On positive real zeros of theta and L-functions associated with real, even and primitive characters

Let D range over the positive fundamental discriminants. Let θ(t; xD), t > 0, denote the theta function associated with the real, even and primitive Dirichlet character of conductor D. On the one hand, we prove that there are infinitely many positive discriminants D for which θ(t; xD) has at least one positive real zero. On the other hand, we prove that for a given positive real number t0, there are at least ≥ X= log13=2 X positive fundamental discriminants D ≥X for which θ(t0; xD)= 0.