The completed standard $L$-function of modular forms on $G_2$

TL;DR

This paper explicitly defines and proves the functional equation for the completed standard L-function of certain non-generic automorphic representations of the exceptional group G_2, using Rankin-Selberg integrals.

Contribution

It provides a complete and refined study of the standard L-function for non-generic cuspidal automorphic representations of G_2, including explicit definitions and proof of the functional equation.

Findings

Explicit definition of the completed standard L-function for G_2

Proof of the functional equation under a nonzero Fourier coefficient assumption

Analysis of Rankin-Selberg integrals for G_2 representations

Abstract

The goal of this paper is to provide a complete and refined study of the standard $L$-functions $L(\pi,\operatorname{Std},s)$ for certain non-generic cuspidal automorphic representations $\pi$ of $G_2(\mathbb{A})$. For a cuspidal automorphic representation $\pi$ of $G_2(\mathbb{A})$ that corresponds to a modular form $\varphi$ of level one and of even weight on $G_2$, we explicitly define the completed standard $L$-function, $\Lambda(\pi,\operatorname{Std},s)$. Assuming that a certain Fourier coefficient of $\varphi$ is nonzero, we prove the functional equation $\Lambda(\pi,\operatorname{Std},s) = \Lambda(\pi,\operatorname{Std},1-s)$. Our proof proceeds via a careful analysis of a Rankin-Selberg integral that is due to an earlier work of Gurevich and Segal.