Landau-Siegel Zeros of Triple Product L-functions

Abstract

Let $F$ be a number field. Let ${\pi }_1,{\pi }_2$ be cuspidal automorphic representations of $G{L}_2({\mathbb{A}}_F)$, and let $\pi$ be a cuspidal automorphic representation of either $G{L}_2({\mathbb{A}}_F)$ or $G{L}_3({\mathbb{A}}_F)$. When $({\pi }_1,{\pi }_2,\pi )$ is of general type, we show that the triple product $L$-function $L(s,{\pi }_1\times {\pi }_2\times \pi )$ on either $GL(2)\times GL(2)\times GL(2)$ or $GL(2)\times GL(2)\times GL(3)$ has a standard zero-free region with no exceptional Landau-Siegel zero. Moreover, when $({\pi }_1,{\pi }_2,\pi )$ is not of general type, we give precise conditions when $L(s,{\pi }_1\times {\pi }_2\times \pi )$ could possibly have exceptional Landau-Siegel zeros.

21 pages. This preprint is now superseded by arXiv:2601.04189