Determination of cusp forms by central values of Rankin–Selberg L-functions^*

Abstract

Let f be a fixed holomorphic Hecke eigen cusp form of weight k for $SL\left( {2,{\mathbb Z}} \right)$, and let ${\mathcal U} = \left\{ {{u_j}:j \geqslant 1} \right\}$ be an orthonormal basis of Hecke–Maass cusp forms for $SL\left( {2,{\mathbb Z}} \right)$. We prove an asymptotic formula for the twisted first moment of the Rankin–Selberg L-functions $L\left( {s,f \otimes {u_j}} \right)$ at $s = \frac{1}{2}$ as u _j runs over ${\mathcal U}$. It follows that f is uniquely determined by the central values of the family of Rankin–Selberg L-functions $\left\{ {L\left( {s,f \otimes {u_j}} \right):{u_j} \in {\mathcal U}} \right\}$.