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

Let Hk(Γ)Hk(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)Γ=SL(2,Z), and denote by L(s,sym2f) the corresponding symmetric square L  -function associated to f∈Hk(Γ)f∈Hk(Γ). In this paper, the lower bound of the higher moment of L(1/2,sym2f) is established, i.e., for any even number r∈Z+r∈Z+,∑f∈Hk(Γ)ωf−1L(1/2,sym2f)r≫r(logk)r(r+1)2 holds for k→∞k→∞, where ωf=k−12π2L(1,sym2f).