Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)

Abstract

Let ff be a self-dual Hecke-Maass eigenform for the group SL3(Z)S{L}_{3}\left({\mathbb{Z}}). For 12<σ<1\frac{1}{2}\lt \sigma \lt 1 fixed we define m(σ)m\left(\sigma ) (≥2\ge 2) as the supremum of all numbers mm such that ∫1T∣L(s,f)∣mdt≪f,εT1+ε,\underset{1}{\overset{T}{\int }}| L\left(s,f){| }^{m}{\rm{d}}t{\ll }_{f,\varepsilon }{T}^{1+\varepsilon }, where L(s,f)L\left(s,f) is the Godement-Jacquet L-function related to ff. In this paper, we first show the lower bound of m(σ)m\left(\sigma ) for 23<σ<1\frac{2}{3}\lt \sigma \lt 1. Then we establish asymptotic formulas for the second, fourth and sixth powers of L(s,f)L\left(s,f) as applications.

Keywords

• power moments
• l-function
• automorphic form
• 11f03
• 11f66