Tatuzawa's theorem for Rankin-Selberg $L$-functions

TL;DR

This paper establishes a new zero-free region for Rankin-Selberg $L$-functions twisted by automorphic representations, generalizing Tatuzawa's refinement and extending previous results to broader cases with explicit bounds.

Contribution

It introduces a novel zero-free region for all twists of Rankin-Selberg $L$-functions, including a new standard zero-free region for twists of $L(s, ext{pi} imes ilde{ ext{pi}})$, extending prior work.

Findings

Proves a zero-free region for all $ ext{GL}(1)$-twists of $L(s, ext{pi} imes ext{pi}')$.

Shows at most one zero in a specified region, with an explicit effective bound.

Extends earlier results by Humphries and Thorner to more general cases.

Abstract

Let $\pi$ and $\pi'$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ over a number field $F$. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,\pi\times\pi')$, generalizing Tatuzawa's refinement of Siegel's work on Dirichlet $L$-functions. As a corollary, we show that for all $\varepsilon>0$, there exists an effectively computable constant $c>0$ depending only on $(n,n',[F:\mathbb{Q}],\varepsilon)$ such that $L(s,\pi\times\pi')$ has at most one zero (necessarily simple) in the region \[ \mathrm{Re}(s)\geq 1-c/(C(\pi)C(\pi')(|\mathrm{Im}(s)|+1))^{\varepsilon}, \] where $C(\pi)$ and $C(\pi')$ are the analytic conductors. A crucial component of our proof is a new standard zero-free region for any twist of $L(s,\pi\times\widetilde{\pi})$ by an idele class character $\chi$ apart from a possible single exceptional zero (necessarily real and simple) that can occur only when $\pi\otimes\chi^2=\pi$. This extends earlier work of Humphries and Thorner.