2).

######  Proposition 4.7.

For any $(s_{0},s)\in\mathbb{C}^{2}$ except on the line $2s-s_{0}=1$, we have

$\displaystyle\mathcal{L}^{(p)}_{\pm}\left(s_{0},s;\Phi\right)\ \ =\ \ $
$\displaystyle\mathcal{P}_{p}(s_{0},s;\Phi)\,\zeta(2s-s_{0})\,L(s_{0},\Phi)$
$\displaystyle\hskip 20.0pt\ +\ \frac{p^{s_{0}+2s-1}}{\phi(p)}\ \left(\ \
\sideset{}{{}^{+}}{\sum}_{\chi\ (\bmod\,p)}\ \ \mp\ \
\sideset{}{{}^{-}}{\sum}_{\chi\ (\bmod\,p)}\right)\
\tau(\chi)L(2s-s_{0},\chi)L\left(s_{0},\Phi\otimes\overline{\chi}\right),$
(4.34)

where
$\tau(\chi):=\sideset{}{{}^{*}}{\sum}\limits_{x\,(\bmod\,p)}\chi(x)e(x/p)$ is
the Gauss sum of $\chi\,(\bmod\,p)$ and

$\displaystyle\mathcal{P}_{p}(s_{0},s;\Phi)\ \ :=\ \ $
$\displaystyle\frac{p^{2s+s_{0}-1}}{\phi(p)}\,(1-p^{-(2s-s_{0})})\left(-1+\lambda_{\Phi}(p)p^{1-s_{0}}-\overline{\lambda_{\Phi}}(p)p^{1-2s_{0}}+p^{1-3s_{0}}\right)$
$\displaystyle\hskip 40.0pt\ +\ \overline{\lambda_{\Phi}}(p)\ -\ p^{-s_{0}}.$
(4.35)

###### Proof.

The computations below will be first carried out on the region of absolute
convergence (4.10). Then our desired conclusion will follow from analytic
continuation. In (4.15) we either have $d=1$ or $d=p$, and correspondingly
$r\mid p$ or $r\mid 1$.

Case (1): $d=1$.  We take $\ell=1$ in this case and then

$\displaystyle\mathcal{L}^{(p)}_{\pm}\left(s_{0},s;\Phi\right)\bigg{|}_{d=1}\
\ =\ \ $ $\displaystyle\ \sum_{r\in\\{1,p\\}}\
\frac{\mu(r)}{r^{s_{0}}}\cdot\overline{\lambda_{\Phi}}(p/r)\cdot\zeta(2s-s_{0})L\left(s_{0},\mp
r;\Phi\right)$ $\displaystyle\ \ =\ \ $
$\displaystyle\left(\,\overline{\lambda_{\Phi}}(p)-p^{-s_{0}}\right)\zeta(2s-s_{0})L\left(s_{0},\Phi\right).$

Case (2): $d=p$. We have

$\displaystyle\mathcal{L}^{(p)}_{\pm}\left(s_{0},s;\Phi\right)\bigg{|}_{d=p}\
=\ p^{2s_{0}-1}\ \ \sideset{}{{}^{*}}{\sum}_{\ell\ (\bmod\,p)}\
\zeta\left(2s-s_{0},\frac{\ell}{p}\right)L\left(s_{0};\mp\frac{\overline{\ell}}{p};\Phi\right).$
(4.36)

The Hurwitz zeta function in (4.36) can be rewritten as

$\displaystyle\zeta\left(2s-s_{0},\frac{\ell}{p}\right)\ \ $ $\displaystyle=\
\ \frac{p^{2s-s_{0}}}{\phi(p)}\ \sum_{\chi\ (\bmod\,p)}\
\overline{\chi}(\ell)L(2s-s_{0},\chi)$

with (4.13) and the othogonality relation for Dirichlet characters. Hence, we
have

$\displaystyle\mathcal{L}^{(p)}_{\pm}\left(s_{0},s;\Phi\right)\bigg{|}_{d=p}\
$ $\displaystyle=\ \frac{p^{2s+s_{0}-1}}{\phi(p)}\ \sum_{\chi\ (\bmod\,p)}\
L(2s-s_{0},\chi)\ \sideset{}{{}^{*}}{\sum}_{\ell\ (\bmod\,p)}\
\overline{\chi}(\ell)L\left(s_{0};\mp\frac{\overline{\ell}}{p};\Phi\right)$
$\displaystyle=\ \frac{p^{2s+s_{0}-1}}{\phi(p)}\ \sum_{\chi\ (\bmod\,p)}\
L(2s-s_{0},\chi)\ \sum_{n=1}^{\infty}\ \frac{\lambda_{\Phi}(n)}{n^{s_{0}}}\
\sideset{}{{}^{*}}{\sum}_{\ell\ (\bmod\,p)}\
\overline{\chi}(\ell)e\left(\mp\frac{n\bar{\ell}}{p}\right).$ (4.37)

If $(n,p)=1$, then the replacement $\ell\to\mp n\bar{\ell}$ $(\bmod\,p)$
implies

$\displaystyle\sideset{}{{}^{*}}{\sum}_{\ell\ (\bmod\,p)}\
\overline{\chi}(\ell)e\left(\mp\frac{n\bar{\ell}}{p}\right)\ \ =\ \
\overline{\chi}(\mp n)\tau(\chi).$ (4.38)

(When $\chi=\chi_{0}$, equation (4.38) is equal to $-1$.) If $p\mid n$, then

$\displaystyle\sideset{}{{}^{*}}{\sum}_{\ell\ (\bmod\,p)}\
\overline{\chi}(\ell)e\left(\mp\frac{n\bar{\ell}}{p}\right)\ \ =\ \
\sideset{}{{}^{*}}{\sum}_{\ell\ (\bmod\,p)}\ \overline{\chi}(\ell)\ \ =\ \
\phi(p)\cdot 1_{\chi=\chi_{0}}.$

As a result, the contribution from $\chi=\chi_{0}$ in (4.6) is equal to

$\displaystyle\mathcal{L}^{(p)}_{\pm}\left(s_{0},s;\Phi\right)\bigg{|}_{d=p,\,\chi=\chi_{0}}\
\ $ $\displaystyle=\ \
\frac{p^{2s+s_{0}-1}}{\phi(p)}\,L(2s-s_{0},\chi_{0})\left\\{-\sum_{(n,p)=1}\
\frac{\lambda_{\Phi}(n)}{n^{s_{0}}}\ +\ \phi(p)\,\sum_{p\mid n}\
\frac{\lambda_{\Phi}(n)}{n^{s_{0}}}\right\\}$ $\displaystyle=\ \
\frac{p^{2s+s_{0}-1}}{\phi(p)}\,L(2s-s_{0},\chi_{0})\left\\{\phi(p)L(s_{0},\Phi)\
-\ p\sum_{(n,p)=1}\ \frac{\lambda_{\Phi}(n)}{n^{s_{0}}}\right\\}.$

It follows from (4.25) that

$\displaystyle\mathcal{L}^{(p)}_{\pm}\left(s_{0},s;\Phi\right)\bigg{|}_{d=p,\,\chi=\chi_{0}}\
\ $ $\displaystyle=\ \
\frac{p^{2s+s_{0}-1}}{\phi(p)}\,L(2s-s_{0},\chi_{0})\,L(s_{0},\Phi)$
$\displaystyle\hskip 60.0pt\,\cdot\,\left\\{\phi(p)\ -\
p\left(1-\lambda_{\Phi}(p)p^{-s_{0}}+\overline{\lambda_{\Phi}}(p)p^{-2s_{0}}-p^{-3s_{0}}\right)\right\\}$
$\displaystyle=\ \
\frac{p^{2s+s_{0}-1}}{\phi(p)}\,(1-p^{-(2s-s_{0})})\left(-1+\lambda_{\Phi}(p)p^{1-s_{0}}-\overline{\lambda_{\Phi}}(p)p^{1-2s_{0}}+p^{1-3s_{0}}\right)$
$\displaystyle\hskip 60.0pt\cdot\,\zeta(2s-s_{0})\,L(s_{0},\Phi).$

On the other hand, the contributions from $\chi\neq\chi_{0}$ in (4.6) is equal
to

$\displaystyle\mathcal{L}^{(p)}_{\pm}\left(s_{0},s;\Phi\right)\bigg{|}_{d=p,\,\chi\neq\chi_{0}}\
\ $ $\displaystyle=\ \ \frac{p^{2s+s_{0}-1}}{\phi(p)}\
\sideset{}{{}^{*}}{\sum}_{\chi\,(\bmod\,p)}\ \chi(\mp
1)\tau(\chi)L(2s-s_{0},\chi)\,\sum_{(n,p)=1}\
\frac{\lambda_{\Phi}(n)\overline{\chi}(n)}{n^{s_{0}}}$ $\displaystyle=\ \
\frac{p^{2s+s_{0}-1}}{\phi(p)}\ \sideset{}{{}^{*}}{\sum}_{\chi\,(\bmod\,p)}\
\chi(\mp 1)\tau(\chi)L(2s-s_{0},\chi)L(s_{0},\Phi\otimes\overline{\chi}).$

The conclusion of Proposition 4.7 follows readily from

$\displaystyle\mathcal{L}^{(p)}_{\pm}\left(s_{0},s;\Phi\right)\ \ =\ \
\mathcal{L}^{(p)}_{\pm}\left(s_{0},s;\Phi\right)\bigg{|}_{d=1}\ +\
\mathcal{L}^{(p)}_{\pm}\left(s_{0},s;\Phi\right)\bigg{|}_{d=p,\,\chi=\chi_{0}}\
+\
\mathcal{L}^{(p)}_{\pm}\left(s_{0},s;\Phi\right)\bigg{|}_{d=p,\,\chi\neq\chi_{0}}$

and analytic continuation.

∎

It is certainly possible, but combinatorially involved, to obtain a version of
(4.7) when $a$ is composite, i.e., rewrite
$\mathcal{L}^{(a)}_{\pm}\left(s_{0},s;\Phi\right)$ in terms of automorphic
$L$-functions twisted by primitive Dirichlet characters. If $\Phi\cong
1\boxplus 1\boxplus 1$ or $\phi\boxplus 1$ with $\phi$ being a cusp form of
$GL(2)$, the key technicalities behind seem to have been worked out in the
recent work [DHKL20]. We will not pursue this task in order to maintain a
reasonable length for this article.

##  5\. Concluding Remarks

###  5.1. Relation to the 4th moment of Dirichlet $L$-functions

We specialize to $s=1/2$, $\operatorname{Re}s_{0}=1/2$ and
$\Phi=(E_{\min}^{(3)})^{{}^{*}}(\ *\ ;(0,0,0))$ in (4.7). Once again, we
restrict ourselves to prime conductors only. In this case, we have

$\displaystyle\mathcal{L}^{(p)}_{\pm}\left(s_{0},1/2;\Phi\right)\ \ =\ \ $
$\displaystyle\mathcal{P}_{p}(s_{0},s)\ \zeta(1-s_{0})\zeta(s_{0})^{3}$
$\displaystyle\hskip 20.0pt\ +\ \frac{p^{s_{0}}}{\phi(p)}\ \left(\ \
\sideset{}{{}^{+}}{\sum}_{\chi\ (\bmod\,p)}\ \ \mp\ \
\sideset{}{{}^{-}}{\sum}_{\chi\ (\bmod\,p)}\right)\
\tau(\chi)L(1-s_{0},\chi)L\left(s_{0},\overline{\chi}\right)^{3}.$ (5.1)

Recall the functional equation for the primitive Dirichlet $L$-functions:

$\displaystyle L(s_{0},\overline{\chi})\ \ =\ \
i^{-a_{\chi}}\,\frac{\tau(\overline{\chi})}{\sqrt{p}}\ p^{\frac{1}{2}-s_{0}}\
\frac{\Gamma_{\mathbb{R}}(1-s_{0}+a_{\chi})}{\Gamma_{\mathbb{R}}(s_{0}+a_{\chi})}\
L(1-s_{0},\chi),$ (5.2)

where $a_{\chi}=0$ if $\chi(-1)=1$ and $a_{\chi}=1$ if $\chi(-1)=-1$. Using
also the fact that $|\tau(\chi)|=\sqrt{p}$, it follows that

$\displaystyle\sideset{}{{}^{+}}{\sum}_{\chi\ (\bmod\,p)}\
\tau(\chi)L(1-s_{0},\chi)L\left(s_{0},\overline{\chi}\right)^{3}\ \ $
$\displaystyle=\ \ p^{1-s_{0}}\
\frac{\Gamma_{\mathbb{R}}(1-s_{0})}{\Gamma_{\mathbb{R}}(s_{0})}\ \
\sideset{}{{}^{+}}{\sum}_{\chi\ (\bmod\,p)}\
L(1-s_{0},\chi)^{2}L\left(s_{0},\overline{\chi}\right)^{2},$
$\displaystyle\sideset{}{{}^{-}}{\sum}_{\chi\ (\bmod\,p)}\
\tau(\chi)L(1-s_{0},\chi)L\left(s_{0},\overline{\chi}\right)^{3}\ \ $
$\displaystyle=\ \ -ip^{1-s_{0}}\
\frac{\Gamma_{\mathbb{R}}(2-s_{0})}{\Gamma_{\mathbb{R}}(1+s_{0})}\ \
\sideset{}{{}^{-}}{\sum}_{\chi\ (\bmod\,p)}\
L(1-s_{0},\chi)^{2}L\left(s_{0},\overline{\chi}\right)^{2}.$ (5.3)

The dual moment of

$\displaystyle 2\ \sum_{j=1}^{\infty}\ H(\mu_{j})\
\frac{\lambda_{j}(a)\Lambda(1/2,\phi_{j})^{3}}{\langle\phi_{j},\phi_{j}\rangle}+\frac{1}{2\pi}\,\int_{\mathbb{R}}\
H(i\mu)\
\frac{\sigma_{-2\mu}(a)a^{-\mu}|\Lambda(1/2+i\mu)|^{6}}{|\Lambda(1+2i\mu)|^{2}}\
d\mu$ (5.4)

is given by

$\displaystyle p^{-1/2}\ \sum_{\pm}\ \ \int_{(1/2)}\
\mathcal{L}^{(p)}_{\pm}\left(s_{0},s;\Phi\right)(\mathcal{F}_{\Phi}^{(\pm)}H)\left(s_{0},s\right)\
\frac{ds_{0}}{2\pi i},$ (5.5)

which is readily observed to be a sum of the following three weighted moments
of $GL(1)$ $L$-functions:

$\displaystyle\hskip 60.0ptp^{-1/2}\,\int_{(1/2)}\ \mathcal{P}_{p}(s_{0},s)\
|\zeta(s_{0})|^{4}\
\frac{\Gamma_{\mathbb{R}}(1-s_{0})}{\Gamma_{\mathbb{R}}(s_{0})}\cdot\sum_{\pm}\
(\mathcal{F}_{\Phi}^{(\pm)}H)\left(s_{0},s\right)\ \frac{ds_{0}}{2\pi i},$
$\displaystyle\hskip 40.0pt\frac{p^{1/2}}{\phi(p)}\ \
\sideset{}{{}^{+}}{\sum}_{\chi\ (\bmod\,p)}\ \int_{(1/2)}\
\left|L(s_{0},\chi)\right|^{4}\
\frac{\Gamma_{\mathbb{R}}(1-s_{0})}{\Gamma_{\mathbb{R}}(s_{0})}\cdot\sum_{\pm}\
(\mathcal{F}_{\Phi}^{(\pm)}H)\left(s_{0},s\right)\ \frac{ds_{0}}{2\pi i},$
$\displaystyle\frac{p^{1/2}}{\phi(p)}\ \ \sideset{}{{}^{-}}{\sum}_{\chi\
(\bmod\,p)}\ \int_{(1/2)}\
\left|L(s_{0},\chi)\right|^{4}\,\frac{\Gamma_{\mathbb{R}}(2-s_{0})}{\Gamma_{\mathbb{R}}(1+s_{0})}\cdot
i\left\\{(\mathcal{F}_{\Phi}^{(+)}H)\left(s_{0},s\right)-(\mathcal{F}_{\Phi}^{(-)}H)\left(s_{0},s\right)\right\\}\
\frac{ds_{0}}{2\pi i}.$

######  Remark 5.1.

While the twists appeared on each side of the reciprocity formula (1.1),
namely $\left\\{\lambda_{f}(p)\right\\}$ and $\left\\{\chi\bmod\,p\right\\}$,
are seemingly unrelated as discussed in Section 1.1, the additive harmonics

$\displaystyle\mathcal{E}_{p}\ :=\ \left\\{\,e(\mp
a_{1}\bar{\ell}/p):\,a_{1}\geq 1,\ \ell\ (\bmod\,p),\ (\ell,p)=1\,\right\\}$
(5.7)

obtained in our approach connects the mentioned multiplicative twistings.
Indeed, $\mathcal{E}_{p}$ naturally relates to $\left\\{\chi\bmod\,p\right\\}$
by the orthogonality relation (see Proposition 4.7), whereas $\mathcal{E}_{p}$
pertains to $\left\\{\lambda_{f}(p)\right\\}$ via our period integral
construction, more precisely, the arithmetic factor originating from the
unipotent translate of the Whittaker function (see Corollary 3.2).

Additive harmonics of various shapes are involved in the methods of [Y11,
BHKM20] as well, but certainly in rather different ways, e.g., Kloosterman
sums, Kuznetsov/ Poisson/ Voronoi formulae, $\delta$-symbol expansion etc.

###  5.2. The Twisted Cubic Moment Conjecture

In [Kw23a+], we stated and proved the untwisted version of the following
conjecture:

######  Conjecture 5.2.

For any integer $a\geq 1$, the full set of the main terms for

$\displaystyle\sum_{j=1}^{\infty}\ H(\mu_{j})\
\frac{\lambda_{j}(a)\Lambda(1/2-\alpha_{1},\phi_{j})\Lambda(1/2-\alpha_{2},\phi_{j})\Lambda(1/2-\alpha_{3},\phi_{j})}{\langle\phi_{j},\phi_{j}\rangle}$
$\displaystyle\hskip 120.0pt+\frac{1}{4\pi}\ \int_{\mathbb{R}}\ H(i\mu)\
\frac{\sigma_{-2\mu}(a)a^{-\mu}\prod\limits_{i=1}^{3}\Lambda(1/2+i\mu-\alpha_{i})\Lambda(1/2+i\mu+\alpha_{i})}{|\Lambda(1+2i\mu)|^{2}}\
d\mu$ (5.8)

is given by

$\displaystyle\hskip
20.0pt\frac{1}{2}\cdot\sum_{\epsilon_{1},\epsilon_{2},\epsilon_{3}=\pm 1}\
\zeta(1-\epsilon_{1}\alpha_{1}-\epsilon_{2}\alpha_{2})\zeta(1-\epsilon_{1}\alpha_{1}-\epsilon_{3}\alpha_{3})\zeta(1-\epsilon_{2}\alpha_{2}-\epsilon_{3}\alpha_{3})$
$\displaystyle\hskip 60.0pt\ \cdot\ a^{-\frac{1}{2}}\ \prod_{p\mid a}\
\left\\{\tau_{-\epsilon\cdot\alpha}\left(p^{o_{p}(a)}\right)\ -\
\frac{\tau_{-\epsilon\cdot\alpha}\left(p^{o_{p}(a)-1}\right)}{p}\right\\}$
$\displaystyle\hskip
140.0pt\cdot\int_{(0)}\frac{H(\mu)}{|\Gamma(\mu)|^{2}}\cdot\prod_{i=1}^{3}\
\prod_{\pm}\ \Gamma_{\mathbb{R}}\left(1/2\pm\mu-\epsilon_{i}\alpha_{i}\right)\
\frac{d\mu}{2\pi i},$ (5.9)

where
$\epsilon\cdot\alpha:=(\epsilon_{1}\alpha_{1},\epsilon_{2}\alpha_{2},\epsilon_{3}\alpha_{3})$,
$\tau_{\alpha}(n):=\sum\limits_{d_{1}d_{2}d_{3}=n}d_{1}^{-\alpha_{1}}d_{2}^{-\alpha_{2}}d_{3}^{-\alpha_{3}}$,
and $(\phi_{j})_{j=1}^{\infty}$ is an orthogonal basis of even Hecke-
normalized Maass cusp forms of $SL_{2}(\mathbb{Z})$ which satisfy
$\Delta\phi_{j}=(1/4-\mu_{j}^{2})\,\phi_{j}$. Note:
$(\epsilon_{1},\epsilon_{2},\epsilon_{3})=(+1,+1,+1)$ corresponds to the
$0$-swap term.

In this article and [Kw23a+], we have established all of the archimedean
functional relations for showing agreement with the prediction (5.2), as well
as the non-archimedean counterparts for the $0$\- and $3$-swap terms. The
identities for the $1,2$-swap terms should follow in a similar fashion using
the method of this article. (In [Kw23a+], this is very much true for the
archimedean case.) As indicated by (5.1) and the main theorem of [Kw23a+], the
full set of main terms for the fourth moment of the Dirichlet $L$-functions à
la CFKRS should also be visible in the full spectral identity for (5.2). This
offers an alternative approach to assemble the main terms, distinct from the
one of [Y11]. We shall leave these to a future work.

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