Twisted Moments of Rankin-Selberg $L$-functions in the Prime-Power Level Aspect

Abstract

We compute the twisted first and second moments of the shifted central values of the Rankin-Selberg $L$-functions given by $L\left(\frac12+ω, f\otimes g\right)$ as $f$ varies over primitive forms of prime power level $p^ν$ with $ν\geq 3$. Here $ω$ is a bounded shift and $g$ is a fixed primitive form of level relatively prime to $p$.