Central Values of $GL(2)\times GL(3)$ Rankin-Selberg $L$-functions with Applications

Abstract

Let $f$ be a normalized holomorphic cusp form for $S{L}_2(\mathbb{Z})$ of weight $k$ with $k\equiv 0\mathrm{mod}4$. By the Kuznetsov trace formula for $G{L}_3(\mathbb{R})$, we obtain the first moment of central values of $L(s,f\otimes \varphi )$, where $\varphi$ varies over Hecke-Maass cusp forms for $S{L}_3(\mathbb{Z})$. As an application, we obtain a non-vanishing result for $L(1/2,f\otimes \varphi )$ and show that such $f$ is determined by $\{L(1/2,f\otimes \varphi )\}$ as $\varphi$ varies.

18 pages