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

Abstract

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

36 pages