Landau-Siegel Zeros of Triple Product L-functions

Abstract

Let $F$ be a number field. Let $π_1,π_2$ be cuspidal automorphic representations of $GL_2(\mathbb{A}_F)$, and let $π$ be a cuspidal automorphic representation of either $GL_2(\mathbb{A}_F)$ or $GL_3(\mathbb{A}_F)$. When $(π_1,π_2,π)$ is of general type, we show that the triple product $L$-function $L(s,π_1 \times π_2 \times π)$ 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 $(π_1,π_2,π)$ is not of general type, we give precise conditions when $L(s,π_1 \times π_2 \times π)$ could possibly have exceptional Landau-Siegel zeros.

Theorems & Definitions (23)

• Theorem A
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem B
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Theorem C
• Lemma 2.1