Exceptional zeros of ${\mathrm{GL}}_3\times {\mathrm{GL}}_3$ Rankin-Selberg $L$-functions

Abstract

Let $\chi$ be an idele class character over a number field $F$, and let $\pi ,{\pi }^{\prime }$ be any two cuspidal automorphic representations of ${\mathrm{GL}}_2({\mathbb{A}}_F)$. We prove that 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 except possibly when it is divisible by the $L$-function of a real idele class character. In particular, no such zero exists if $\pi$ is non-dihedral and ${\pi }^{\prime }$ is not a twist of $\pi$. Until now, this was only known when $\pi ={\pi }^{\prime }$, $\pi$ is self-dual, and $\chi$ is trivial.

12 pages. This preprint is now superseded by arXiv:2601.04189. arXiv admin note: substantial text overlap with arXiv:2404.06482