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

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