Second moments of twisted Koecher-Maass series

Abstract

Imai considered the twisted Koecher-Maass series for Siegel cusp forms of degree 2, twisted by Maass cusp forms and Eisenstein series, and used them to prove the converse theorem for Siegel modular forms. They do not have Euler products, and it is not even known whether they converge absolutely for Re(s)>1. Hence the standard convexity arguments do not apply to give bounds. In this paper, we obtain the average version of the second moments of the twisted Koecher-Maass series, using Titchmarsh’s method of Mellin inversion. When the Siegel modular form is a Saito Kurokawa lift of some half integral weight modular form, a theorem of Duke and Imamoglu says that the twisted Koecher Maass series is the Rankin-Selberg L-function of the half-integral weight form and Maass form of weight 1/2. Hence as a corollary, we obtain the average version of the second moment result for the Rankin-Selberg L-functions attached to half integral weight forms.