Mixed moment of $GL(2)$ and $GL(3)$ $L$-functions in the level aspect

Abstract

In this paper, we establish an asymptotic formula of mixed moments of $L(1/2,f)$ and $L(1/2,{\mathrm{sym}}^{2}f)$ for cusp forms $f$ of weight $k$, prime level $q$ and real primitive nebentypus $\chi$ modulo $q$ in the level aspect with a power-saving error term. As an application, we deduce the simultaneous non-vanishing of $L(1/2,f)$ and $L(1/2,{\mathrm{sym}}^{2}f)$ for sufficiently large prime levels.

15 pages

References in corpus (3)

• Oscillatory integrals with uniformity in parameters
• A Mean Value Result for a Product of GL(2) and GL(3) L-Functions
• Mixed moment of $GL(2)$ and $GL(3)$ $L$-functions