On the mean values of L-functions in orthogonal and symplectic families

Hybrid Euler–Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet L-functions, in the context of the calculation of moments and connections with Random Matrix Theory. According to the Katz–Sarnak classification, these are believed to represent families of L-function with unitary symmetry. We here extend the formalism to families with orthogonal and symplectic symmetry. Specifically, we establish formulae for real quadratic Dirichlet L-functions and for the L-functions associated with primitive Hecke eigenforms of weight 2 in terms of partial Euler and Hadamard products. We then prove asymptotic formulae for some moments of these partial products and make general conjectures based on results for the moments of characteristic polynomials of random matrices.