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

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 $1/2.$