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

Abstract

Let $\pi$ and ${\pi }^{\prime }$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n^{\prime })$ over a number field $F$. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,\pi \times {\pi }^{\prime })$, generalizing Tatuzawa's refinement of Siegel's work on Dirichlet $L$-functions. As a corollary, we show that for all $\epsilon >0$, there exists an effectively computable constant $c>0$ depending only on $(n,n^{\prime },[F:\mathbb{Q}],\epsilon )$ such that $L(s,\pi \times {\pi }^{\prime })$ has at most one zero (necessarily simple) in the region \[ \mathrm{Re}(s)\geq 1-c/(C(π)C(π')(|\mathrm{Im}(s)|+1))^{\varepsilon}, \] where $C(\pi )$ and $C({\pi }^{\prime })$ are the analytic conductors. A crucial component of our proof is a new standard zero-free region for any twist of $L(s,\pi \times \tilde{\pi })$ 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.

21 pages, LaTeX2e; v3: Small typographical errors fixed