A mean value of a triple product of $L$-functions

TL;DR

This paper investigates the mean value of certain $L$-functions related to dihedral Maass forms, providing an asymptotic with power savings that could advance understanding of $L^4$-norms.

Contribution

It introduces a new asymptotic analysis of a mean value of $L$-functions similar to those used in bounding $L^4$-norms of Maass forms.

Findings

Derived an asymptotic with power savings for the mean value of $L$-functions.

Provides insights that may lead to an asymptotic for the $L^4$-norm.

Builds on Luo's bounds for dihedral Maass forms.

Abstract

Luo has proven an optimal upper bound for the $L^4$-norm of dihedral Maass forms of large eigenvalue, by bounding a mean value of triple product $L$-functions. Motivated by this result, we study a mean value of $L$-functions having similar shape, and obtain for it an asymptotic with power savings. Our work may be helpful in eventually obtaining an asymptotic for the $L^4$-norm.