Mixed moment of GL(2) and GL(3)L‐functions

Abstract

Let frakturf run over the set H4k of primitive cusp forms of level one and weight 4k, k∈N. We prove an explicit formula for the mixed moment of the Hecke L‐function L(f,1/2) and the symmetric square L‐function L(prefixsym2f,1/2), relating it to the dual mixed moment of a double Dirichlet series and the Riemann zeta function weighted by the 3F2 hypergeometric function. Analysing the corresponding special functions by means of the Liouville–Green approximation followed by the saddle point method, we prove that the initial mixed moment is bounded above by log3k.

On Motohashi’s formula

“…This possibility does not seem to be within the scope of Michel-Venkatesh's period formalism. Such moment was recently studied in [1,2]. It should be possible to generalize these results over general number fields in the light of this possible application of Theorem 2.10.…”

“…in any Siegel domain. (2) Its Fourier inversion with respect to y in L 2 (GL 2 (F)\GL 2 (A), ω −1 ) converges normally for (x, y) ∈ (GL 2 (F)\GL 2 (A)) 2 , and takes the form…”

“…where KK (2) (x; y) is defined in (2.2). In order to rewrite E 2 , we notice that * L n(u) L d(t1,t2) −1 R g Ψ (ξ) du.…”

“…1 In fact, this follows also from the theory of zeta function for finitely regularizable functions in[26, §2.3] 2. We do not know how to work directly with the original region.…”

Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions

“…More generally, the transform can be expressed in terms of hypergeometric functions of special types. Recently, the articles [BF18, BF21], [BBFR20], [BFW21+] brought in the powerful asymptotic analysis of hypergeometric functions into the study of moments which lead to sharp estimates. Our moment formula (1.3) should immediately open the door for such aspect for the family of GLp3q ˆGLp2q L-functions.…”