Mixed moment of $GL(2)$ and $GL(3)$ $L$-functions

Abstract

Let $\mathfrak{f}$ run over the space $H_{4k}$ of primitive cusp forms of level one and weight $4k$, $k\in N$. We prove an explicit formula for the mixed moment of the Hecke $L$-function $L(\mathfrak{f},1/2)$ and the symmetric square $L$-function $L(sy{m}^2\mathfrak{f},1/2)$, relating it to the dual mixed moment of the double Dirichlet series and the Riemann zeta function weighted by the ${}_3{F}_2$ hypergeometric function. Analysing the corresponding special functions by the means of the Liouville-Green approximation followed by the saddle point method, we prove that the initial mixed moment is bounded by ${\log }^{3}k$.

arXiv comment38 pages

References in corpus (11)

• The Weyl bound for Dirichlet $L$-functions of cube-free conductor
• Weyl-type hybrid subconvexity bounds for twisted $L$-functions and Heegner points on shrinking sets
• A generalized cubic moment and the Petersson formula for newforms
• Twisted moments of L-functions and spectral reciprocity
• A Twisted Motohashi Formula and Weyl-Subconvexity for $L$-functions of Weight Two Cusp Forms
• On the reciprocity law for the twisted second moment of Dirichlet $L$-functions
• The reciprocity law for the twisted second moment of Dirichlet L-functions
• Level Reciprocity in the twisted second moment of Rankin-Selberg L-functions
• On the mean value of symmetric square L-functions
• A Mean Value Result for a Product of GL(2) and GL(3) L-Functions
• A note on the second moment of automorphic L-functions

Cited by in corpus (3)

• Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions I: The Cuspidal Case
• Mixed moment of $GL(2)$ and $GL(3)$ $L$-functions in the level aspect
• Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions II: The Eisenstein Case