Low-lying zeros of dihedral L-functions

Assuming the grand Riemann hypothesis, we investigate the distribution of the lowlying zeros of the L-functions L(s, psi), where psi is a character of the ideal class group of the imaginary quadratic field Q(root-D) (D squarefree, D > 3, D equivalent to 3 (mod 4)). We prove that, in the vicinity of the central point s = 1/2, the average distribution of these zeros (for D --> infinity) is governed by the symplectic distribution. By averaging over D, we go beyond the natural bound of the support of the Fourier transform of the test function. This problem is naturally linked with the question of counting primes p of the form 4p = m(2) + Dn(2) and sieve techniques are applied.