The central value of the Rankin-Selberg $L$-functions

TL;DR

This paper establishes an asymptotic formula for the average of products of Rankin-Selberg $L$-functions involving Maass forms on $SL(3,\mathbb{Z})$ and $SL(2,\mathbb{Z})$, leading to nonvanishing results at the central point.

Contribution

It provides a new asymptotic formula for averages of Rankin-Selberg $L$-functions involving Maass forms, advancing understanding of their nonvanishing at the central point.

Findings

Asymptotic formula for average of $L$-function products

Nonvanishing results at the central value $1/2$

Insights into the distribution of $L$-values for Maass forms

Abstract

Let $f$ be a Maass form for $SL(3, \mathbb{Z})$ which is fixed and $u_j$ be an orthonormal basis of even Maass forms for $SL(2, \mathbb{Z}),$ we prove an asymptotic formula for the average of the product of the Rankin-Selberg $L$-function of $f$ and $u_j$ and the $L$-function of $u_j$ at the central value 1/2. This implies simultaneous nonvanishing results of these $L$-functions at $1/2.$