abconsymbol=c

# Exceptional zeros of Rankin–Selberg $L$-functions and joint Sato–Tate
distributions

Jesse Thorner  Department of Mathematics, University of Illinois, Urbana, IL
61801, USA<EMAIL_ADDRESS>
###### Abstract.

Let $\chi$ be an idele class character over a number field $F$, and let
$\pi,\pi^{\prime}$ be non-dihedral twist-inequivalent cuspidal automorphic
representations of $\mathrm{GL}_{2}(\mathbb{A}_{F})$. We prove the following
results.

  1. (1)

If $m,n\geq 0$ are integers, $m+n\geq 1$, $F$ is totally real, $\chi$
corresponds with a ray class character, and $\pi,\pi^{\prime}$ correspond with
primitive non-CM holomorphic Hilbert cusp forms, then the Rankin–Selberg
$L$-function
$L(s,\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi))$
has a standard zero-free region with no exceptional Landau–Siegel zero. When
$m,n\geq 1$ and $m+n\geq 4$, this is new even for $F=\mathbb{Q}$. As an
application, we establish the strongest known unconditional effective rates of
convergence in the Sato–Tate distribution for $\pi$ and the joint Sato–Tate
distribution for $\pi$ and $\pi^{\prime}$.

  2. (2)

The Rankin–Selberg $L$-function
$L(s,\mathrm{Sym}^{2}(\pi)\times(\mathrm{Sym}^{2}(\pi^{\prime})\otimes\chi))$
has a standard zero-free region with no exceptional Landau–Siegel zero. Until
now, this was only known when $\pi=\pi^{\prime}$, $\pi$ is self-dual, and
$\chi$ is trivial.

##  1\. Introduction and statement of the main results

Let $F$ be a number field, $\mathbb{A}_{F}$ be the ring of adeles over $F$,
and $\mathfrak{F}_{n}$ be the set of cuspidal automorphic representations
$\pi$ of $\mathrm{GL}_{n}(\mathbb{A}_{F})$ with unitary central character
$\omega_{\pi}$. Let $C(\pi)$ be the analytic conductor of $\pi$ (see (3.1)),
$L(s,\pi)$ be the standard $L$-function, and
$\widetilde{\pi}\in\mathfrak{F}_{n}$ be the contragredient of $\pi$. The
generalized Riemann hypothesis (GRH) predicts that if $\pi\in\mathfrak{F}_{n}$
and $\mathrm{Re}(s)>\frac{1}{2}$, then $L(s,\pi)\neq 0$. Jacquet and Shalika
[22] proved that if $\mathrm{Re}(s)\geq 1$, then $L(s,\pi)\neq 0$, extending
the work of Hadamard and de la Vallée Poussin for the Riemann zeta function.
Let $|\cdot|$ denote the idelic norm, and let $t\in\mathbb{R}$. By replacing
$\pi$ with $\pi\otimes|\cdot|^{it}$ and varying $t$, we find that it is
equivalent to prove that for all $\pi\in\mathfrak{F}_{n}$ and $\sigma\geq 1$,
we have that $L(\sigma,\pi)\neq 0$. By [18, Theorem A.1], there exists an
effectively computable constant $\Cl[abcon]{ZFR1}=\Cr{ZFR1}(n)>0$ such that if
$\pi\neq\widetilde{\pi}$, then

(1.1) $L(\sigma,\pi)\neq 0,\qquad\sigma\geq 1-\Cr{ZFR1}/\log C(\pi).$

If $\pi=\widetilde{\pi}$, then $L(\sigma,\pi)$ has at most one zero
$\beta_{\pi}$ (necessarily simple) in the interval (1.1). If no such zero
exists, then $L(s,\pi)$ has no exceptional zero or no Landau–Siegel zero.

Given $(\pi,\pi^{\prime})\in\mathfrak{F}_{n}\times\mathfrak{F}_{n^{\prime}}$,
let $L(s,\pi\times\pi^{\prime})$ be the Rankin–Selberg $L$-function, as
defined by Jacquet, Piatetski-Shapiro, and Shalika [21]. Shahidi [41] proved
that if $\mathrm{Re}(s)\geq 1$, then $L(s,\pi\times\pi^{\prime})\neq 0$.
Replacing $\pi$ with $\pi\otimes|\cdot|^{it}$ and varying $t\in\mathbb{R}$, we
find that it suffices to prove that if
$(\pi,\pi^{\prime})\in\mathfrak{F}_{n}\times\mathfrak{F}_{n^{\prime}}$ and
$\sigma\geq 1$, then $L(\sigma,\pi\times\pi^{\prime})\neq 0$. The only fully
uniform improvement is Brumley’s narrow zero-free region ([5] and [26,
Appendix])—there exist effectively computable constants
$\Cl[abcon]{Brumley}=\Cr{Brumley}(n,n^{\prime},F,\varepsilon)>0$ and
$\Cl[abcon]{Brumley2}=\Cr{Brumley2}(n,n^{\prime},F,\varepsilon)>0$ such that

$L(\sigma,\pi\times\pi^{\prime})\neq 0,\qquad\sigma\geq
1-\Cr{Brumley}/(C(\pi)C(\pi^{\prime}))^{\Cr{Brumley2}}.$

Harcos and the author [14, Theorem 1.1] proved that for all $\varepsilon>0$,
there exists an ineffective constant
$\Cl[abcon]{Siegel}=\Cr{Siegel}(\pi,\pi^{\prime},\varepsilon)>0$ such that if
$\chi\in\mathfrak{F}_{1}$, then

$L(\sigma,\pi\times(\pi^{\prime}\otimes\chi))\neq 0,\qquad\sigma\geq
1-\Cr{Siegel}/C(\chi)^{\varepsilon}.$

Langlands conjectured that $L(s,\pi\times\pi^{\prime})$ factors as a product
of standard $L$-functions (i.e., $L(s,\pi\times\pi^{\prime})$ is modular). As
of now, this is only known in special cases, most notably when
$\pi\in\mathfrak{F}_{2}$ and
$\pi^{\prime}\in\mathfrak{F}_{2}\cup\mathfrak{F}_{3}$ [25, 39]. Hoffstein and
Ramakrishnan [17] proved that if all Rankin–Selberg $L$-functions are modular
and $\pi\in\cup_{n=2}^{\infty}\mathfrak{F}_{n}$, then $L(s,\pi)$ has no zero
in (1.1), eliminating exceptional zeros. Modularity is known only in special
cases, so unconditionally eliminating exceptional zeros remains a difficult
and fruitful problem. For a Rankin–Selberg $L$-function
$L(s,\pi\times\pi^{\prime})$, we say that $L(s,\pi\times\pi^{\prime})$ has no
exceptional zero if there exists an effectively computable constant
$\Cl[abcon]{ZFR11}=\Cr{ZFR11}(n,n^{\prime})>0$ such that

$L(\sigma,\pi\times\pi^{\prime})\neq 0,\qquad\sigma\geq
1-\Cr{ZFR11}/\log(C(\pi)C(\pi^{\prime})).$

Further discussion requires symmetric power lifts. Let $v$ be a place of $F$,
and let $F_{v}$ be the completion of $F$ relative to $v$. Given
$\pi\in\mathfrak{F}_{2}$, we express $\pi$ as a restricted tensor product
$\bigotimes_{v}\pi_{v}$ of smooth, admissible representations of
$\mathrm{GL}_{2}(F_{v})$. If $m\geq 0$, then
$\mathrm{Sym}^{m}\colon\mathrm{GL}_{2}(\mathbb{C})\to\mathrm{GL}_{m+1}(\mathbb{C})$
is the $(m+1)$-dimensional irreducible representation of
$\mathrm{GL}_{2}(\mathbb{C})$ on symmetric tensors of rank $m$. If $P(x,y)$ is
a homogeneous degree $m$ polynomial in two variables and
$g\in\mathrm{GL}_{2}(\mathbb{C})$, then
$\mathrm{Sym}^{m}(g)\in\mathrm{GL}_{m+1}(\mathbb{C})$ is the matrix giving the
change in coefficients of $P$ under the change of variables by $g$. Let
$\varphi_{v}$ be the two-dimensional representation of the Deligne–Weil group
attached to $\pi_{v}$ and $\mathrm{Sym}^{m}(\pi_{v})$ be the smooth admissible
representation of $\mathrm{GL}_{m+1}(F_{v})$ attached to the representation
$\mathrm{Sym}^{m}\circ\varphi_{v}$. By the local Langlands correspondence,
$\mathrm{Sym}^{m}(\pi_{v})$ is well-defined for every place $v$ of $F$.

Langlands conjectured that
$\mathrm{Sym}^{m}(\pi)=\bigotimes_{v}\mathrm{Sym}^{m}(\pi_{v})$ is an
automorphic representation of $\mathrm{GL}_{m+1}(\mathbb{A}_{F})$, possibly
not cuspidal. This is known for $m\leq 4$ [11, 23, 25]. Let

$\mathrm{RA}_{F}=\\{\pi\in\mathfrak{F}_{2}\colon\textup{$\pi$ non-dihedral and
regular algebraic}\\}$

(see [4, Section 7] for definitions). Such $\pi$ correspond with primitive
non-CM holomorphic Hilbert cusp forms whose weights are integers at least 2.
When $F=\mathbb{Q}$, these are the classical non-CM holomorphic newforms.
Newton and Thorne [33, 34, 35] proved that if $m\geq 0$, $F$ is totally real,
and $\pi\in\mathrm{RA}_{F}$, then
$\mathrm{Sym}^{m}(\pi)\in\mathfrak{F}_{m+1}$.

Given $\pi,\pi^{\prime}\in\mathfrak{F}_{n}$, we write $\pi\sim\pi^{\prime}$ if
there exists $\psi\in\mathfrak{F}_{1}$ such that
$\pi^{\prime}=\pi\otimes\psi$. Otherwise, we write $\pi\not\sim\pi^{\prime}$.
Let $\mathbbm{1}\in\mathfrak{F}_{1}$ be the trivial representation, whose
$L$-function is the Dedekind zeta function $\zeta_{F}(s)$. As of now, the
following list gives all of the conditions under which we know that
exceptional zeros do not exist. (See also [13, 30].)

######  Proposition 1.1.

  1. (i)

[17] If $\pi\in\cup_{n=1}^{\infty}\mathfrak{F}_{n}$ is not self-dual, then
$L(s,\pi)$ has no exceptional zero.

  2. (ii)

[2, 16, 17] If $\pi\in\mathfrak{F}_{2}\cup\mathfrak{F}_{3}$, then $L(s,\pi)$
has no exceptional zero.

  3. (iii)

[39] Let $\pi,\pi^{\prime}\in\mathfrak{F}_{2}$. Any exceptional zero of the
modular Rankin–Selberg $L$-function $L(s,\pi\times\pi^{\prime})$ is the
exceptional zero of an abelian $L$-function dividing it, if any exist. If
$\pi,\pi^{\prime}$ are non-dihedral and $\pi\not\sim\pi^{\prime}$, then no
exceptional zero exists.

  4. (iv)

[39] Let $\pi\in\mathfrak{F}_{2}$ be self-dual, in which case
$\omega_{\pi}^{2}=\mathbbm{1}$. Any exceptional zero of

$L(s,\mathrm{Sym}^{2}(\pi)\times\mathrm{Sym}^{2}(\pi))/\zeta_{F}(s)=L(s,\mathrm{Sym}^{2}(\pi)\otimes\omega_{\pi})L(s,\mathrm{Sym}^{4}(\pi))$

is the exceptional zero of an abelian $L$-function dividing it, if any exist.
No exceptional zero exists when $\pi$ is not dihedral, tetrahedral, or
octahedral.

  5. (v)

[27] Let $(\pi,\pi^{\prime})\in\mathfrak{F}_{2}\times\mathfrak{F}_{3}$. If
$\pi$ is non-dihedral, then assume that
$\pi^{\prime}\not\sim\mathrm{Sym}^{2}(\pi)$. The modular Rankin–Selberg
$L$-function $L(s,\pi\times\pi^{\prime})$ has no exceptional zero.

  6. (vi)

[44] Let $\pi\in\mathrm{RA}_{\mathbb{Q}}$ have trivial central character. If
$m\geq 0$ is an integer and $t\in\mathbb{R}$, then
$L(s,\mathrm{Sym}^{m}(\pi)\otimes|\cdot|^{it})$ has no exceptional zero.

  7. (vii)

[27] Let $\pi\in\mathfrak{F}_{n}$. If there exists
$\psi\in\mathfrak{F}_{1}-\\{\mathbbm{1}\\}$ such that $\pi=\pi\otimes\psi$,
then $L(s,\pi)$ has no exceptional zero.

  8. (viii)

[19] If $\pi\in\mathfrak{F}_{n}$ and $t\in\mathbb{R}-\\{0\\}$, then
$L(s,\pi\times(\widetilde{\pi}\otimes|\cdot|^{it}))$ has no exceptional zero.
(See also [13].)

  9. (ix)

[5, Appendix] Let
$(\pi,\pi^{\prime})\in\mathfrak{F}_{n}\times\mathfrak{F}_{n^{\prime}}$. If
exactly one of $\pi$ and $\pi^{\prime}$ is self-dual, then
$L(s,\pi\times\pi^{\prime})$ has no exceptional zero. (See also [30].)

Here, we produce a large class of Rankin–Selberg $L$-functions
$L(s,\pi\times\pi^{\prime})$ where
$(\pi,\pi^{\prime})\in\mathfrak{F}_{n}\times\mathfrak{F}_{n^{\prime}}$, $n$
and $n^{\prime}$ can be arbitrarily large, $L(s,\pi\times\pi^{\prime})$ is not
known to be modular, $L(s,\pi\times\pi^{\prime})$ has no exceptional zero, and
$L(s,\pi\times\pi^{\prime})$ is not accounted for by Proposition 1.1 or minor
variations to its proof. To state our results, we define

(1.2)
$\mathfrak{F}_{n}^{*}=\\{\pi\in\mathfrak{F}_{n}\colon\omega_{\pi}\textup{ is
trivial on the diagonally embedded positive reals}\\}.$

For all $\pi\in\mathfrak{F}_{n}$, there exists a unique pair
$(\pi^{*},t_{\pi})\in\mathfrak{F}_{n}^{*}\times\mathbb{R}$ such that
$\pi=\pi^{*}\otimes|\cdot|^{it_{\pi}}$. It follows that
$L(s,\pi)=L(s+it_{\pi},\pi^{*})$.

######  Theorem 1.2.

Let $m,n\geq 0$ be integers, $M=\max\\{m,n\\}\geq 1$, and $F$ be a totally
real number field. Let $\pi,\pi^{\prime}\in\mathrm{RA}_{F}$ and
$\chi=\chi^{*}|\cdot|^{it_{\chi}}\in\mathfrak{F}_{1}$. Define

(1.3) $y_{\pi}=\begin{cases}0&\parbox{238.49231pt}{if $\pi$ has squarefree
conductor or $\pi$ corresponds with a non-CM elliptic curve over
$\mathbb{Q}$,}\vspace{1mm}\\\ 2&\mbox{otherwise,}\end{cases}$

and similarly for $\pi^{\prime}$. Let $\chi^{*}$ correspond with a ray class
character. Define

$\mathcal{L}(s)=\begin{cases}\displaystyle\frac{L(s,\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi))}{L(s,\chi\omega_{\pi}^{m}\psi^{m})}&\mbox{if
$m=n$, $\psi\in\mathfrak{F}_{1}$, and $\pi^{\prime}=\pi\otimes\psi$,}\\\
L(s,\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi))&\mbox{if
$m\neq n$ or $\pi\not\sim\pi^{\prime}$.}\end{cases}$

There exists an absolute, effectively computable constant $\Cl[abcon]{main}>0$
such that

$\mathcal{L}(\sigma)\neq 0,\qquad\sigma\geq
1-\frac{\Cr{main}}{M^{2}[M^{\max\\{y_{\pi},y_{\pi^{\prime}}\\}}\log(MC(\pi)C(\pi^{\prime}))+\log
C(\chi)]}.$

######  Remark.

Theorem 1.2 is new when $m+n\geq 4$, as a comparison with Proposition 1.1
shows. We restrict to totally real $F$ only to ensure the cuspidal automorphy
of all symmetric powers of all $\pi\in\mathrm{RA}_{F}$ via [33, 34, 35]. In
light of (3.1), Theorem 1.2 is uniform in $F$.

Our next result follows immediately from Theorem 1.2.

######  Corollary 1.3.

Let $m,n\geq 0$ be integers, $M=\max\\{m,n\\}\geq 1$, $F$ be a totally real
number field, and
$\pi,\pi^{\prime}\in\mathrm{RA}_{F}\cap\mathfrak{F}_{2}^{*}$. Let $y_{\pi}$
and $y_{\pi^{\prime}}$ be as in (1.3). Let $\chi\in\mathfrak{F}_{1}^{*}$
correspond with a ray class character. If $\pi\not\sim\pi^{\prime}$ and
$t\in\mathbb{R}$, then
$L(\sigma+it,\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi))\neq
0$ when

$\sigma\geq
1-\frac{\Cr{main}}{M^{2}[M^{\max\\{y_{\pi},y_{\pi^{\prime}}\\}}\log(MC(\pi)C(\pi^{\prime}))+\log(C(\chi)(3+|t|)^{[F:\mathbb{Q}]})]}.$

######  Remark.

Our proof shows that if $m=0$, then $C(\pi)$ can be replaced by $1$ and
$y_{\pi}$ by $0$. If $n=0$, then $C(\pi^{\prime})$ can be replaced by $1$ and
$y_{\pi^{\prime}}$ by $0$.

Let $\pi,\pi^{\prime}\in\mathfrak{F}_{2}$ and $\chi\in\mathfrak{F}_{1}$. Under
the hypotheses of Proposition 1.1(iv) or Theorem 1.2, any exceptional zero of
$L(s,\mathrm{Sym}^{2}(\pi)\times(\mathrm{Sym}^{2}(\pi^{\prime})\otimes\chi))$
comes from an abelian $L$-function dividing it. Our next result establishes
this conclusion in full generality. For a cleaner presentation and proof, we
introduce
$\mathrm{Ad}(\pi)=\mathrm{Sym}^{2}(\pi)\otimes\overline{\omega}_{\pi}$, the
self-dual automorphic representation of $\mathrm{GL}_{3}(\mathbb{A}_{F})$
whose $L$-function is $L(s,\pi\times\widetilde{\pi})/\zeta_{F}(s)$.

######  Theorem 1.4.

Let $\pi,\pi^{\prime}\in\mathfrak{F}_{2}$ and $\chi\in\mathfrak{F}_{1}$.
Suppose that at least one of $\pi$ and $\pi^{\prime}$ is non-dihedral. Define
$\mathcal{T}_{\pi}=\\{\eta\in\mathfrak{F}_{1}\colon\eta=\eta^{*}|\cdot|^{it_{\eta}},~{}\mathrm{Ad}(\pi)=\mathrm{Ad}(\pi)\otimes\eta^{*}\\}$
and

$\mathscr{L}(s)=\begin{cases}L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi))/\zeta_{F}(s+it_{\chi})&\mbox{if
$\pi\sim\pi^{\prime}$ and
$\chi=\chi^{*}|\cdot|^{it_{\chi}}\in\mathcal{T}_{\pi}$,}\vspace{1mm}\\\
L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi))/L(s,\chi)&\mbox{if
$\pi\sim\pi^{\prime}$ and $\chi\notin\mathcal{T}_{\pi}$,}\vspace{1mm}\\\
L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi))&\mbox{if
$\pi\not\sim\pi^{\prime}$.}\end{cases}$

There exists an absolute, effectively computable constant $\Cl[abcon]{thm2}>0$
such that

(1.4) $\mathscr{L}(\sigma)\neq 0,\qquad\sigma\geq
1-\Cr{thm2}/\log(C(\pi)C(\pi^{\prime})C(\chi)).$

If $\pi$ and $\pi^{\prime}$ are dihedral, then any zero of
$L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi))$ in the
interval (1.4) is the exceptional zero of an abelian $L$-function that divides
$L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi))$.

######  Remark.

Our proof handles 11 different cases. For
$L(s,\mathrm{Sym}^{2}(\pi)\times(\mathrm{Sym}^{2}(\pi^{\prime})\otimes\chi))$,
replace all instances of $\chi$ in Theorem 1.4 with
$\chi\omega_{\pi}\omega_{\pi^{\prime}}$. So doing, we recover Proposition
1.1(iv). Note that if there exists $\psi\in\mathfrak{F}_{1}$ such that
$\pi^{\prime}=\pi\otimes\psi$, then
$\omega_{\pi^{\prime}}=\omega_{\pi}\psi^{2}$.

Our results rely crucially on the existence of carefully constructed auxiliary
Dirichlet series with non-negative Dirichlet coefficients. The usual method
for constructing such Dirichlet series, taking the Rankin–Selberg convolution
of a carefully chosen isobaric automorphic representation with its dual,
underlies the proof of each part in Proposition 1.1. This method is not
available for the proofs of our main results, the twist-inequivalent cases in
Theorems 1.2 and 1.4, because of insufficient progress towards the modularity
of Rankin–Selberg $L$-functions (see Section 4.1 for discussion). The novelty
in our proofs, therefore, is the construction of the auxiliary Dirichlet
series in (4.1) and the unconditional verification of its useful analytic
properties in the very different settings of Theorems 1.2 and 1.4.

### Outline

In Section 2, we discuss the application of Theorem 1.2 to Theorem 2.1, an
unconditional and effective bound on the rate of convergence in the Sato–Tate
distribution for $\pi\in\mathrm{RA}_{F}$ and the joint Sato–Tate distribution
for twist-inequivalent pairs $\pi,\pi^{\prime}\in\mathrm{RA}_{F}$. In Section
3, we review analytic properties of $L$-functions. In Sections 4 and 5, we
prove Theorem 1.2 and compare the proof with past approaches. In Sections 6
and 7, we prove Theorem 1.4. In Section 8, we prove Theorem 2.1.

### Acknowledgements

The author thanks Farrell Brumley, Gergely Harcos, and Jeffrey Hoffstein for
helpful conversations and Laurent Clozel for providing the strategy for Lemma
4.2. The author is partially funded by the Simons Foundation.

##  2\. Application to Sato–Tate and joint Sato–Tate distributions

Let $F$ be a totally real number field. Let $v\nmid\infty$ denote a non-
archimedean place of $F$ and $v\mid\infty$ otherwise. Let
$\pi=\bigotimes_{v}\pi_{v}\in\mathrm{RA}_{F}\cap\mathfrak{F}_{2}^{*}$, and let
$S_{\pi}=\\{v\nmid\infty\colon\textup{$\pi_{v}$ ramified}\\}$. Since
$\pi\in\mathrm{RA}_{F}$, the central character
$\omega_{\pi}=\bigotimes_{v}\omega_{\pi,v}$ corresponds with a ray class
character over $F$. For $v\nmid\infty$, let $q_{v}$ be the cardinality of the
residue field of the local ring of integers $\mathcal{O}_{v}\subseteq F_{v}$
and $\varpi_{v}$ be a uniformizer. Fix a root of unity $\zeta$ such that
$\zeta^{2}\in\mathrm{im}(\omega_{\pi})$. Let $v\nmid\infty$ satisfy $v\notin
S_{\pi}$ and $\omega_{\pi,v}(\varpi_{v})=\zeta^{2}$. Let $a_{\pi}(v)$ be the
eigenvalue of the Hecke operator

$\Big{[}\mathrm{GL}_{2}(\mathcal{O}_{F_{v}})\Big{(}\begin{matrix}\varpi_{v}&0\\\
0&1\end{matrix}\Big{)}\mathrm{GL}_{2}(\mathcal{O}_{F_{v}})\Big{]}$

on $\pi_{v}^{\mathrm{GL}_{2}(\mathcal{O}_{F_{v}})}$. Since
$\mathrm{Sym}^{m}(\pi)\in\mathfrak{F}_{m+1}$ for all $m\geq 0$ [33, 34, 35],
we have that

$\frac{a_{\pi}(v)}{2\zeta}\in[-1,1].$

The Sato–Tate conjecture for $\pi$ states that if
$\mathrm{d}\mu_{\mathrm{ST}}=(2/\pi)\sqrt{1-t^{2}}~{}\mathrm{d}t$ is the
Sato–Tate measure, $I\subseteq[-1,1]$ is a fixed subinterval, and
$\mathbf{1}_{I}(t)$ is the indicator function of $I$, then

(2.1) $\lim_{x\to\infty}\sum_{\begin{subarray}{c}v\notin S_{\pi},~{}q_{v}\leq
x\\\
\omega_{\pi,v}(\varpi_{v})=\zeta^{2}\end{subarray}}\mathbf{1}_{I}\Big{(}\frac{a_{\pi}(v)}{2\zeta}\Big{)}\Big{/}\sum_{\begin{subarray}{c}v\notin
S_{\pi},~{}q_{v}\leq x\\\
\omega_{\pi,v}(\varpi_{v})=\zeta^{2}\end{subarray}}1=\mu_{\mathrm{ST}}(I).$

This was proved when $F=\mathbb{Q}$ by Barnet-Lamb, Geraghty, Harris, and
Taylor [4] and for all totally real $F$ by Barnet-Lamb, Gee, and Geraghty [3].

Let $\pi,\pi^{\prime}\in\mathrm{RA}_{F}\cap\mathfrak{F}_{2}^{*}$. Let
$I,I^{\prime}\subseteq[-1,1]$ be fixed subintervals, and let $\zeta$ (resp.
$\zeta^{\prime}$) be a fixed root of unity such that
$\zeta^{2}\in\mathrm{im}(\omega_{\pi})$ (resp.
${\zeta^{\prime}}^{2}\in\mathrm{im}(\omega_{\pi^{\prime}})$). Generalizing a
question of Katz and Mazur originally posed for non-CM elliptic curves, one
might ask whether

(2.2) $\lim_{x\to\infty}\sum_{\begin{subarray}{c}v\notin S_{\pi}\cup
S_{\pi^{\prime}},~{}q_{v}\leq x\\\ \omega_{\pi,v}(\varpi_{v})=\zeta^{2}\\\
\omega_{\pi^{\prime},v}(\varpi_{v})=\zeta^{\prime
2}\end{subarray}}\mathbf{1}_{I}\Big{(}\frac{a_{\pi}(v)}{2\zeta}\Big{)}\mathbf{1}_{I^{\prime}}\Big{(}\frac{a_{\pi^{\prime}}(v)}{2\zeta^{\prime}}\Big{)}\Big{/}\sum_{\begin{subarray}{c}v\notin
S_{\pi}\cup S_{\pi^{\prime}},~{}q_{v}\leq x\\\
\omega_{\pi,v}(\varpi_{v})=\zeta^{2}\\\
\omega_{\pi^{\prime},v}(\varpi_{v})=\zeta^{\prime
2}\end{subarray}}1=\mu_{\mathrm{ST}}(I)\mu_{\mathrm{ST}}(I^{\prime}).$

Building on work of Harris [15], Wong [46] proved (2.2) when
$\omega_{\pi}=\omega_{\pi^{\prime}}=\mathbbm{1}$ and
$\pi\not\sim\pi^{\prime}$. Using Theorem 1.2, we prove variants of (2.1) and
(2.2) with effective rates of convergence.

######  Theorem 2.1.

Let $F$ be a totally real number field and
$\pi,\pi^{\prime}\in\mathrm{RA}_{F}\cap\mathfrak{F}_{2}^{*}$. Let $\zeta$
(resp. $\zeta^{\prime}$) be a fixed root of unity such that $\zeta^{2}$ (resp.
$\zeta^{\prime 2}$) lies in the image of $\omega_{\pi}$ (resp.
$\omega_{\pi^{\prime}}$). Let $S_{\pi}$ (resp. $S_{\pi^{\prime}}$) be the set
of places $v\nmid\infty$ of $F$ such that $\pi_{v}$ (resp. $\pi_{v}^{\prime}$)
is ramified. Let $y_{\pi}$ and $y_{\pi^{\prime}}$ be as in (1.3). There exists
an absolute, effectively computable constant $\Cl[abcon]{ST1}>0$ such that the
following results are true.

  1. (1)

If $I\subseteq[-1,1]$ is a closed subinterval and $x\geq 2$, then

$\displaystyle\Big{|}\sum_{\begin{subarray}{c}v\notin S_{\pi},~{}q_{v}\leq
x\\\
\omega_{\pi,v}(\varpi_{v})=\zeta^{2}\end{subarray}}\mathbf{1}_{I}\Big{(}\frac{a_{\pi}(v)}{2\zeta}\Big{)}\Big{/}\sum_{\begin{subarray}{c}v\notin
S_{\pi},~{}q_{v}\leq x\\\
\omega_{\pi,v}(\varpi_{v})=\zeta^{2}\end{subarray}}1-\mu_{\mathrm{ST}}(I)\Big{|}\leq\Cr{ST1}\sqrt{[F:\mathbb{Q}]}\Big{(}\frac{\log(C(\pi)\log
x)}{\sqrt{\log x}}\Big{)}^{\frac{2}{2+y_{\pi}}}.$

  2. (2)

Assume that $\pi\not\sim\pi^{\prime}$. Let $I,I^{\prime}\subseteq[-1,1]$ be
closed subintervals and $x\geq 2$. If there are infinitely many $v\nmid\infty$
such that $\omega_{\pi,v}(\varpi_{v})=\zeta^{2}$ and
$\omega_{\pi^{\prime},v}(\varpi_{v})=\zeta^{\prime 2}$, then

$\displaystyle\Big{|}\sum_{\begin{subarray}{c}v\notin S_{\pi}\cup
S_{\pi^{\prime}},~{}q_{v}\leq x\\\ \omega_{\pi,v}(\varpi_{v})=\zeta^{2}\\\
\omega_{\pi^{\prime},v}(\varpi_{v})=\zeta^{\prime
2}\end{subarray}}\mathbf{1}_{I}\Big{(}\frac{a_{\pi}(v)}{2\zeta}\Big{)}\mathbf{1}_{I^{\prime}}\Big{(}\frac{a_{\pi^{\prime}}(v)}{2\zeta^{\prime}}\Big{)}\Big{/}\sum_{\begin{subarray}{c}v\notin
S_{\pi}\cup S_{\pi^{\prime}},~{}q_{v}\leq x\\\
\omega_{\pi,v}(\varpi_{v})=\zeta^{2}\\\
\omega_{\pi^{\prime},v}(\varpi_{v})=\zeta^{\prime
2}\end{subarray}}1-\mu_{\mathrm{ST}}(I)\mu_{\mathrm{ST}}(I^{\prime})\Big{|}$
$\displaystyle\leq\Cr{ST1}\sqrt{[F:\mathbb{Q}]}\Big{(}\frac{\log(C(\pi)C(\pi^{\prime})\log
x)}{\sqrt{\log x}}\Big{)}^{\frac{2}{2+\max\\{y_{\pi},y_{\pi^{\prime}}\\}}}.$

######  Remark.

In [44, Theorem 1.1], the author proved Theorem 2.1(1) when $F=\mathbb{Q}$ and
$\omega_{\pi}=\mathbbm{1}$. Theorem 2.1(1) is new in all other cases. Theorem
2.1(2) is new in all cases; in [44, Theorem 1.2], where $F=\mathbb{Q}$ and
$\omega_{\pi}=\omega_{\pi^{\prime}}=\mathbbm{1}$, the author proved a version
of Theorem 2.1(2) with an error term that only decays as a power of $\log\log
x$ because of the potential existence of exceptional zeros. These are now
precluded by Theorem 1.2. Throughout [44], it was assumed that
$y_{\pi}=y_{\pi^{\prime}}=0$.

As in Theorem 1.2, we restrict to totally real $F$ only to ensure the cuspidal
automorphy of all symmetric powers of all $\pi\in\mathrm{RA}_{F}$ via [33, 34,
35]. By our definition for $C(\pi)$ and $C(\pi^{\prime})$ in (3.1), Theorem
2.1 is uniform in $F$.

We will only detail the proof of Theorem 2.1(2), since Theorem 2.1(1) is
easier to prove.

##  3\. Analytic properties of $L$-functions

Let $F$ be a number field with absolute discriminant $D_{F}$. For each place
$v$ of $F$, we write $v\mid\infty$ (resp. $v\nmid\infty$) if $v$ is
archimedean (resp. non-archimedean). If $v\nmid\infty$, then $q_{v}$ is the
cardinality of the residue field of the local ring of integers
$\mathcal{O}_{v}\subseteq F_{v}$, and $\varpi_{v}$ is the uniformizer. The
properties of $L$-functions given here rely on [12, 21, 29]. In our use of
$f=O(g)$ or $f\ll g$, the implied constant is always absolute and effectively
computable.

Let $\pi\in\mathfrak{F}_{n}$. Recall $\mathfrak{F}_{n}^{*}$ in (1.2). There
exists a unique pair
$(\pi^{*},t_{\pi})\in\mathfrak{F}_{n}^{*}\times\mathbb{R}$ such that
$\pi=\pi^{*}\otimes|\cdot|^{it_{\pi}}$, in which case
$L(s,\pi)=L(s+it_{\pi},\pi^{*})$. Therefore, in our discussion of standard
$L$-functions and Rankin–Selberg $L$-functions, it suffices to restrict to
$\mathfrak{F}_{n}^{*}$.

###  3.1. Standard $L$-functions

Let $\pi\in\mathfrak{F}_{n}^{*}$, let $\widetilde{\pi}\in\mathfrak{F}_{n}^{*}$
be the contragredient, and let $\omega_{\pi}$ the central character. We
express $\pi$ as a restricted tensor product $\bigotimes_{v}\pi_{v}$ of smooth
admissible representations of $\mathrm{GL}_{n}(F_{v})$. Let $\delta_{\pi}=1$
if $\pi=\mathbbm{1}$ and $\delta_{\pi}=0$ otherwise. Let

$S_{\pi}=\\{v\nmid\infty\colon\textup{$\pi_{v}$ ramified}\\},\qquad
S_{\pi}^{\infty}=S_{\pi}\cup\\{v\mid\infty\\}.$

Let $N_{\pi}$ be the norm of the conductor of $\pi$. If $v\nmid\infty$, then
there are $n$ Satake parameters $(\alpha_{j,\pi}(v))_{j=1}^{n}$ such that

$L(s,\pi)=\prod_{v\nmid\infty}L(s,\pi_{v}),\qquad
L(s,\pi_{v})=\prod_{j=1}^{n}\frac{1}{1-\alpha_{j,\pi}(v)q_{v}^{-s}}$

converges absolutely for $\mathrm{Re}(s)>1$. If $v\in S_{\pi}$, then at least
one of the $\alpha_{j,\pi}(p)$ equals zero.

If $v\mid\infty$, then $(\mu_{j,\pi}(v))_{j=1}^{n}$ are the Langlands
parameters at $v$, from which we define

$L(s,\pi_{\infty})=\prod_{v\mid\infty}L(s,\pi_{v})=\prod_{v\mid\infty}\prod_{j=1}^{n}\Gamma_{v}(s+\mu_{j,\pi}(v)),\qquad\Gamma_{v}(s)=\begin{cases}\pi^{-s/2}\Gamma(s/2)&\mbox{if
$F_{v}=\mathbb{R}$,}\\\ 2(2\pi)^{-s}\Gamma(s)&\mbox{if
$F_{v}=\mathbb{C}$.}\end{cases}$

The completed $L$-function
$\Lambda(s,\pi)=(s(1-s))^{\delta_{\pi}}(D_{F}^{n}N_{\pi})^{\frac{s}{2}}L(s,\pi)L(s,\pi_{\infty})$
is entire of order $1$. Since,
$\\{\alpha_{j,\widetilde{\pi}}(v)\\}=\\{\overline{\alpha_{j,\pi}(v)}\\}$,
$N_{\widetilde{\pi}}=N_{\pi}$, and
$\\{\mu_{j,\widetilde{\pi}}(v)\\}=\\{\overline{\mu_{j,\pi}(v)}\\}$, we have
that $L(s,\widetilde{\pi})=\overline{L(\overline{s},\pi)}$. The analytic
conductor is

(3.1)
$C(\pi)=D_{F}^{n}N_{\pi}\prod_{v\mid\infty}\prod_{j=1}^{n}(|\mu_{j,\pi}(v)|+3)^{[F_{v}:\mathbb{R}]}.$

By [28, 32] there exists $\theta_{n}\in[0,1/2-1/(n^{2}+1)]$ such that

(3.2) $|\alpha_{j,\pi}(v)|\leq
q_{v}^{\theta_{n}},\qquad\mathrm{Re}(\mu_{j,\pi}(v))\geq-\theta_{n}.$

We define $a_{\pi}(v^{\ell})$ by the Dirichlet series identity

$-\frac{L^{\prime}}{L}(s,\pi)=\sum_{v}\sum_{\ell=1}^{\infty}\frac{\sum_{j=1}^{n}\alpha_{j,\pi}(v)^{\ell}\log
q_{v}}{q_{v}^{\ell
s}}=\sum_{v\nmid\infty}\sum_{\ell=1}^{\infty}\frac{a_{\pi}(v^{\ell})\log
q_{v}}{q_{v}^{\ell s}}.$

###  3.2. Rankin–Selberg $L$-functions

Let $\pi\in\mathfrak{F}_{n}^{*}$ and
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}^{*}$. Let
$\delta_{\pi\times\pi^{\prime}}=1$ if $\pi^{\prime}=\widetilde{\pi}$ and
$\delta_{\pi\times\pi^{\prime}}=0$ otherwise. For each $v\notin
S_{\pi}^{\infty}\cup S_{\pi^{\prime}}^{\infty}$, define

$L(s,\pi_{v}\times\pi_{v}^{\prime})=\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n^{\prime}}\frac{1}{1-\alpha_{j,\pi}(v)\alpha_{j^{\prime},\pi^{\prime}}(v)q_{v}^{-s}}.$

Jaquet, Piatetski-Shapiro, and Shalika proved the following theorem.

######  Theorem 3.1.

[21] If
$(\pi,\pi^{\prime})\in\mathfrak{F}_{n}^{*}\times\mathfrak{F}_{n^{\prime}}^{*}$,
then there exist

  1. (1)

complex numbers
$(\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(v))_{j=1}^{n}{}_{j^{\prime}=1}^{n^{\prime}}$
for each $v\in S_{\pi}\cup S_{\pi^{\prime}}$, from which we define

$\displaystyle
L(s,\pi_{v}\times\pi_{v}^{\prime})=\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n^{\prime}}\frac{1}{1-\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(v)q_{v}^{-s}},\quad
L(s,\widetilde{\pi}_{v}\times\widetilde{\pi}_{v}^{\prime})=\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n^{\prime}}\frac{1}{1-\overline{\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(v)}q_{v}^{-s}};$

  2. (2)

complex numbers
$(\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v))_{j=1}^{n}{}_{j^{\prime}=1}^{n^{\prime}}$
for each $v\mid\infty$, from which we define

$\displaystyle
L(s,\pi_{v}\times\pi_{v}^{\prime})=\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n^{\prime}}\Gamma_{v}(s+\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)),\quad
L(s,\widetilde{\pi}_{v}\times\widetilde{\pi}_{v}^{\prime})=\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n^{\prime}}\Gamma_{v}(s+\overline{\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)});$

  3. (3)

a conductor, an integral ideal whose norm is denoted
$N_{\pi\times\pi^{\prime}}=N_{\widetilde{\pi}\times\widetilde{\pi}^{\prime}}$;
and

  4. (4)

a complex number $W(\pi\times\pi^{\prime})$ of modulus $1$

such that the Rankin–Selberg $L$-functions

$L(s,\pi\times\pi^{\prime})=\prod_{v\nmid\infty}L(s,\pi_{v}\times\pi_{v}^{\prime}),\qquad
L(s,\widetilde{\pi}\times\widetilde{\pi}^{\prime})=\prod_{v\nmid\infty}L(s,\widetilde{\pi}_{v}\times\widetilde{\pi}_{v}^{\prime})$

converge absolutely for $\mathrm{Re}(s)>1$, the completed $L$-functions

$\displaystyle\Lambda(s,\pi\times\pi^{\prime})$
$\displaystyle=(s(1-s))^{\delta_{\pi\times\pi^{\prime}}}(D_{F}^{n^{\prime}n}N_{\pi\times\pi^{\prime}})^{\frac{s}{2}}L(s,\pi\times\pi^{\prime})\prod_{v\mid\infty}L(s,\pi_{v}\times\pi_{v}^{\prime})$
$\displaystyle\Lambda(s,\widetilde{\pi}\times\widetilde{\pi}^{\prime})$
$\displaystyle=(s(1-s))^{\delta_{\pi\times\pi^{\prime}}}(D_{F}^{n^{\prime}n}N_{\pi\times\pi^{\prime}})^{\frac{s}{2}}L(s,\widetilde{\pi}\times\widetilde{\pi}^{\prime})\prod_{v\mid\infty}L(s,\widetilde{\pi}_{v}\times\widetilde{\pi}_{v}^{\prime})$

are entire of order $1$, and
$\Lambda(s,\pi\times\pi^{\prime})=W(\pi\times\pi^{\prime})\Lambda(1-s,\widetilde{\pi}\times\widetilde{\pi}^{\prime})$.

It follows from Theorem 3.1 that

(3.3)
$L(s,\widetilde{\pi}\times\widetilde{\pi}^{\prime})=\overline{L(\overline{s},\pi\times\pi)}.$

We can explicitly determine the numbers
$\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)$ at all $v\mid\infty$ (resp.
$\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(v)$ at all $v\in S_{\pi}\cup
S_{\pi^{\prime}}$) using the archimedean case of the local Langlands
correspondence [32, Section 3.1] (resp. [42, Appendix]). These descriptions
and (3.2) yield the bounds

(3.4) $|\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(v)|\leq
q_{v}^{\theta_{n}+\theta_{n^{\prime}}},\qquad\mathrm{Re}(\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v))\geq-(\theta_{n}+\theta_{n^{\prime}}).$

If $\ell\geq 1$ is an integer and $v\nmid\infty$, then we define

(3.5)
$a_{\pi\times\pi^{\prime}}(v^{\ell})=\begin{cases}a_{\pi}(v^{\ell})a_{\pi^{\prime}}(v^{\ell})&\mbox{if
$v\notin S_{\pi}\cup S_{\pi^{\prime}}$,}\\\
\sum_{j=1}^{n}\sum_{j^{\prime}=1}^{n^{\prime}}\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(v)^{\ell}&\mbox{if
$v\in S_{\pi}\cup S_{\pi^{\prime}}$,}\end{cases}\qquad
a_{\widetilde{\pi}\times\widetilde{\pi}^{\prime}}(v^{\ell})=\overline{a_{\pi\times\pi^{\prime}}(v^{\ell})}.$

We have the Dirichlet series identity

(3.6)
$-\frac{L^{\prime}}{L}(s,\pi\times\pi^{\prime})=\sum_{v\nmid\infty}\sum_{\ell=1}^{\infty}\frac{a_{\pi\times\pi^{\prime}}(v^{\ell})\log
q_{v}}{q_{v}^{\ell s}},\qquad\mathrm{Re}(s)>1.$

The numbers $a_{\pi\times\widetilde{\pi}}(v^{\ell})$ and
$a_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(v^{\ell})$ are non-negative
(see Lemma 3.3 below). By [42, Appendix], we have the bound

(3.7) $|a_{\pi\times\pi^{\prime}}(v^{\ell})|^{2}\leq
a_{\pi\times\widetilde{\pi}}(v^{\ell})a_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(v^{\ell}).$

Let $\pi=\pi^{*}\otimes|\cdot|^{it_{\pi}}\in\mathfrak{F}_{n}$,
$\pi^{\prime}={\pi^{\prime}}^{*}\otimes|\cdot|^{it_{\pi^{\prime}}}\in\mathfrak{F}_{n^{\prime}}$,
and $\chi=\chi^{*}\otimes|\cdot|^{it_{\chi}}\in\mathfrak{F}_{1}$. We define
the analytic conductor

$C(\pi\times\pi^{\prime})=D_{F}^{n^{\prime}n}N_{\pi^{*}\times{\pi^{*}}^{\prime}}\prod_{v\mid\infty}\prod_{j=1}^{n}\prod_{j=1}^{n^{\prime}}(|\mu_{j,j^{\prime},\pi^{*}\times{\pi^{*}}^{\prime}}(v)+i(t_{\pi}+t_{\pi^{\prime}})|+3)^{[F_{v}:\mathbb{Q}]}.$

Using the bound $N_{\pi^{*}\times{\pi^{\prime}}^{*}}\mid
N_{\pi^{*}}^{n^{\prime}}N_{{\pi^{\prime}}^{*}}^{n}$ [7, Theorem 2] and the
archimedean case of the local Langlands correspondence [32, Section 3] (see
also [18, Appendix]), we find that

(3.8) $\log C(\pi\times(\pi^{\prime}\otimes\chi))\ll n^{\prime}\log
C(\pi)+n\log C(\pi^{\prime})+n^{\prime}nC(\chi).$

######  Lemma 3.2.

Let $J\geq 1$. For $j\in\\{1,\ldots,J\\}$, let
$(\pi_{j},\pi_{j}^{\prime},\chi_{j})\in\mathfrak{F}_{n_{j}}\times\mathfrak{F}_{n_{j}^{\prime}}\times\mathfrak{F}_{1}$.
Define

$\mathfrak{Q}=\prod_{j=1}^{J}C(\pi_{j})^{n_{j}^{\prime}}C(\pi_{j}^{\prime})^{n_{j}}C(\chi)^{n_{j}^{\prime}n_{j}},\quad\mathfrak{S}=\bigcup_{j=1}^{J}(S_{\pi_{j}}\cup
S_{\pi_{j}^{\prime}}\cup S_{\chi}),\quad
D(s)=\prod_{j=1}^{J}L(s,\pi_{j}\times(\pi^{\prime}_{j}\otimes\chi_{j})).$

Assume that $D(s)$ is holomorphic on $\mathbb{C}-\\{1\\}$ with a pole of order
$r\geq 1$ at $s=1$. Write

(3.9)
$a_{D}(v^{\ell})=\sum_{j=1}^{J}a_{\pi_{j}\times(\pi_{j}^{\prime}\otimes\chi}(v^{\ell}),\qquad-\frac{D^{\prime}}{D}(s)=\sum_{v}\sum_{\ell=1}^{\infty}\frac{a_{D}(v^{\ell})\log
q_{v}}{q_{v}^{\ell s}}.$

Assume that if $\ell\geq 1$ and $v\notin\mathfrak{S}$, then
$\mathrm{Re}(a_{D}(v^{\ell}))\geq 0$. Let $Q\geq\mathfrak{Q}$ satisfy
$|\mathfrak{S}|\ll\log Q$. There exists an effectively computable constant
$\Cl[abcon]{GHL}=\Cr{GHL}(n_{1},\ldots,n_{J},n_{1}^{\prime},\ldots,n_{J}^{\prime})>0$
such that $D(\sigma)$ has at most $r$ zeros in the interval
$[1-\Cr{GHL}/(r\log Q),1)$ and no zeros in the interval $[1,\infty)$. If for
all $v\in\mathfrak{S}$ and $\ell\geq 1$, there holds
$\mathrm{Re}(a_{D}(v^{\ell}))\geq 0$ or
$|a_{D}(v^{\ell})|\leq\sum_{j=1}^{J}n_{j}^{\prime}n_{j}$, then the constant
$\Cr{GHL}$ is absolute.

###### Proof.

The proof is identical to [20, Lemma 5.9] except in how we handle the
contribution from the $a_{D}(v^{\ell})$ with $v\in\mathfrak{S}$. If for all
$v\in\mathfrak{S}$ and all $\ell\geq 1$ we have that
$\mathrm{Re}(a_{D}(v^{\ell}))\geq 0$ or
$|a_{D}(v^{\ell})|\leq\sum_{j=1}^{J}n_{j}^{\prime}n_{j}$, then their
contribution is $O(r\log Q)$. Otherwise, we use (3.4). ∎

###  3.3. Isobaric sums

Let $r\geq 1$ be an integer. For $1\leq j\leq r$, let
$\pi_{j}\in\mathfrak{F}_{d_{j}}$. Langlands associated to
$(\pi_{1},\ldots,\pi_{r})$ an automorphic representation of
$\mathrm{GL}_{d_{1}+\cdots+d_{r}}(\mathbb{A}_{F})$, the isobaric sum
$\Pi=\pi_{1}\boxplus\cdots\boxplus\pi_{r}$. Its $L$-function is
$L(s,\Pi)=\prod_{j=1}^{r}L(s,\pi_{j})$, and its contragredient is
$\widetilde{\pi}_{1}\boxplus\cdots\boxplus\widetilde{\pi}_{r}$. Let
$\mathfrak{A}_{n}$ be the set of isobaric automorphic representations of
$\mathrm{GL}_{n}(\mathbb{A}_{F})$. If
$\Pi=\pi_{1}\boxplus\cdots\boxplus\pi_{r}$ and
$\Pi^{\prime}=\pi_{1}^{\prime}\boxplus\cdots\boxplus\pi_{r^{\prime}}^{\prime}$,
then

$L(s,\Pi\times\Pi^{\prime})=\prod_{j=1}^{r}\prod_{k=1}^{r^{\prime}}L(s,\pi_{j}\times\pi_{k}^{\prime}).$

######  Lemma 3.3.

[17] If $\Pi\in\mathfrak{A}_{n}$, then
$-\frac{L^{\prime}}{L}(s,\Pi\times\widetilde{\Pi})$ has non-negative Dirichlet
coefficients.

###  3.4. Symmetric power lifts from $\mathrm{GL}_{2}$

Let $\pi\in\mathfrak{F}_{2}$. Define $\mathrm{Sym}^{0}(\pi)=\mathbbm{1}$ and
$\mathrm{Sym}^{1}(\pi)=\pi$. If is conjectured that if $m\geq 2$, then
$\mathrm{Sym}^{m}(\pi)\in\mathfrak{A}_{m+1}$ with contragredient
$\mathrm{Sym}^{m}(\widetilde{\pi})=\mathrm{Sym}^{m}(\pi)\otimes\overline{\omega}_{\pi}^{m}$.
This is known for $m\leq 4$ [11, 25, 23]. If
$\mathrm{Sym}^{m}(\pi)\in\mathfrak{A}_{m+1}$, then for each $v\in S_{\pi}$,
there exist complex numbers $(\alpha_{j,\mathrm{Sym}^{m}(\pi)}(v))_{j=0}^{m}$
such that

(3.10)
$L(s,\mathrm{Sym}^{m}(\pi_{v}))^{-1}=\begin{cases}\prod_{j=0}^{m}(1-\alpha_{1,\pi}(v)^{j}\alpha_{2,\pi}(v)^{m-j}q_{v}^{-s})&\mbox{if
$v\nmid\infty$ and $v\notin S_{\pi}$,}\\\
\prod_{j=0}^{m}(1-\alpha_{j,\mathrm{Sym}^{m}(\pi)}(v)q_{v}^{-s})&\mbox{if
$v\nmid\infty$ and $v\in S_{\pi}$.}\end{cases}$

######  Theorem 3.4 ([35]).

If $F$ is totally real, $j\geq 0$, and $\pi\in\mathrm{RA}_{F}$, then
$\mathrm{Sym}^{j}(\pi)\in\mathfrak{F}_{j+1}$.

######  Lemma 3.5.

Let $j,k,u\geq 0$ be integers, $\pi\in\mathfrak{F}_{2}$, and
$\psi_{1},\psi_{2},\chi\in\mathfrak{F}_{1}$. If
$\mathrm{Sym}^{u}(\pi)\in\mathfrak{F}_{u+1}$ for
$u\in\\{j,k\\}\cup\\{j+k-2r\colon 0\leq r\leq\min\\{j,k\\}\\}$ and

$\mathrm{Sym}^{j}(\pi\otimes\psi_{1})\boxtimes\mathrm{Sym}^{k}(\pi\otimes\psi_{2})\otimes\chi=\mathop{\boxplus}_{r=0}^{\min\\{j,k\\}}\mathrm{Sym}^{j+k-2r}(\pi)\otimes\chi\psi_{1}^{j}\psi_{2}^{k}\omega_{\pi}^{r}\in\mathfrak{A}_{(j+1)(k+1)},$

then

$L(s,\mathrm{Sym}^{j}(\pi\otimes\psi_{1})\boxtimes(\mathrm{Sym}^{k}(\pi\otimes\psi_{2})\otimes\chi))=L(s,\mathrm{Sym}^{j}(\pi\otimes\psi_{2})\times(\mathrm{Sym}^{k}(\pi\otimes\psi_{2})\otimes\chi)).$

In particular, if $v\nmid\infty$, then the following identities hold:

(3.11) $\displaystyle
a_{\mathrm{Sym}^{j}(\pi)\times(\mathrm{Sym}^{k}(\pi)\otimes\chi)}(v^{\ell})$
$\displaystyle=\sum_{r=0}^{\min\\{j,k\\}}a_{\mathrm{Sym}^{j+k-2r}(\pi)\otimes\chi\omega_{\pi}^{r}}(v^{\ell}),$
(3.12) $\displaystyle
a_{\mathrm{Sym}^{j}(\pi)\times\mathrm{Sym}^{j}(\widetilde{\pi})}(v^{\ell})$
$\displaystyle=1+\sum_{r=1}^{j}a_{\mathrm{Sym}^{2r}(\pi)\otimes\omega_{\pi}^{-r}}(v^{\ell}).$

###### Proof.

These follow from the Clebsch–Gordan identities. ∎

######  Lemma 3.6.

Let $F$ be totally real, let $\pi\in\mathrm{RA}_{F}$, and let $y_{\pi}$ be as
in (1.3). If $m\geq 1$, then $S_{\mathrm{Sym}^{m}(\pi)}\subseteq S_{\pi}$ and
$\log C(\mathrm{Sym}^{m}(\pi))\ll m\log(mC(\pi))+m^{1+y_{\pi}}\log N_{\pi}$.

###### Proof.

Write the analytic conductor $C(\mathrm{Sym}^{m}(\pi))$ as
$D_{F}^{m+1}N_{\mathrm{Sym}^{m}(\pi)}K_{\mathrm{Sym}^{m}(\pi)}$. The bound
$\log K_{\mathrm{Sym}^{m}(\pi)}\ll m\log(mK_{\pi})$ follows from the explicit
description of $\pi_{v}$ at $v\mid\infty$ given by Moreno and Shahidi [31]. If
$y_{\pi}=0$, then the bound $\log N_{\mathrm{Sym}^{m}(\pi)}\ll m\log N_{\pi}$
follows from the local computations in [9, Section 3] and [10, Appendix].
Otherwise, for $p$ a rational prime, it suffices to bound
$\mathrm{ord}_{p}(N_{\mathrm{Sym}^{m}(\pi)})$. By Lemma 3.5 and the identity
$\widetilde{\pi}=\pi\otimes\overline{\omega}_{\pi}$, the relation
$L(s,\mathrm{Sym}^{m+1}(\pi)\times\widetilde{\pi})=L(s,\mathrm{Sym}^{m+2}(\pi)\otimes\overline{\omega}_{\pi})L(s,\mathrm{Sym}^{m}(\pi))$
holds. Also, we have that
$\mathrm{Sym}^{m+2}(\pi)=(\mathrm{Sym}^{m+2}(\pi)\otimes\overline{\omega}_{\pi})\otimes\omega_{\pi}$.
Bushnell and Henniart [7] proved that if $\pi\in\mathfrak{F}_{n}$ and
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$, then
$N_{\pi\times\pi^{\prime}}\mid N_{\pi}^{n^{\prime}}N_{\pi^{\prime}}^{n}$. It
follows that

$N_{\mathrm{Sym}^{m+2}(\pi)}\mid
N_{\omega_{\pi}}^{m+3}N_{\mathrm{Sym}^{m+2}(\pi)\otimes\overline{\omega}_{\pi}}\qquad\textup{and}\qquad
N_{\mathrm{Sym}^{m+2}(\pi)\otimes\overline{\omega}_{\pi}}N_{\mathrm{Sym}^{m}(\pi)}\mid
N_{\mathrm{Sym}^{m+1}(\pi)}^{2}N_{\pi}^{m+2}.$

Since $N_{\omega_{\pi}}\mid N_{\pi}$ [40, Theorem A], the bound

$\mathrm{ord}_{p}(N_{\mathrm{Sym}^{m+2}(\pi)})\leq
2\mathrm{ord}_{p}(N_{\mathrm{Sym}^{m+1}(\pi)})-\mathrm{ord}_{p}(N_{\mathrm{Sym}^{m}(\pi)})+(2m+5)\mathrm{ord}_{p}(N_{\pi})$

holds. By induction on $m$, we conclude that
$\mathrm{ord}_{p}(N_{\mathrm{Sym}^{m}(\pi)})\leq
m^{3}\mathrm{ord}_{p}(N_{\pi})$, as desired. ∎

######  Lemma 3.7.

Let $F$ be totally real, and let $\ell\geq 1$ and $m,n\geq 0$ be integers. Let
$\pi,\pi^{\prime}\in\mathrm{RA}_{F}$. If $v\nmid\infty$, then
$|a_{\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi)}(v^{\ell})|\leq(m+1)(n+1)$.

###### Proof.

Using Theorem 3.4, we proceed as in [1, Corollary 7.1.15] and conclude that at
every place $v\nmid\infty$ and every $\ell\geq 1$, we have that if $j,k\geq 0$
are integers, then

(3.13) $|a_{\mathrm{Sym}^{j}(\pi)}(v^{\ell})|\leq
j+1,\qquad|a_{\mathrm{Sym}^{k}(\pi^{\prime})}(v^{\ell})|\leq k+1.$

Note that
$a_{\mathrm{Sym}^{m}(\pi)\times\mathrm{Sym}^{m}(\widetilde{\pi})}(v^{\ell})\geq
0$ and
$a_{\mathrm{Sym}^{n}(\pi^{\prime})\times\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})}(v^{\ell})\geq
0$ by Lemma 3.3, and

$|a_{\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi)\otimes\chi)}(v^{\ell})|^{2}\leq
a_{\mathrm{Sym}^{m}(\pi)\times\mathrm{Sym}^{m}(\widetilde{\pi})}(v^{\ell})a_{\mathrm{Sym}^{n}(\pi^{\prime})\times\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})}(v^{\ell})$

by (3.7). Combining Lemma 3.5 and (3.13), we find that

$a_{\mathrm{Sym}^{m}(\pi)\times\mathrm{Sym}^{m}(\widetilde{\pi})}(v^{\ell})\leq(m+1)^{2},\qquad
a_{\mathrm{Sym}^{n}(\pi^{\prime})\times\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})}(v^{\ell})\leq(n+1)^{2}.$

The lemma now follows. ∎

###  3.5. The symmetric square lift from $\mathrm{GL}_{n}$

Let $(\pi,\chi)\in\mathfrak{F}_{n}\times\mathfrak{F}_{1}$. Let $S$ be a set of
places containing $S_{\pi}^{\infty}\cup S_{\chi}^{\infty}$ and all places
dividing $2$. The $\chi$-twist of the partial $L$-function of the symmetric
square representation
$\mathrm{Sym}^{2}\colon\mathrm{GL}_{n}(\mathbb{C})\to\mathrm{GL}_{n(n+1)/2}(\mathbb{C})$
is

(3.14) $L^{S}(s,\pi;\mathrm{Sym}^{2}\otimes\chi)=\prod_{v\notin
S}~{}\prod_{1\leq j\leq k\leq
n}\frac{1}{1-\chi_{v}(\varpi_{v})\alpha_{j,\pi}(v)\alpha_{k,\pi}(v)q_{v}^{-s}}.$

######  Theorem 3.8.

[43, Theorem 7.1] Let $(\pi,\chi)\in\mathfrak{F}_{n}\times\mathfrak{F}_{1}$.
If $S$ is a set of places containing $S_{\pi}^{\infty}\cup S_{\chi}^{\infty}$
and all places dividing $2$, then $L^{S}(s,\pi;\mathrm{Sym}^{2}\otimes\chi)$
is holomorphic on $\mathbb{C}$ except possibly for simple poles at
$s\in\\{0,1\\}$. If $\chi^{n}\omega_{\pi}^{2}\neq\mathbbm{1}$, then there is
no pole.

##  4\. Preliminaries for the proof of Theorem 1.2

Let $(\pi_{1},\pi_{2})\in\mathfrak{F}_{n_{1}}\times\mathfrak{F}_{n_{2}}$.
Assume that $L(s,\pi_{1}\times\pi_{2})$ is an irreducible $L$-function that is
not the $L$-function of a real-valued ray class character. The only known
strategy to prove that $L(\sigma,\pi_{1}\times\pi_{2})$ has no exceptional
zero is to construct a Dirichlet series $D(s)$, integers
$\ell_{1},\ell_{2}\geq 0$ and $k\geq 1$ satisfying $\ell_{1}+\ell_{2}>k$, and
a fixed $t\in(0,1)$ such that

  1. (1)

at each unramified $v\nmid\infty$ and $\ell\geq 1$, we have
$\mathrm{Re}(a_{D}(v^{\ell}))\geq 0$ (see (3.9)),

  2. (2)

$D(s)$ is holomorphic everywhere except for a pole of order $k$ at $s=1$, and

  3. (3)

$D(\sigma)(\sigma-1)^{k}L(\sigma,\pi_{1}\times\pi_{2})^{-\ell_{1}}L(\sigma,\widetilde{\pi}_{1}\times\widetilde{\pi}_{2})^{-\ell_{2}}$
is holomorphic at each $t<\sigma\leq 1$.

A real zero of $L(s,\pi_{1}\times\pi_{2})$ is a zero of $D(s)$ with
multiplicity at least $\ell_{1}+\ell_{2}>k$, so the existence of a exceptional
zero of $L(s,\pi_{1}\times\pi_{2})$ contradicts Lemma 3.2 applied to $D(s)$.

###  4.1. Earlier work

First, let $\pi_{1}\in\cup_{n=2}^{\infty}\mathfrak{F}_{n}$ and
$\pi_{2}=\mathbbm{1}$. Hoffstein and Ramakrishnan [17, Proof of Theorem B]
showed that sufficient progress towards the modularity of Rankin–Selberg
$L$-functions suffices to prove that $L(s,\pi_{1})$ has no exceptional zero.
Assume that $L(s,\pi_{1}\times\widetilde{\pi}_{1})$ is modular so that there
exists an isobaric automorphic representation
$\pi_{1}\boxtimes\widetilde{\pi}_{1}$ such that
$L(s,\pi_{1}\times\widetilde{\pi}_{1})=L(s,\pi_{1}\boxtimes\widetilde{\pi}_{1})$.
By [17, Lemma 4.4], there exists a cuspidal constituent
$\tau\notin\\{\mathbbm{1},\pi_{1},\widetilde{\pi}_{1}\\}$ of
$\pi_{1}\boxtimes\widetilde{\pi}_{1}$. If $L(s,\pi_{1}\times\tau)$ is modular,
then by [16, Proofs of Lemma 4.4 and Claim 4.5],
$L(s,\pi_{1}\times\tau)/L(s,\pi_{1})$ is entire. Therefore, subject to the
modularity of $L(s,\pi_{1}\times\widetilde{\pi}_{1})$ and
$L(s,\pi_{1}\times\tau)$, Hoffstein and Ramakrishnan prove that if
$\Pi=\mathbbm{1}\boxplus\widetilde{\tau}\boxplus\pi_{1}$, then
$D(s)=L(s,\Pi\times\widetilde{\Pi})$ satisfies the above criteria (1)–(3).

Proposition 1.1(ii–vi) follow by proving the existence of
$\pi_{1}\boxtimes\widetilde{\pi}_{1}$ and the existence of a cuspidal
constituent $\tau$ of $\pi_{1}\boxtimes\widetilde{\pi}_{1}$ such that there
exists an effectively computable constant $\Cl[abcon]{HR}=\Cr{HR}(n)\in(0,1)$
such that $L(\sigma,\pi_{1}\times\tau)/L(\sigma,\pi_{1})$ is holomorphic at
each $\sigma\in(\Cr{HR},1)$. The holomorphy condition is most complicated in
the proof of part (iv), where $\pi\in\mathfrak{F}_{2}$ is self-dual and not
dihedral or tetrahedral or octahedral, $\pi_{1}=\mathrm{Sym}^{4}(\pi)$,
$\tau=\mathrm{Sym}^{2}(\pi)$, and
$L(\sigma,\pi_{1}\times\tau)/L(\sigma,\pi_{1})=L(\sigma,\mathrm{Sym}^{3}(\pi);\mathrm{Sym}^{2})$.
Bump and Ginzburg [6] proved that this is holomorphic at each
$\sigma\in(1/2,1)$. Proposition 1.1(i,vii,viii,xi) are proved by constructing
a $D(s)$ satisfying (1)–(3) above using hypotheses related to self-duality or
twist-equivalence.

Let $F$ be totally real, $\pi,\pi^{\prime}\in\mathrm{RA}_{F}$,
$\chi\in\mathfrak{F}_{1}$, and $\pi\not\sim\pi^{\prime}$. Let
$\pi_{1}=\mathrm{Sym}^{m}(\pi)$ and
$\pi_{2}=\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi$. By Theorems 3.4 and 3.5,
$L(s,\pi_{1}\times\widetilde{\pi}_{1})$ and
$L(s,\pi_{2}\times\widetilde{\pi}_{2})$ are modular. By Lemma 3.5 again, if
$\tau=\mathrm{Ad}(\pi)$, then $L(s,\pi_{1}\times\tau)/L(s,\pi_{1})$ is entire.
If there exists $\pi_{0}\in\mathfrak{F}_{(m+1)(n+1)}$ such that such that
$L(s,\pi_{1}\times\pi_{2})=L(s,\pi_{0})$, then we can apply the
Hoffstein–Ramakrishnan strategy to
$\Pi=\mathbbm{1}\boxplus\mathrm{Ad}(\pi)\boxplus\pi_{0}$ and
$D(s)=L(s,\Pi\times\widetilde{\Pi})$. No such $\pi_{0}$ is known to exist yet
for $m,n\geq 2$. Even if such a $\pi_{0}$ exists, the zero-free region that
follows from Lemma 3.2 applied to $D(s)$ is insufficient for Theorem 2.1(2).

###  4.2. Ideas for our proofs

Let $F$ be totally real and $m,n\geq 0$ satisfy $\max\\{m,n\\}\geq 1$. Let
$\pi,\pi^{\prime}\in\mathrm{RA}_{F}$ and
$\chi=\chi^{*}|\cdot|^{it_{\chi}}\in\mathfrak{F}_{1}$. Suppose that
$\pi\not\sim\pi^{\prime}$. If $j<0$ or $k<0$, and $\xi\in\mathfrak{F}_{1}$,
then
$L(s,\mathrm{Sym}^{j}(\pi)\times(\mathrm{Sym}^{k}(\pi^{\prime})\otimes\xi))$
is the constant function $1$. We introduce

(4.1) $\displaystyle\mathcal{D}(s)$
$\displaystyle=L(s,\mathrm{Sym}^{m}(\pi)\times\mathrm{Sym}^{m}(\widetilde{\pi}))^{4}L(s,\mathrm{Sym}^{n}(\pi^{\prime})\times\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime}))^{2}$
$\displaystyle\cdot
L(s,\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi))^{4}L(s,\mathrm{Sym}^{m}(\widetilde{\pi})\times(\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})\otimes\overline{\chi}))^{4}$
$\displaystyle\cdot
L(s,\mathrm{Sym}^{m-2}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi\omega_{\pi}))^{2}L(s,\mathrm{Sym}^{m-2}(\widetilde{\pi})\times(\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})\otimes\overline{\chi}\,\overline{\omega}_{\pi}))^{2}$
$\displaystyle\cdot
L(s,\mathrm{Sym}^{m+2}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi\overline{\omega}_{\pi}))^{2}L(s,\mathrm{Sym}^{m+2}(\widetilde{\pi})\times(\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})\otimes\overline{\chi}\omega_{\pi}))^{2}$
$\displaystyle\cdot
L(s,\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2})L(s,\mathrm{Ad}(\pi))^{3}$
$\displaystyle\cdot\prod_{k=1}^{n}[L(s,\mathrm{Ad}(\pi)\times(\mathrm{Sym}^{2k}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{k}))^{3}L(s,\mathrm{Sym}^{4}(\pi)\times(\mathrm{Sym}^{2k}(\pi^{\prime})\otimes\overline{\omega}_{\pi}^{2}\overline{\omega}_{\pi^{\prime}}^{k}))].$

The Euler product that defines $\mathcal{D}(s)$ for $\mathrm{Re}(s)>1$ has
degree $(2m+4n+6)^{2}$.

####  4.2.1. Non-negativity of Dirichlet coefficients

Unlike in [2, 16, 17, 27, 39, 44], when $m\geq 2$ and $n\geq 2$, it is not yet
known whether there exists an isobaric automorphic representation $\Pi$ such
that $\mathcal{D}(s)=L(s,\Pi\times\widetilde{\Pi})$. Therefore, we cannot use
Lemma 3.3 to establish the non-negativity of the Dirichlet coefficients
$a_{\mathcal{D}}(v^{\ell})\log q_{v}$ of $-(\mathcal{D}^{\prime}/D)(s)$
(recall (3.9)).

######  Lemma 4.1.

Let $\pi,\pi^{\prime}\in\mathrm{RA}_{F}$ and $\chi\in\mathfrak{F}_{1}$. If
$v\notin S_{\pi}\cup S_{\pi^{\prime}}\cup S_{\chi}$, then
$a_{\mathcal{D}}(v^{\ell})\geq 0$.

###### Proof.

Fix $\ell\geq 1$ and a place $v\nmid\infty$ of $F$ such that $v\notin
S_{\pi}\cup S_{\pi^{\prime}}\cup S_{\chi}$ We use (3.5) throughout the proof
without further mention. Since $v$ and $\ell$ are fixed, we suppress
$(v^{\ell})$ throughout (e.g., $a_{\mathcal{D}}=a_{\mathcal{D}}(v^{\ell})$ and
$a_{\mathrm{Ad}(\pi)}=a_{\mathrm{Ad}(\pi)}(v^{\ell})$). A direct calculation
shows that

(4.2) $\displaystyle a_{\mathcal{D}}$
$\displaystyle=4a_{\mathrm{Sym}^{m}(\pi)\times\mathrm{Sym}^{m}(\widetilde{\pi})}+2a_{\mathrm{Sym}^{n}(\pi^{\prime})\times\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})}+4a_{\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi)}$
$\displaystyle\qquad+4a_{\mathrm{Sym}^{m}(\widetilde{\pi})\times(\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})\otimes\overline{\chi})}+2a_{\mathrm{Sym}^{m-2}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi\omega_{\pi})}$
$\displaystyle\qquad+2a_{\mathrm{Sym}^{m-2}(\widetilde{\pi})\times(\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})\otimes\overline{\chi}\,\overline{\omega}_{\pi})}+2a_{\mathrm{Sym}^{m+2}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi\overline{\omega}_{\pi})}$
$\displaystyle\qquad+2a_{\mathrm{Sym}^{m+2}(\widetilde{\pi})\times(\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})\otimes\overline{\chi}\omega_{\pi})}+a_{\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}}+3a_{\mathrm{Ad}(\pi)}$
$\displaystyle\qquad+\sum_{k=1}^{n}[3a_{\mathrm{Ad}(\pi)\times(\mathrm{Sym}^{2k}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{k})}+a_{\mathrm{Sym}^{4}(\pi)\times(\mathrm{Sym}^{2k}(\pi^{\prime})\otimes\overline{\omega}_{\pi}^{2}\overline{\omega}_{\pi^{\prime}}^{k})}].$

By Lemma 3.5, we find that
$(a_{\mathrm{Ad}(\pi)}+1)^{2}=2+3a_{\mathrm{Ad}(\pi)}+a_{\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}}$
and

(4.3) $\displaystyle
2a_{\mathrm{Sym}^{n}(\pi^{\prime})\times\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})}+3a_{\mathrm{Ad}(\pi)}+a_{\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}}$
$\displaystyle\qquad+\sum_{k=1}^{n}[3a_{\mathrm{Ad}(\pi)\times(\mathrm{Sym}^{2k}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{k})}+a_{\mathrm{Sym}^{4}(\pi)\times(\mathrm{Sym}^{2k}(\pi^{\prime})\otimes\overline{\omega}_{\pi}^{2}\overline{\omega}_{\pi^{\prime}}^{k})}]$
$\displaystyle=2a_{\mathrm{Sym}^{n}(\pi^{\prime})\times\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})}+(3a_{\mathrm{Ad}(\pi)}+a_{\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}})\sum_{k=0}^{n}a_{\mathrm{Sym}^{2k}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{k}}$
$\displaystyle=2a_{\mathrm{Sym}^{n}(\pi^{\prime})\times\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})}+(3a_{\mathrm{Ad}(\pi)}+a_{\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}})a_{\mathrm{Sym}^{n}(\pi^{\prime})\times\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})}$
$\displaystyle=(a_{\mathrm{Ad}(\pi)}+1)^{2}|a_{\mathrm{Sym}^{n}(\pi^{\prime})}|^{2}.$

By Lemma 3.5 again, we find that

$a_{\mathrm{Ad}(\pi)\times\mathrm{Sym}^{m}(\pi)}=a_{\mathrm{Sym}^{m-2}(\pi)\otimes\omega_{\pi}}+a_{\mathrm{Sym}^{m}(\pi)}+a_{\mathrm{Sym}^{m+2}(\pi)\otimes\overline{\omega}_{\pi}},$

hence

$\displaystyle
2a_{\mathrm{Sym}^{m-2}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\omega_{\pi}\chi)}+2a_{\mathrm{Sym}^{m+2}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\overline{\omega}_{\pi}\chi)}+4a_{\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi)}$
$\displaystyle=2(a_{\mathrm{Sym}^{m-2}(\pi)\otimes\omega_{\pi}}+a_{\mathrm{Sym}^{m}(\pi)}+a_{\mathrm{Sym}^{m+2}(\pi)\otimes\overline{\omega}_{\pi}})a_{\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi}+2a_{\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi)}$
$\displaystyle=2a_{\mathrm{Ad}(\pi)\times\mathrm{Sym}^{m}(\pi)\otimes\chi}a_{\mathrm{Sym}^{n}(\pi^{\prime})}+2a_{\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi)}$
$\displaystyle=2a_{\mathrm{Ad}(\pi)}a_{\mathrm{Sym}^{m}(\pi)}a_{\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi}+2a_{\mathrm{Sym}^{m}(\pi)}a_{\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi}$
$\displaystyle=2(a_{\mathrm{Ad}(\pi)}+1)a_{\mathrm{Sym}^{m}(\pi)}a_{\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi}.$

A dual calculation shows that

$\displaystyle
2a_{\mathrm{Sym}^{m-2}(\widetilde{\pi})\times(\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})\otimes\overline{\omega}_{\pi}\overline{\chi})}+2a_{\mathrm{Sym}^{m+2}(\widetilde{\pi})\times(\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})\otimes\omega_{\pi}\overline{\chi})}+4a_{\mathrm{Sym}^{m}(\widetilde{\pi})\times(\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})\otimes\overline{\chi})}$
$\displaystyle=2(a_{\mathrm{Ad}(\widetilde{\pi})}+1)\overline{a_{\mathrm{Sym}^{m}(\pi)}a_{\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi}}.$

Since $\mathrm{Ad}(\pi)$ is self-dual, we have that
$\mathrm{Ad}(\widetilde{\pi})=\mathrm{Ad}(\pi)$,
$a_{\mathrm{Ad}(\pi)}\in\mathbb{R}$, and

(4.4) $\displaystyle
2a_{\mathrm{Sym}^{m-2}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\omega_{\pi}\chi)}+2a_{\mathrm{Sym}^{m-2}(\widetilde{\pi})\times(\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})\otimes\overline{\omega}_{\pi}\overline{\chi})}$
$\displaystyle\qquad+4a_{\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi)}+4a_{\mathrm{Sym}^{m}(\widetilde{\pi})\times(\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})\otimes\overline{\chi})}$
$\displaystyle\qquad+2a_{\mathrm{Sym}^{m+2}(\widetilde{\pi})\times(\mathrm{Sym}^{n}(\widetilde{\pi}^{\prime})\otimes\omega_{\pi}\overline{\chi})}+2a_{\mathrm{Sym}^{m+2}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\overline{\omega}_{\pi}\chi)}$
$\displaystyle=4\mathrm{Re}((a_{\mathrm{Ad}(\pi)}+1)a_{\mathrm{Sym}^{m}(\pi)}a_{\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi}).$

Inserting (4.3) and (4.4) into (4.2), we find that $a_{\mathcal{D}}$ equals

$\displaystyle=4a_{\mathrm{Sym}^{m}(\pi)\times\mathrm{Sym}^{m}(\widetilde{\pi})}+(a_{\mathrm{Ad}(\pi)}+1)^{2}|a_{\mathrm{Sym}^{n}(\pi^{\prime})}|^{2}+4\mathrm{Re}((a_{\mathrm{Ad}(\pi)}+1)a_{\mathrm{Sym}^{m}(\pi)}a_{\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi})$
$\displaystyle=4|a_{\mathrm{Sym}^{m}(\pi)}|^{2}+(a_{\mathrm{Ad}(\pi)}+1)^{2}|a_{\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi}|^{2}+4\mathrm{Re}((a_{\mathrm{Ad}(\pi)}+1)a_{\mathrm{Sym}^{m}(\pi)}a_{\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi}).$

This factors as
$|2\overline{a_{\mathrm{Sym}^{m}(\pi)}}+a_{\mathrm{Ad}(\pi)}a_{\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi}+a_{\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi}|^{2}\geq
0$. ∎

####  4.2.2. Order the of pole at $s=1$

Unlike in [2, 16, 17, 27, 39, 44], $\mathcal{D}(s)$ has several factors for
which the basic theory of Rankin–Selberg $L$-functions cannot determine
whether a pole exists. For certain choices of $m$ and $n$, there might be
factors in $\mathcal{D}(s)$ of the form
$L(s,\mathrm{Sym}^{j}(\pi)\times(\mathrm{Sym}^{j}(\pi^{\prime})\otimes\xi))$,
where $\xi=\xi^{*}\otimes|\cdot|^{it_{\xi}}\in\mathfrak{F}_{1}$. When do these
factors have poles? In order to answer this question, we rely on the fact that
$\pi,\pi^{\prime}\in\mathrm{RA}_{F}$, in which case $\pi$ and $\pi^{\prime}$
have associated $\ell$-adic representations for each prime $\ell$. In order to
use these representations, we must restrict $\xi^{*}$ to be associated to a
ray class character, so that its image has finite order. For a set $U$ of
places of $F$, let $\delta(U)$ (resp. $\overline{\delta}(U)$) denote its
Dirichlet density (resp. its upper Dirichlet density).

######  Lemma 4.2.

Let $F$ be totally real, $\pi,\pi^{\prime}\in\mathrm{RA}_{F}$, and
$\xi\in\mathfrak{F}_{1}$ with $\xi^{*}$ corresponding with a ray class
character. Let $j\geq 2$. If $\pi\not\sim\pi^{\prime}$, then
$L(s,\mathrm{Sym}^{j}(\pi)\times(\mathrm{Sym}^{j}(\pi^{\prime})\otimes\xi))$
is entire.

###### Proof.

We prove the contrapositive. Let $F$ be totally real, $j\geq 2$,
$\pi,\pi^{\prime}\in\mathrm{RA}_{F}$, and
$\xi=\xi^{*}|\cdot|^{it_{\xi}}\in\mathfrak{F}_{1}$ with $\xi^{*}$
corresponding with a ray class character. It suffices to let
$\pi,\pi^{\prime}\in\mathrm{RA}_{F}\cap\mathfrak{F}_{2}^{*}$ and $t_{\xi}=0$.
Now, $\omega_{\pi}$ and $\omega_{\pi^{\prime}}$ also correspond with ray class
characters. We have that
$\mathrm{Sym}^{j}(\pi),\mathrm{Sym}^{j}(\pi^{\prime})\otimes\xi\in\mathfrak{F}_{j+1}$
by Theorem 3.4. If
$L(s,\mathrm{Sym}^{j}(\pi)\times(\mathrm{Sym}^{j}(\pi^{\prime})\otimes\xi))$
has a pole, then
$\mathrm{Sym}^{j}(\pi)=\mathrm{Sym}^{j}(\widetilde{\pi}^{\prime})\otimes\overline{\xi}$.
Since
$\widetilde{\pi}^{\prime}=\pi^{\prime}\otimes\overline{\omega}_{\pi^{\prime}}$
and
$\mathrm{Sym}^{j}(\widetilde{\pi}^{\prime})=\mathrm{Sym}^{j}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{j}$,
it follows that
$\mathrm{Sym}^{j}(\pi)=\mathrm{Sym}^{j}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{j}\overline{\xi}$
and
$\overline{\delta}(\\{v\nmid\infty\colon\mathrm{Sym}^{j}(\pi_{v})=\mathrm{Sym}^{j}(\pi_{v}^{\prime})\otimes(\overline{\omega}_{\pi^{\prime}}^{j}\overline{\xi})_{v}\\})=1$.
By the Chebotarev density theorem,
$\delta(\\{v\nmid\infty\colon\textup{$(\overline{\omega}_{\pi^{\prime}}^{j}\overline{\xi})_{v}$
is trivial}\\})>0$, so

(4.5)
$\overline{\delta}(\\{v\nmid\infty\colon\mathrm{Sym}^{j}(\pi_{v})=\mathrm{Sym}^{j}(\pi_{v}^{\prime})\\})>0.$

Since $\pi\in\mathrm{RA}_{F}$, if $v\nmid\infty$ and $\ell$ is the residue
characteristic of $F_{v}$, then there corresponds with $\pi_{v}$ a continuous
$\ell$-adic representation
$\rho_{\ell,\pi}\colon\mathrm{Gal}(\overline{F}/F)\to\mathrm{GL}_{2}(F_{v})$
such that if $v\notin S_{\pi}$ and $\sigma_{v}$ denotes the Frobenius
conjugacy class at $v$ in $\mathrm{Gal}(\overline{F}/F)/\ker\rho_{\ell,\pi}$,
then $\mathrm{Tr}\,\rho_{\ell,\pi}(\sigma_{v})=a_{\pi}(v)$ and
$\det\rho_{\ell,\pi}(\sigma_{v})=\omega_{\pi}(\varpi_{v})$. Consequently,
(4.5) implies that

(4.6)
$\overline{\delta}(\\{v\nmid\infty\colon\mathrm{Tr}((\mathrm{Sym}^{j}\circ\rho_{\ell,\pi})(\sigma_{v}))=\mathrm{Tr}((\mathrm{Sym}^{j}\circ\rho_{\ell,\pi^{\prime}})(\sigma_{v}))\\})>0.$

By the work of Rajan [36, Corollary 5.1 and Remark 5.1], (4.6) implies that
$\pi\sim\pi^{\prime}$. ∎

##  5\. Proof of Theorem 1.2

###  5.1. Case 1: $\pi\not\sim\pi^{\prime}$

Let $m,n\geq 0$, $M=\max\\{m,n\\}\geq 1$, and
$\chi=\chi^{*}|\cdot|^{it_{\chi}}\in\mathfrak{F}_{1}$. Assume that $\chi^{*}$
corresponds with a ray class character. Let $y_{\pi}$ and $y_{\pi^{\prime}}$
be as in (1.3), and let $Y=\max\\{y_{\pi},y_{\pi^{\prime}}\\}$. Consider
$\mathcal{D}(s)$ in (4.1). By Theorem 3.4, the hypotheses of Lemma 3.6 are
satisfied. By Lemmata 3.6, 3.7, and 4.1, the Dirichlet coefficients
$a_{\mathcal{D}}(v^{\ell})$ of $-(\mathcal{D}^{\prime}/\mathcal{D})(s)$ are
non-negative when $v\notin S_{\pi}\cup S_{\pi^{\prime}}\cup S_{\chi}$, and
$|a_{\mathcal{D}}(v^{\ell})|\leq(m+4n+5)^{2}$ otherwise. By Lemma 4.2, we find
that the only pole of $\mathcal{D}(s)$ is at $s=1$, with order 3. By (3.8) and
Lemmata 3.2 and 3.6, there exists an absolute, effectively computable constant
$\Cl[abcon]{GHL2}>0$ such that $\mathcal{D}(s)$ has no real zeros in the
interval $[1,\infty)$ and at most 3 real zeros in

$I=[1-\Cr{GHL2}/\\{M^{2+Y}\log(MC(\pi)C(\pi^{\prime}))+M^{2}\log
C(\chi)\\},1).$

If
$L(s,\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi))$
has a zero in $I$, then by (3.3) and (4.1), $\mathcal{D}(s)$ has a zero in $I$
with order at least 4, a contradiction. Theorem 1.2 follows.

If $n=0$, then
$L(s,\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi))=L(s,\mathrm{Sym}^{m}(\pi)\otimes\chi)$,
and we do not need to invoke Lemma 4.2. Therefore, we do not need to restrict
$\chi^{*}\in\mathfrak{F}_{1}$ to correspond with a ray class character. Thus,
the proof in the preceding paragraph shows that there exists an absolute,
effectively computable constant $\Cl[abcon]{GHL22}>0$ such that if
$\xi\in\mathfrak{F}_{1}$, then

(5.1) $L(\sigma,\mathrm{Sym}^{m}(\pi)\otimes\xi)\neq 0,\qquad\sigma\geq
1-\Cr{GHL22}/(m^{2+y_{\pi}}\log(mC(\pi))+m^{2}\log C(\xi)).$

###  5.2. Case 2: $\pi\sim\pi^{\prime}$

If $m,n\geq 1$ and $\psi\in\mathfrak{F}_{1}$ satisfies
$\pi^{\prime}=\pi\otimes\psi$, then by Lemma 3.5,

$L(\sigma,\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi))=\prod_{r=0}^{\min\\{m,n\\}}L(\sigma,\mathrm{Sym}^{m+n-2r}(\pi)\otimes\chi\psi^{n}\omega_{\pi}^{r}).$

By (5.1), each factor is non-zero in $I$ except possibly when $m=n=r$. The
corresponding factor is $L(s,\chi\psi^{m}\omega_{\pi}^{m})$. In this final
case, Theorem 1.2 follows.

##  6\. Proof of Theorem 1.4: non-dihedral and twist-inequivalent cases

Define $\mathfrak{F}_{2}^{\flat}=\\{\pi\in\mathfrak{F}_{2}\colon\textup{$\pi$
non-dihedral}\\}$. We require the following result.

######  Lemma 6.1.

[11, 23, 24, 25, 37]

  1. (i)

If $\pi\in\mathfrak{F}_{2}$, then $\mathrm{Ad}(\pi)\in\mathfrak{A}_{3}$,
$\mathrm{Sym}^{3}(\pi)\in\mathfrak{A}_{4}$, and
$\mathrm{Sym}^{4}(\pi)\in\mathfrak{A}_{5}$.

  2. (ii)

If $\pi\in\mathfrak{F}_{2}$, then $\mathrm{Ad}(\pi)\notin\mathfrak{F}_{3}$ if
and only if $\pi\notin\mathfrak{F}_{2}^{\flat}$, in which case there exists an
idele class character $\xi_{\pi}$ of a quadratic extension $K_{\pi}/F$ such
that $\pi=\mathrm{Ind}_{K_{\pi}}^{F}(\xi_{\pi})$ and
$\mathrm{Ad}(\pi)=\mathrm{Ind}_{K_{\pi}}^{F}(\xi_{\pi}^{2})\otimes\omega_{\pi}\boxplus\xi_{\pi}|_{F}\omega_{\pi}$.

  3. (iii)

Let $\pi\in\mathfrak{F}_{2}^{\flat}$, in which case
$\mathrm{Ad}(\pi)\in\mathfrak{F}_{3}$.

    1. (a)

$\mathrm{Sym}^{3}(\pi)\notin\mathfrak{F}_{4}$ if and only if there exists
$\mu_{\pi}\in\mathfrak{F}_{1}-\\{\mathbbm{1}\\}$ such that

$\mu_{\pi}^{3}=\mathbbm{1},\quad\mathrm{Ad}(\pi)=\mathrm{Ad}(\pi)\otimes\mu_{\pi},\quad\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}=\mathrm{Ad}(\pi)\boxplus\mu_{\pi}\boxplus\overline{\mu}_{\pi}.$

    2. (b)

If $\mathrm{Sym}^{3}(\pi)\in\mathfrak{F}_{4}$, then
$\mathrm{Sym}^{4}(\pi)\notin\mathfrak{F}_{5}$ if and only if there exists
$\eta_{\pi}\in\mathfrak{F}_{1}-\\{\mathbbm{1}\\}$ and a dihedral
$\nu_{\pi}\in\mathfrak{F}_{2}$ such that

$\eta_{\pi}^{2}=\mathbbm{1},\quad\mathrm{Sym}^{3}(\pi)=\mathrm{Sym}^{3}(\pi)\otimes\eta_{\pi},\quad\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}=\nu_{\pi}\boxplus\mathrm{Ad}(\pi)\otimes\eta_{\pi}.$

In this section, we prove the hardest cases of Theorem 1.4, when
$\pi,\pi^{\prime}\in\mathfrak{F}_{2}^{\flat}$ and $\pi\not\sim\pi^{\prime}$.
Consider $\mathcal{D}(s)$ in (4.1) with $m=n=2$ and $\chi$ replaced by
$\chi\overline{\omega}_{\pi}\overline{\omega}_{\pi^{\prime}}$ (in order to
express $\mathcal{D}(s)$ in terms of $\mathrm{Ad}(\pi)$ and
$\mathrm{Ad}(\pi^{\prime})$). Since
$\mathrm{Ad}(\pi)\boxtimes\mathrm{Ad}(\pi)=\mathbbm{1}\boxplus\mathrm{Ad}(\pi)\boxplus\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}$
(per Lemma 3.5), the resulting Dirichlet series is

(6.1) $\displaystyle\mathcal{D}_{0}(s)$
$\displaystyle=\zeta_{F}(s)^{6}L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi))^{4}L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi}))^{4}$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi))^{7}L(s,\mathrm{Ad}(\pi^{\prime}))^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\chi)^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi})^{2}$
$\displaystyle\cdot
L(s,\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2})^{5}L(s,\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{2})^{2}L(s,\mathrm{Ad}(\pi)\times\mathrm{Ad}(\pi^{\prime}))^{3}$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi)\times(\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{2}))^{3}L(s,\mathrm{Ad}(\pi^{\prime})\times(\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}))$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi^{\prime})\times(\mathrm{Sym}^{4}(\pi)\otimes\chi\overline{\omega}_{\pi}^{2}))^{2}L(s,\mathrm{Ad}(\pi^{\prime})\times(\mathrm{Sym}^{4}(\pi)\otimes\overline{\chi}\,\overline{\omega}_{\pi}^{2}))^{2}$
$\displaystyle\cdot
L(s,\mathrm{Sym}^{4}(\pi)\times(\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi}^{2}\overline{\omega}_{\pi^{\prime}}^{2})).$

As in Lemma 4.1, we see that at unramified places $v\nmid\infty$, the
Dirichlet coefficients of $-(\mathcal{D}_{0}^{\prime}/\mathcal{D}_{0})(s)$ are
non-negative. It remains to show that each factor other than
$\zeta_{F}(s)^{3}$ is entire. Per Theorem 3.1, only
$L(s,\mathrm{Sym}^{4}(\pi)\times(\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi}^{2}\overline{\omega}_{\pi^{\prime}}^{2}))$
and the twists of $L(s,\mathrm{Ad}(\pi)\times\mathrm{Ad}(\pi^{\prime}))$ might
have poles.

######  Lemma 6.2.

Let $\pi,\pi^{\prime}\in\mathfrak{F}_{2}^{\flat}$ and
$\xi=\xi^{*}|\cdot|^{it_{\xi}}\in\mathfrak{F}_{1}$. If

  * •

${\xi^{*}}^{3}\neq\mathbbm{1}$, or

  * •

${\xi^{*}}^{3}=\mathbbm{1}$, $\xi^{*}\neq\mathbbm{1}$,
$\mathrm{Ad}(\pi)\neq\mathrm{Ad}(\pi)\otimes\xi^{*}$, and
$\mathrm{Ad}(\pi^{\prime})\neq\mathrm{Ad}(\pi^{\prime})\otimes\xi^{*}$, or

  * •

$\xi^{*}=\mathbbm{1}$ and $\pi\not\sim\pi^{\prime}$,

then $L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\xi))$ is
entire.

###### Proof.

We prove the contrapositive. If
$L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\xi))$ has a pole,
then $\mathrm{Ad}(\pi)=\mathrm{Ad}(\pi^{\prime})\otimes\xi^{*}$ since
$\mathrm{Ad}(\pi)$ and $\mathrm{Ad}(\pi^{\prime})$ are self-dual. This
equality implies that
$\mathbbm{1}=\omega_{\mathrm{Ad}(\pi)}=\omega_{\mathrm{Ad}(\pi^{\prime})\otimes\xi^{*}}={\xi^{*}}^{3}$.
Since ${\xi^{*}}^{3}=\mathbbm{1}$, $\xi^{*}$ is not self-dual unless
$\xi^{*}=\mathbbm{1}$. Therefore, if $\xi^{*}\neq\mathbbm{1}$, then
$\mathrm{Ad}(\pi)=\mathrm{Ad}(\pi)\otimes\xi^{*}$ or
$\mathrm{Ad}(\pi^{\prime})=\mathrm{Ad}(\pi^{\prime})\otimes\xi^{*}$. If
$\xi^{*}=\mathbbm{1}$, then $\mathrm{Ad}(\pi)=\mathrm{Ad}(\pi^{\prime})$. It
now follows from [37, Theorem 4.1.2] that $\pi\sim\pi^{\prime}$. ∎

When $\pi\not\sim\pi^{\prime}$ and $\pi,\pi^{\prime}\notin\mathrm{RA}_{F}$, we
do not know whether
$L(s,\mathrm{Sym}^{4}(\pi)\times(\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi}^{2}\overline{\omega}_{\pi^{\prime}}^{2}))$
has a pole. Because of lack of progress towards the modularity of
Rankin–Selberg $L$-functions, we have no analogue of [38, Theorem 4.1.2].
Fortunately, as we will see, $\mathcal{D}_{0}(s)$ was constructed in such a
way that this lack of progress will not hinder our proof!

We proceed to find integers $\ell\geq 1$ and $k\geq 1$ such that $2\ell>k$ and

(6.2)
$(s-1)^{k}\mathcal{D}_{0}(s)L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi))^{-\ell}L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi}))^{-\ell}$

is entire. Once we find such a $k$ and $\ell$, then in light of (3.3), the
proof of Theorem 1.4 proceeds just as in the twist-inequivalent case of
Theorem 1.2, mutatis mutandis.

###  6.1.
$\mathrm{Sym}^{3}(\pi),\mathrm{Sym}^{3}(\pi^{\prime})\notin\mathfrak{F}_{4}$

We now apply Lemma 6.1(iiia) to
$\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}$ and
$\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{2}$.
It follows that if $k=10$, $\ell=6$, and

$\displaystyle D_{1}(s)$
$\displaystyle=\zeta_{F}(s)^{6}L(s,\mu_{\pi}\mu_{\pi^{\prime}})L(s,\overline{\mu}_{\pi}\overline{\mu}_{\pi^{\prime}})L(s,\mu_{\pi}\overline{\mu}_{\pi^{\prime}})L(s,\overline{\mu}_{\pi}\mu_{\pi^{\prime}}),$
$\displaystyle D_{2}(s)$
$\displaystyle=L(s,\mathrm{Ad}(\pi))^{12}L(s,\mathrm{Ad}(\pi^{\prime}))^{4}L(s,\mathrm{Ad}(\pi)\otimes\mu_{\pi^{\prime}})^{4}L(s,\mathrm{Ad}(\pi)\otimes\overline{\mu}_{\pi^{\prime}})^{4}L(s,\mu_{\pi})^{5}$
$\displaystyle\cdot
L(s,\overline{\mu}_{\pi})^{5}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\mu_{\pi})^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\mu}_{\pi})^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\chi)^{2}$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi})^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\chi\mu_{\pi})^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi}\mu_{\pi})^{2}L(s,\mu_{\pi^{\prime}})^{2}L(s,\overline{\mu}_{\pi^{\prime}})^{2}$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi^{\prime})\otimes\chi\overline{\mu}_{\pi})^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi}\,\overline{\mu}_{\pi})^{2}L(s,\mathrm{Ad}(\pi)\times\mathrm{Ad}(\pi^{\prime}))^{8},$

then (6.2) equals $D_{1}(s)D_{2}(s)(s-1)^{10}$. Since
$\mu_{\pi}^{3}=\mu_{\pi^{\prime}}^{3}=\mathbbm{1}$, this is entire by Lemma
6.2.

###  6.2. $\mathrm{Sym}^{3}(\pi)\notin\mathfrak{F}_{4}$,
$\mathrm{Sym}^{3}(\pi^{\prime})\in\mathfrak{F}_{4}$, and
$\mathrm{Sym}^{4}(\pi^{\prime})\notin\mathfrak{F}_{5}$

Decompose $\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}$ and
$\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{2}$
using Lemma 6.1(iiia,iiib). We conclude that if $k=6$, $\ell=6$, and

$\displaystyle D_{3}(s)$
$\displaystyle=L(s,\mathrm{Ad}(\pi))^{12}L(s,\mathrm{Ad}(\pi^{\prime}))^{2}L(s,\nu_{\pi^{\prime}})^{2}L(s,\mathrm{Ad}(\pi)\times\nu_{\pi^{\prime}})^{4}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\eta_{\pi^{\prime}})^{2}$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\eta_{\pi^{\prime}}))^{4}L(s,\mu_{\pi})^{5}L(s,\overline{\mu}_{\pi})^{5}L(s,\nu_{\pi^{\prime}}\otimes\mu_{\pi})L(s,\nu_{\pi^{\prime}}\otimes\overline{\mu}_{\pi})$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi^{\prime})\otimes\mu_{\pi})L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\mu}_{\pi})L(s,\mathrm{Ad}(\pi^{\prime})\otimes\mu_{\pi}\eta_{\pi^{\prime}})L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\mu}_{\pi}\eta_{\pi^{\prime}})$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi)\times\mathrm{Ad}(\pi^{\prime}))^{4}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\chi)^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi})^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\chi\mu_{\pi})^{2}$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi}\mu_{\pi})^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\chi\overline{\mu}_{\pi})^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi}\,\overline{\mu}_{\pi})^{2},$

then (6.2) equals $\zeta_{F}(s)^{6}(s-1)^{6}D_{3}(s)$. Since
$\mu_{\pi}^{3}=\eta_{\pi^{\prime}}^{2}=\mathbbm{1}$, this is entire by Lemma
6.2.

###  6.3.
$\mathrm{Sym}^{3}(\pi),\mathrm{Sym}^{3}(\pi^{\prime})\in\mathfrak{F}_{4}$ and
$\mathrm{Sym}^{4}(\pi),\mathrm{Sym}^{4}(\pi^{\prime})\notin\mathfrak{F}_{5}$

We apply Lemma 6.1(iiib) to
$\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}$ and
$\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{2}$.
If $k=7$, $\ell=4$, and $D_{4}(s)$ equals

$\displaystyle
L(s,\mathrm{Ad}(\pi))^{7}L(s,\mathrm{Ad}(\pi^{\prime}))^{2}L(s,\nu_{\pi}\times\mathrm{Ad}(\pi^{\prime}))L(s,\mathrm{Ad}(\pi)\otimes\eta_{\pi})^{5}L(s,\mathrm{Ad}(\pi)\times\nu_{\pi^{\prime}})^{3}$
$\displaystyle\cdot
L(s,\nu_{\pi^{\prime}})^{2}L(s,\nu_{\pi})^{5}L(s,\mathrm{Ad}(\pi)\times(\nu_{\pi^{\prime}}\otimes\eta_{\pi}))L(s,\mathrm{Ad}(\pi^{\prime})\otimes\eta_{\pi^{\prime}})^{2}L(s,\mathrm{Ad}(\pi^{\prime})\times(\nu_{\pi}\otimes\eta_{\pi^{\prime}}))$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\eta_{\pi}))L(s,\mathrm{Ad}(\pi^{\prime})\otimes\chi)^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi})^{2}L(s,\mathrm{Ad}(\pi^{\prime})\times(\nu_{\pi}\otimes\chi))^{2}$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\eta_{\pi^{\prime}}))^{3}L(s,\mathrm{Ad}(\pi^{\prime})\times(\nu_{\pi}\otimes\overline{\chi}))^{2}L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\eta_{\pi}\eta_{\pi^{\prime}}))$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi)\times\mathrm{Ad}(\pi^{\prime}))^{3}L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi\eta_{\pi}))^{2}L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi}\eta_{\pi}))^{2},$

then (6.2) equals
$\zeta_{F}(s)^{6}L(s,\nu_{\pi}\times\nu_{\pi^{\prime}})(s-1)^{7}D_{4}(s)$.
Note that $L(s,\nu_{\pi}\times\nu_{\pi^{\prime}})$ is either entire or has a
simple pole at $s=1$ since $\nu_{\pi}$ and $\nu_{\pi^{\prime}}$ are cuspidal.
Therefore, by Lemma 6.2,
$\zeta_{F}(s)^{6}L(s,\nu_{\pi}\times\nu_{\pi^{\prime}})(s-1)^{7}D_{4}(s)$ is
entire.

###  6.4. $\mathrm{Sym}^{3}(\pi)\in\mathfrak{F}_{4}$,
$\mathrm{Sym}^{4}(\pi)\in\mathfrak{F}_{5}$, and
$\pi^{\prime}\in\mathfrak{F}_{2}^{\flat}$

####  6.4.1. $\mathrm{Sym}^{3}(\pi^{\prime})\notin\mathfrak{F}_{4}$

We apply Lemma 6.1(iiia) to
$\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{2}$.
If $k=6$, $\ell=4$, and $D_{5}(s)$ equals

$\displaystyle
L(s,\mathrm{Ad}(\pi))^{7}L(s,\mathrm{Ad}(\pi^{\prime}))^{4}L(s,\mu_{\pi^{\prime}})^{2}L(s,\overline{\mu}_{\pi^{\prime}})^{2}L(s,\mathrm{Ad}(\pi)\otimes\mu_{\pi^{\prime}})^{3}L(s,\mathrm{Ad}(\pi)\otimes\overline{\mu}_{\pi^{\prime}})^{3}$
$\displaystyle\cdot
L(s,\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2})^{5}L(s,\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}\mu_{\pi^{\prime}})L(s,\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}\overline{\mu}_{\pi^{\prime}})L(s,\mathrm{Ad}(\pi^{\prime})\otimes\chi)^{2}$
$\displaystyle\cdot
L(s,\mathrm{Sym}^{4}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi\overline{\omega}_{\pi}^{2}))^{2}L(s,\mathrm{Sym}^{4}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi}\overline{\omega}_{\pi}^{2}))^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi})^{2}$
$\displaystyle\cdot
L(s,\mathrm{Sym}^{4}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\overline{\omega}_{\pi}^{2}))^{2}L(s,\mathrm{Ad}(\pi)\times\mathrm{Ad}(\pi^{\prime}))^{6},$

then (6.2) equals $\zeta_{F}(s)^{6}(s-1)^{6}D_{5}(s)$. Since
$\mu_{\pi^{\prime}}^{3}=\mathbbm{1}$ and $\mu_{\pi^{\prime}}\neq\mathbbm{1}$,
it follows from Lemma 6.2 that $\zeta_{F}(s)^{6}(s-1)^{6}D_{5}(s)$ is entire.

####  6.4.2. $\mathrm{Sym}^{3}(\pi^{\prime})\in\mathfrak{F}_{4}$ and
$\mathrm{Sym}^{4}(\pi^{\prime})\notin\mathfrak{F}_{5}$

We apply Lemma 6.1(iiib) to
$\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{2}$
and conclude that if $k=6$, $\ell=4$, and

$\displaystyle D_{6}(s)$
$\displaystyle=L(s,\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2})^{5}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\eta_{\pi^{\prime}})^{2}L(s,\mathrm{Ad}(\pi)\times\nu_{\pi^{\prime}})^{3}L(s,\mathrm{Ad}(\pi^{\prime}))^{2}$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\eta_{\pi^{\prime}}))^{3}L(s,\mathrm{Sym}^{4}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\overline{\omega}_{\pi}^{2}\eta_{\pi^{\prime}}))L(s,\nu_{\pi^{\prime}})^{2}$
$\displaystyle\cdot
L(s,\mathrm{Sym}^{4}(\pi)\times(\nu_{\pi^{\prime}}\otimes\overline{\omega}_{\pi}^{2}))L(s,\mathrm{Sym}^{4}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\overline{\omega}_{\pi}^{2}))L(s,\mathrm{Ad}(\pi))^{7}$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi^{\prime})\otimes\chi)^{2}L(s,\mathrm{Sym}^{4}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi\overline{\omega}_{\pi}^{2}))^{2}L(s,\mathrm{Ad}(\pi)\times\mathrm{Ad}(\pi^{\prime}))^{3}$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi})^{2}L(s,\mathrm{Sym}^{4}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi}\,\overline{\omega}_{\pi}^{2}))^{2},$

then (6.2) equals $\zeta_{F}(s)^{6}(s-1)^{6}D_{6}(s)$. By Lemma 6.2, this is
entire.

####  6.4.3. $\mathrm{Sym}^{3}(\pi^{\prime})\in\mathfrak{F}_{4}$ and
$\mathrm{Sym}^{4}(\pi^{\prime})\in\mathfrak{F}_{5}$

If $k=7$, $\ell=4$, and

$\displaystyle D_{7}(s)$
$\displaystyle=L(s,\mathrm{Ad}(\pi^{\prime})\otimes\chi)^{2}L(s,\mathrm{Ad}(\pi^{\prime})\otimes\overline{\chi})^{2}L(s,\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2})^{5}L(s,\mathrm{Ad}(\pi)\times\mathrm{Ad}(\pi^{\prime}))^{3}$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi)\times(\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{2}))^{3}L(s,\mathrm{Ad}(\pi^{\prime})\times(\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}))L(s,\mathrm{Ad}(\pi))^{7}$
$\displaystyle\cdot
L(s,\mathrm{Ad}(\pi^{\prime})\times(\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}))^{4}L(s,\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi^{\prime}}^{2})^{2}L(s,\mathrm{Ad}(\pi^{\prime}))^{2},$

then (6.2) equals
$D_{8}(s)=\zeta_{F}(s)^{6}(s-1)^{7}L(s,\mathrm{Sym}^{4}(\pi)\times(\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi}^{2}\overline{\omega}_{\pi^{\prime}}^{2}))D_{7}(s)$.
The $L$-function
$L(s,\mathrm{Sym}^{4}(\pi)\times(\mathrm{Sym}^{4}(\pi^{\prime})\otimes\overline{\omega}_{\pi}^{2}\overline{\omega}_{\pi^{\prime}}^{2}))$
has a pole of order $0$ or $1$ at $s=1$ because both $\mathrm{Sym}^{4}(\pi)$
and $\mathrm{Sym}^{4}(\pi^{\prime})$ are cuspidal. Therefore, by Lemma 6.2,
$D_{8}(s)$ is entire.

##  7\. Proof of Theorem 1.4: The remaining cases

For the remaining cases,
$L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi))$ factors
via Lemmata 3.5 and 6.1.

###  7.1. $\pi,\pi^{\prime}\in\mathfrak{F}_{2}-\mathfrak{F}_{2}^{\flat}$

By Lemma 6.1(ii),
$L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi))$ is a
product of $\mathrm{GL}_{m}\times\mathrm{GL}_{n}$ $L$-functions with $m,n\leq
2$. By Proposition 1.1(i,ii,iii), Theorem 1.4 follows.

###  7.2. $\pi\in\mathfrak{F}_{2}^{\flat}$ and
$\pi^{\prime}\in\mathfrak{F}_{2}-\mathfrak{F}_{2}^{\flat}$.

By Lemma 6.1(ii), $\mathrm{Ad}(\pi)$ is cuspidal and

$L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi))=L(s,\mathrm{Ind}_{K_{\pi^{\prime}}}^{F}(\xi_{\pi^{\prime}}^{2})\times(\mathrm{Ad}(\pi)\otimes\chi\overline{\omega}_{\pi^{\prime}}))L(s,\mathrm{Ad}(\pi)\otimes\chi\overline{\omega}_{\pi^{\prime}}\xi_{\pi^{\prime}}|_{F}).$

Note that
$\mathrm{Ad}(\pi)\not\sim\mathrm{Ad}(\mathrm{Ind}_{K_{\pi^{\prime}}}^{F}(\xi_{\pi^{\prime}}^{2}))$.
Now, Theorem 1.4 follows from Proposition 1.1(ii,v).

###  7.3. $\pi,\pi^{\prime}\in\mathfrak{F}_{2}^{\flat}$ and
$\pi\sim\pi^{\prime}$

In this case, we have that $\mathrm{Ad}(\pi)=\mathrm{Ad}(\pi^{\prime})$, and
$\mathrm{Ad}(\pi)\in\mathfrak{F}_{3}$. By Lemma 3.5, we have the factorization

$L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi))=L(s,\chi)L(s,\mathrm{Ad}(\pi)\otimes\chi)L(s,\mathrm{Sym}^{4}(\pi)\otimes\chi\overline{\omega}_{\pi}^{2})$

unless $\mathrm{Sym}^{3}(\pi)\notin\mathfrak{F}_{4}$ and
$\chi=\chi^{*}\otimes|\cdot|^{it_{\chi}}$ satisfies
$\mathrm{Ad}(\pi)\otimes\chi^{*}=\mathrm{Ad}(\pi)$, in which case

$L(s,\mathrm{Ad}(\pi)\times(\mathrm{Ad}(\pi^{\prime})\otimes\chi))=\zeta_{F}(s+it_{\chi})L(s,\mathrm{Ad}(\pi)\otimes|\cdot|^{it_{\chi}})L(s,\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}|\cdot|^{it_{\chi}}),$

${\chi^{*}}^{3}=\mathbbm{1}$, and $\chi^{*}\neq\mathbbm{1}$. In light of
Proposition 1.1(ii), Theorem 1.4 follows from a determination of whether the
pertinent twist of $L(s,\mathrm{Sym}^{4}(\pi))$ has a exceptional zero.

####  7.3.1. $\mathrm{Sym}^{3}(\pi)\notin\mathfrak{F}_{4}$

Let $\chi=\chi^{*}|\cdot|^{it_{\chi}}\in\mathfrak{F}_{1}$. If
$\chi^{*}\in\\{\mu_{\pi},\overline{\mu}_{\pi}\\}$, then

$L(s,\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}|\cdot|^{it_{\chi}})=L(s,\mathrm{Ad}(\pi)\otimes|\cdot|^{it_{\chi}})L(s,\mu_{\pi}|\cdot|^{it_{\chi}})L(s,\overline{\mu}_{\pi}|\cdot|^{it_{\chi}}).$

by Lemma 6.1(iiia). Since $\mu_{\pi}$ is cubic,
$L(s,\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}|\cdot|^{it_{\chi}})$
has no exceptional zero by Proposition 1.1(i,ii). If
$\chi^{*}\notin\\{\mu_{\pi},\overline{\mu}_{\pi}\\}$, then
$L(s,\mathrm{Sym}^{4}(\pi)\otimes\chi\overline{\omega}_{\pi}^{2})$ factors as
$L(s,\mathrm{Ad}(\pi)\otimes\chi)L(s,\chi\overline{\mu}_{\pi})L(s,\chi\mu_{\pi})$.
Again, by Proposition 1.1(i,ii),
$L(s,\mathrm{Sym}^{4}(\pi)\otimes\chi\overline{\omega}_{\pi}^{2})$ has no
exceptional zero.

####  7.3.2. $\mathrm{Sym}^{3}(\pi)\in\mathfrak{F}_{4}$ and
$\mathrm{Sym}^{4}(\pi)\notin\mathfrak{F}_{5}$

The factorization
$L(s,\mathrm{Sym}^{4}(\pi)\otimes\chi\overline{\omega}_{\pi}^{2})=L(s,\nu_{\pi}\otimes\chi)L(s,\mathrm{Ad}(\pi)\otimes\eta_{\pi}\chi)$
holds by Lemma 6.1(iiib). Since $\nu_{\pi}$ is cuspidal, no exceptional zero
exists per Proposition 1.1(ii).

####  7.3.3. $\mathrm{Sym}^{3}(\pi)\in\mathfrak{F}_{4}$ and
$\mathrm{Sym}^{4}(\pi)\in\mathfrak{F}_{5}$

We have that $\mathrm{Ad}(\pi)\in\mathfrak{F}_{3}$,
$\mathrm{Sym}^{3}(\pi)\in\mathfrak{F}_{4}$, and
$\mathrm{Sym}^{4}(\pi)\in\mathfrak{F}_{5}$. We define
$\Pi_{1}=\mathbbm{1}\boxplus\mathrm{Ad}(\pi)\boxplus\mathrm{Sym}^{4}(\pi)\otimes\chi\overline{\omega}_{\pi}^{2}$.
Note that $\mathrm{Ad}(\pi)$ and
$\mathrm{Sym}^{4}(\pi)\otimes\overline{\omega}_{\pi}^{2}$ are self-dual. By
Lemma 3.5, $D_{9}(s)=L(s,\Pi_{1}\times\widetilde{\Pi}_{1})$ factors as

$\displaystyle\zeta_{F}(s)L(s,\mathrm{Ad}(\pi)\times\mathrm{Ad}(\pi))L(s,\mathrm{Sym}^{4}(\pi)\times\mathrm{Sym}^{4}(\widetilde{\pi}))L(s,\mathrm{Ad}(\pi))^{2}L(s,\mathrm{Sym}^{4}(\pi)\otimes\chi\overline{\omega}_{\pi}^{2})$
$\displaystyle\cdot
L(s,\mathrm{Sym}^{4}(\pi)\otimes\overline{\chi}\,\overline{\omega}_{\pi}^{2})L(s,\mathrm{Ad}(\pi)\times(\mathrm{Sym}^{4}(\pi)\otimes\chi\overline{\omega}_{\pi}^{2}))L(s,\mathrm{Ad}(\pi)\times(\mathrm{Sym}^{4}(\pi)\otimes\overline{\chi}\,\overline{\omega}_{\pi}^{2})).$

Using Lemma 3.3, we apply Lemma 3.2 to $D_{9}(s)$ with $\log
Q\ll\log(C(\pi)C(\chi))$ and $r=3$. It follows that $D_{9}(s)$ has at most $3$
zeros in $I=[1-\Cr{GHL}/\log Q,1)$.

We claim that the functions

$\mathcal{H}(s)=\frac{L(s,\mathrm{Ad}(\pi)\times(\mathrm{Sym}^{4}(\pi)\otimes\chi\overline{\omega}_{\pi}^{2}))}{L(s,\mathrm{Sym}^{4}(\pi)\otimes\chi\overline{\omega}_{\pi}^{2})},\quad\overline{\mathcal{H}(\overline{s})}=\frac{L(s,\mathrm{Ad}(\pi)\times(\mathrm{Sym}^{4}(\pi)\otimes\overline{\chi}\,\overline{\omega}_{\pi}^{2}))}{L(s,\mathrm{Sym}^{4}(\pi)\otimes\overline{\chi}\,\overline{\omega}_{\pi}^{2})}$

are holomorphic when $\mathrm{Re}(s)\geq\frac{56}{65}$. Thus, by (3.3), any
real zero of
$L(\sigma,\mathrm{Sym}^{4}(\pi)\otimes\chi\overline{\omega}_{\pi}^{2})$ in $I$
is a real zero of $D_{9}(\sigma)$ in $I$ of order at least 4, a contradiction.
Therefore, no such zero exists. It suffices for us to prove the claim for
$\mathcal{H}(s)$.

To prove the claim, let $S=S_{\pi}^{\infty}\cup S_{\chi}^{\infty}\cup\\{v\mid
2\\}$, which contains $S_{\mathrm{Sym}^{3}(\pi)}^{\infty}\cup
S_{\chi}^{\infty}\cup\\{v\mid 2\\}$ by Lemma 3.6. By (3.10) and (3.14), if
$\mathrm{Re}(s)>1$, then

(7.1)
$\mathcal{H}(s)=L^{S}(s,\mathrm{Sym}^{3}(\pi);\mathrm{Sym}^{2}\otimes\chi\overline{\omega}_{\pi}^{3})\prod_{v\in
S,~{}v\nmid\infty}\frac{L(s,\mathrm{Ad}(\pi_{v})\times\mathrm{Sym}^{4}(\pi_{v})\otimes\chi_{v}\,\overline{\omega}_{\pi,v}^{2})}{L(s,\mathrm{Sym}^{4}(\pi_{v})\otimes\chi_{v}\,\overline{\omega}_{\pi,v}^{2})}.$

By (3.2) and (3.4), the product over $v\in S$ in (7.1) is holomorphic when
$\mathrm{Re}(s)\geq\frac{56}{65}$. The numerator and denominator of
$\mathcal{H}$ are holomorphic and non-vanishing at $s=1$. Therefore, by
Theorem 3.8,
$L^{S}(s,\mathrm{Sym}^{3}(\pi);\mathrm{Sym}^{2}\otimes\chi\overline{\omega}_{\pi}^{3})$
is holomorphic for $\mathrm{Re}(s)>0$. The claim now follows from the identity
theorem in complex analysis.

##  8\. Proof of Theorem 2.1

Throughout, we assume that $F$ is a totally real field,
$\pi,\pi^{\prime}\in\mathrm{RA}_{F}\cap\mathfrak{F}_{2}^{*}$, and
$\pi\not\sim\pi^{\prime}$. Recall $y_{\pi}$ and $y_{\pi^{\prime}}$ from (1.3).
Let $\zeta$ (resp. $\zeta^{\prime}$) be a fixed root of unity such that
$\zeta^{2}$ (resp. $\zeta^{\prime 2}$) lies in the image of $\omega_{\pi}$
(resp. $\omega_{\pi^{\prime}}$). Let $Y=\max\\{y_{\pi},y_{\pi^{\prime}}\\}$,
$\mathcal{Q}=C(\pi)C(\pi^{\prime})\geq e$, and

$\mathcal{V}=\mathcal{V}(\pi,\pi^{\prime},\zeta,\zeta^{\prime})=\\{v\nmid\infty\colon
v\notin S_{\pi}\cup
S_{\pi^{\prime}},~{}\omega_{\pi,v}(\varpi_{v})=\zeta^{2},~{}\omega_{\pi^{\prime},v}(\varpi_{v})={\zeta^{\prime}}^{2}\\}.$

We will assume that $\mathcal{V}$ is infinite. If $v\in\mathcal{V}$, then

$L(s,\pi_{v})^{-1}=(1-\alpha_{1,\pi}(v)q_{v}^{-s})(1-\alpha_{2,\pi}(v)q_{v}^{-s})=1-a_{\pi}(v)q_{v}^{-s}+\zeta^{2}q_{v}^{-2s}.$

There exists $\theta_{v}\in[0,\pi]$ such that if
$A_{\pi}(v)=\mathrm{diag}(\zeta e^{i\theta_{v}},\zeta e^{-i\theta_{v}})$, then

$L(s,\pi_{v})^{-1}=\det(I_{2\times
2}-A_{\pi}(v)q_{v}^{-s}),\quad\mathrm{tr}\,A_{\pi}(v)=2\zeta\cos\theta_{v}=a_{\pi}(v),\quad\det
A_{\pi}(v)=\zeta^{2}.$

Similarly, if $v\notin S_{\pi^{\prime}}^{\infty}$ satisfies
$\omega_{\pi^{\prime},v}(\varpi_{v})={\zeta^{\prime}}^{2}$, then we define the
angle $\theta_{v}^{\prime}\in[0,\pi]$ associated to $\pi_{v}^{\prime}$. Given
a set $U$, let $\mathbf{1}_{U}$ be its indicator function.

Let $I_{1}$ and $I_{2}$ be subintervals of $[0,\pi]$, and define
$\mathrm{d}\widetilde{\mu}_{\mathrm{ST}}=(2/\pi)(\sin\theta)^{2}\mathrm{d}\theta$.
We will prove Theorem 2.1(2) in the following equivalent form: If $x\geq 2$,
then

(8.1) $\displaystyle\Big{|}\sum_{\begin{subarray}{c}v\in\mathcal{V}\\\
q_{v}\leq
x\end{subarray}}\mathbf{1}_{I}(\theta_{v})\mathbf{1}_{I^{\prime}}(\theta_{v}^{\prime})\Big{/}\sum_{\begin{subarray}{c}v\in\mathcal{V}\\\
q_{v}\leq
x\end{subarray}}1-\widetilde{\mu}_{\mathrm{ST}}(I)\widetilde{\mu}_{\mathrm{ST}}(I^{\prime})\Big{|}$
$\displaystyle\leq\Cr{ST1}\sqrt{[F:\mathbb{Q}]}\Big{(}\frac{\log(C(\pi)C(\pi^{\prime})\log
x)}{\sqrt{\log x}}\Big{)}^{\frac{2}{2+\max\\{y_{\pi},y_{\pi^{\prime}}\\}}}.$

We will not give the details for Theorem 2.1(1) because the proof is far less
complicated.

Recall $\Cr{main}$ from Corollary 1.3. Without loss of generality, we assume
that $\Cr{main}\leq 1$.

######  Lemma 8.1.

There exists an absolute, effectively computable constant $\Cl[abcon]{CDT}\geq
1$ such that

(8.2) $\Big{(}\sum_{\begin{subarray}{c}v\in\mathcal{V}\\\ q_{v}\leq
x\end{subarray}}1\Big{)}^{-1}\ll\mathcal{Q}^{\Cr{CDT}}\frac{\log x}{x},\qquad
x\geq\exp\Big{(}\Big{(}\frac{16\Cr{CDT}+224}{\Cr{main}}\Big{)}^{4}[F:\mathbb{Q}]^{1+\frac{Y}{2}}(\log\mathcal{Q})^{2}\Big{)}.$

###### Proof.

By the proof of [45, Theorem 3.1], there exists an absolute, effectively
computable constant $\Cl[abcon]{CDT2}>0$ such that left hand side of (8.2) is
$\ll(\mathcal{Q}|\mathrm{im}(\omega_{\pi})|\cdot|\mathrm{im}(\omega_{\pi^{\prime}})|)^{\Cr{CDT2}}(\log
x)/x$ in the prescribed range of $x$. It remains to bound
$|\mathrm{im}(\omega_{\pi})|$ and $|\mathrm{im}(\omega_{\pi^{\prime}})|$. By
[45, Lemma 2.12] we have that $|\mathrm{im}(\omega_{\pi})|\leq
e^{O([F:\mathbb{Q}])}C(\omega_{\pi})$. By [40, Theorem A], we have that
$C(\omega_{\pi})\leq C(\pi)\leq\mathcal{Q}$. Since
$[F:\mathbb{Q}]\ll\log\mathcal{Q}$, the lemma follows. ∎

Outside of the range of $x$ in (8.2), Theorem 2.1 is trivial. In this range,
we have that

(8.3) $M=\frac{1}{\sqrt{[F:\mathbb{Q}]}}\Big{(}\frac{\sqrt{\Cr{main}\log
x}}{(16\Cr{CDT}+224)\log(\mathcal{Q}\log x)}\Big{)}^{\frac{2}{2+Y}}\geq 2.$

###  8.1. A prime number theorem

Let $m,n\geq 0$ be integers. We will use Corollary 1.3 to prove a prime number
theorem for the $L$-functions
$L(s,\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi))$.

######  Proposition 8.2.

Let $\chi\in\mathfrak{F}_{1}^{*}$ correspond with a ray class character and
$\mathcal{M}=\max\\{m,n\\}$. If $x\geq 2$ and $\mathcal{M}\geq 1$, then

$\displaystyle\sum_{\begin{subarray}{c}v\notin S_{\pi}\cup
S_{\pi^{\prime}}\cup S_{\chi}\\\ q_{v}\leq
x\end{subarray}}a_{\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi)}(v)$
$\displaystyle\ll x\exp\Big{(}-\frac{(\Cr{main}/8)\log
x}{\mathcal{M}^{2+Y}\log(\mathcal{Q}C(\chi)\mathcal{M})+\mathcal{M}\sqrt{\Cr{main}[F:\mathbb{Q}]\log
x}}\Big{)}(\mathcal{M}^{2+Y}\log(\mathcal{Q}C(\chi)\mathcal{M}x))^{6}.$

###### Proof.

We proceed as in [20, Theorem 5.13]. At most $[F:\mathbb{Q}]$ prime ideals of
$F$ lie above any rational prime. Using the Brun–Titchmarsh theorem, we find
that

(8.4) $\\#\\{v\nmid\infty\colon q_{v}\in(x,x+h]\\}\ll[F:\mathbb{Q}]h/\log
h,\qquad x>0,\quad h>1.$

By Corollary 1.3, Lemma 3.7, and (8.4), if $x\geq 2$, then

$\displaystyle\sum_{\begin{subarray}{c}\ell\geq 1,~{}q_{v}^{\ell}\leq
x\end{subarray}}a_{\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi)}(v^{\ell})\log
q_{v}$ $\displaystyle\ll x\exp\Big{(}-\frac{(\Cr{main}/4)\log
x}{\mathcal{M}^{2+Y}\log(\mathcal{Q}C(\chi)\mathcal{M})+\mathcal{M}\sqrt{\Cr{main}[F:\mathbb{Q}]\log
x}}\Big{)}(\mathcal{M}^{2+Y}\log(\mathcal{Q}C(\chi)\mathcal{M}x))^{6}.$

By Lemma 3.7 and (8.4), the contribution from $\ell\geq 2$ or $v\in
S_{\pi}\cup S_{\pi^{\prime}}\cup S_{\chi}$ is negligible. The proposition now
follows by partial summation. ∎

######  Corollary 8.3.

Let $\chi\in\mathfrak{F}_{1}^{*}$ correspond with a ray class character, and
assume that $S_{\chi}\subseteq S_{\pi}\cup S_{\pi^{\prime}}$ and
$C(\chi)\leq\mathcal{Q}$. If (8.2) and (8.3) are satisfied, then

$\sum_{\begin{subarray}{c}0\leq m,n\leq M\\\
(m,n)\neq(0,0)\end{subarray}}\sum_{\begin{subarray}{c}v\notin S_{\pi}\cup
S_{\pi^{\prime}}\\\ q_{v}\leq
x\end{subarray}}a_{\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi)}(v)\ll\frac{x}{(\mathcal{Q}\log
x)^{\Cr{CDT}+1}}.$

###### Proof.

This follows from Proposition 8.2 and our hypotheses on $\chi$, $x$, and $M$.
∎

###  8.2. Proof of (8.1)

The Chebyshev polynomials of the second kind, defined by the relation
$U_{m}(\cos\theta)=\sin((m+1)\theta)/(\sin\theta)$, form an orthonormal basis
of $L^{2}([0,\pi],\widetilde{\mu}_{\mathrm{ST}})$. We approximate
$\mathbf{1}_{I}(\theta_{v})\mathbf{1}_{I^{\prime}}(\theta_{v}^{\prime})$ with
respect to this basis as follows.

######  Lemma 8.4.

Let $I,I^{\prime}\subseteq[0,\pi]$ be subintervals. For each $M\geq 1$, there
exist functions

$T_{I,I^{\prime},M}^{\pm}(\theta,\theta^{\prime})=\sum_{0\leq m\leq
M}~{}\sum_{0\leq n\leq
M}\alpha_{I,I^{\prime},M}^{\pm}(m,n)U_{m}(\cos\theta)U_{n}(\cos\theta^{\prime})$

such that if $\theta,\theta^{\prime}\in[0,\pi]$, then
$T_{I,I^{\prime},M}^{-}(\theta,\theta^{\prime})\leq\mathbf{1}_{I}(\theta)\mathbf{1}_{I^{\prime}}(\theta^{\prime})\leq
T_{I,I^{\prime},M}^{+}(\theta,\theta^{\prime})$. There holds

$\left|\alpha_{I,I^{\prime},M}^{\pm}(m,n)-\begin{cases}\mu_{\mathrm{ST}}(I)\mu_{\mathrm{ST}}(I^{\prime})&\mbox{if
$m=n=0$,}\\\ 0&\mbox{otherwise}\end{cases}\right|\ll\begin{cases}1/M&\mbox{if
$m=n=0$,}\\\ 1/m&\mbox{if $m\neq 0$ and $n=0$,}\\\ 1/n&\mbox{if $m=0$ and
$n\neq 0$,}\\\ 1/(mn)&\mbox{otherwise.}\end{cases}$

###### Proof.

For $j=1,2$, let $I_{j}=[a_{j},b_{j}]\subseteq[0,\pi]$ and
$J_{j}=[a_{j}/(2\pi),b_{j}/(2\pi)]$. Given an interval $J\subseteq[0,1]$, the
Fourier expansion of $\mathbf{1}_{J}(\theta)$ is
$\mathbf{1}_{J}(\theta)=\sum_{n\in\mathbb{Z}}\widehat{\mathbf{1}}_{J}(n)e^{2\pi
in\theta}$, where $\widehat{\mathbf{1}}_{J}(n)=\int_{J}e^{-2\pi int}dt$.
Cochrane [8, Theorem 1] proved that there exist functions

$S_{J_{1},J_{2},M}^{\pm}(x_{1},x_{2})=\sum_{\begin{subarray}{c}m_{1},m_{2}\in\mathbb{Z}\\\
0\leq|m_{1}|,|m_{2}|\leq
M\end{subarray}}\widehat{S}_{J_{1},J_{2},M}^{\pm}(m_{1},m_{2})e^{2\pi
i(m_{1}x_{1}+m_{2}x_{2})}$

with the following properties.

  1. (1)

If $(x_{1},x_{2})\in[0,1]^{2}$, then
$S_{I_{1},I_{2},M}^{-}(x_{1},x_{2})\leq\mathbf{1}_{J_{1}}(x_{1})\mathbf{1}_{J_{2}}(x_{2})\leq
S_{I_{1},I_{2},M}^{+}(x_{1},x_{2})$.

  2. (2)

The identity
$\widehat{S}_{J_{1},J_{2},M}^{\pm}(m_{1},m_{2})+\widehat{S}_{J_{1},J_{2},M}^{\pm}(-m_{1},-m_{2})=2\mathrm{Re}(\widehat{S}_{J_{1},J_{2},M}^{\pm}(m_{1},m_{2}))$
holds.

  3. (3)

If $0\leq|m_{1}|,|m_{2}|\leq M$, then
$|\widehat{S}_{J_{1},J_{2},M}^{\pm}(m_{1},m_{2})-\widehat{1}_{J_{1}}(m_{1})\widehat{1}_{J_{2}}(m_{2})|\ll
1/M$.

The desired functions $T_{I_{1},I_{2},M}^{\pm}(\theta,\theta^{\prime})$ are
the even functions

$\displaystyle\sum_{t_{1}\in\\{-1,1\\}}\sum_{t_{2}\in\\{-1,1\\}}S_{J_{1},J_{2},M}^{\pm}\Big{(}t_{1}\frac{\theta}{2\pi},t_{2}\frac{\theta^{\prime}}{2\pi}\Big{)}$

Once we introduce $U_{-1}(\cos\theta)=0$ and $U_{-2}(\cos\theta)=-1$, we
express $T_{I_{1},I_{2},M}^{\pm}$ in the Chebyshev basis using the identity
$U_{m}(\cos\theta)-U_{m-2}(\cos\theta)=2\cos(m\theta)$ for $m\geq 0$. ∎

######  Corollary 8.5.

If $x\geq 2$ and $I,I^{\prime}\subseteq[-1,1]$ are subintervals, then

(8.5) $\displaystyle\Big{|}\sum_{\begin{subarray}{c}v\in\mathcal{V}\\\
q_{v}\leq
x\end{subarray}}\mathbf{1}_{I}(\theta_{v})\mathbf{1}_{I^{\prime}}(\theta_{v}^{\prime})\Big{/}\sum_{\begin{subarray}{c}v\in\mathcal{V}\\\
q_{v}\leq
x\end{subarray}}1-\widetilde{\mu}_{\mathrm{ST}}(I)\widetilde{\mu}_{\mathrm{ST}}(I^{\prime})\Big{|}$
$\displaystyle\ll\frac{1}{M}+\sum_{\begin{subarray}{c}0\leq m,n\leq M\\\
(m,n)\neq(0,0)\end{subarray}}\Big{|}\sum_{\begin{subarray}{c}v\in\mathcal{V}\\\
q_{v}\leq
x\end{subarray}}a_{\mathrm{Sym}^{m}(\pi)\times\mathrm{Sym}^{n}(\pi^{\prime})}(v)\Big{|}\Big{/}\sum_{\begin{subarray}{c}v\in\mathcal{V}\\\
q_{v}\leq x\end{subarray}}1.$

###### Proof.

This is a straightforward consequence of Lemma 8.4 and the fact that if
$v\in\mathcal{V}$, then
$U_{m}(\cos\theta_{v})U_{n}(\cos\theta_{v}^{\prime})=\zeta^{-m}\zeta^{\prime-n}a_{\mathrm{Sym}^{m}(\pi)\times\mathrm{Sym}^{n}(\pi^{\prime})}(v)$.
∎

If $G$ is the finite abelian group of characters generated by $\omega_{\pi}$
and $\omega_{\pi^{\prime}}$, then the condition $v\in\mathcal{V}$ can be
expressed using orthogonality of characters in $G$. If $\chi\in G$, then there
exist integers $a,b\geq 0$ such that
$\chi=\omega_{\pi}^{a}\omega_{\pi^{\prime}}^{b}$. Since
$\widetilde{\pi}=\pi\otimes\overline{\omega}_{\pi}$ and
$\widetilde{\pi}^{\prime}=\pi^{\prime}\otimes\overline{\omega}_{\pi^{\prime}}$,
we have that $S_{\chi}\subseteq S_{\pi}\cup S_{\pi^{\prime}}$ and
$C(\chi)\leq\mathcal{Q}$ per [40, Theorem A]. We now observe that

(8.6) $\Big{|}\sum_{\begin{subarray}{c}v\in\mathcal{V}\\\ q_{v}\leq
x\end{subarray}}a_{\mathrm{Sym}^{m}(\pi)\times\mathrm{Sym}^{n}(\pi^{\prime})}(v)\Big{|}\ll\max_{\chi\in
G}\Big{|}\sum_{\begin{subarray}{c}v\notin S_{\pi}\cup S_{\pi^{\prime}}\\\
q_{v}\leq
x\end{subarray}}a_{\mathrm{Sym}^{m}(\pi)\times(\mathrm{Sym}^{n}(\pi^{\prime})\otimes\chi)}(v)\Big{|}.$

Therefore, by Lemma 8.1, Corollary 8.3, and (8.6), we find that (8.5) is, as
desired,

$\ll\frac{1}{M}+\frac{\mathcal{Q}^{\Cr{CDT}}}{(\mathcal{Q}\log
x)^{\Cr{CDT}}}\ll\sqrt{[F:\mathbb{Q}]}\Big{(}\frac{\log(\mathcal{Q}\log
x)}{\sqrt{\log x}}\Big{)}^{\frac{2}{2+Y}}.$

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