On the cubic moment of quadratic dirichlet L-functions

where χD is the quadratic Dirichlet character associated to Q( √ D) as defined in [11]. Besides its own interest, this mean value problem also plays a crucial role in the studies of such as the Lindelöf Conjecture and the folklore non-vanishing conjecture that L ( 1 2 , χD ) = 0. See, for example, [5] [7] [12]. Jutila [7] was the first to obtain the asymptotic formulas for the cases m = 1, 2, and Soundararajan [12] succeeded in the cubic case. For higher moments, a good upper bound has been obtained by Heath-Brown [6] in the quartic case, but their asymptotic formulas are still out of reach. In general, motivated by the fundamental work of Katz and Sarnak [8] on symmetric types associated to families of L-functions, and by calculations of Keating and Snaith [9] based on random matrix theory, Conrey and Farmer have made the following conjecture ∑