Simultaneous non-vanishing of $GL(3)\times GL(2)$ and GL(2) $L$-functions

Abstract: Fix $g$ a Hecke-Maass form for $SL_3(\mathbb{Z})$. Let $q$ be any large prime number. In the family of holomorphic newforms $f$ of level $q$ and fixed weight, we find the average value of the product $L(\half,g\times f)L(\half,f)$. From this we derive a result on the simultaneous non-vanishing of these $L$-functions at the central point.