Jesse Thorner

Affiliation: University of Illinois Urbana-Champaign

Title Of Talk: Tatuzawa's theorem for Rankin--Selberg L-functions

URLS:
https://arxiv.org/abs/2508.10844

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 $\epsilon>0$, there exists an effectively computable constant $c>0$ depending only on $(n,n',[F:\mathbb{Q}],\epsilon)$ such that $L(s,\pi\times\pi')$ has at most one zero (necessarily simple) in the region \[ \mathrm{Re}(s)\geq 1-c/(C(\pi)C(\pi')(|\mathrm{Im}(s)|+1))^{\epsilon}, \] where $C(\pi)$ and $C(\pi')$ are the analytic conductors.

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