Lower bound for the higher moment of symmetric square L -functions

Abstract

Let H k ( Γ ) be the space of all normalized holomorphic Hecke-eigen cusp forms of even integral weight k for the full modular group Γ = SL ( 2 , Z ) , and denote by L ( s , sym 2 f ) the corresponding symmetric square L -function associated to f ∈ H k ( Γ ) . In this paper, the lower bound of the higher moment of L ( 1 / 2 , sym 2 f ) is established, i.e., for any even number r ∈ Z + , ∑ f ∈ H k ( Γ ) ω f − 1 L ( 1 / 2 , sym 2 f ) r ≫ r ( log k ) r ( r + 1 ) 2 holds for k → ∞ , where ω f = k − 1 2 π 2 L ( 1 , sym 2 f ) .