  Let $\chi$ range over the $(p-1)/2$ even Dirichlet characters modulo a prime
$p$ and denote by $\theta (x,\chi)$ the associated theta series. The asymptotic
behaviour of the second and fourth moments proved by Louboutin and the author
implies that there exists at least $ \gg p/ \log p$ characters such that the
associated theta function does not vanish at a fixed point. Constructing a
suitable mollifier, we improve this result and show that there exists at least
$ \gg p/ \sqrt{\log p}$ characters such that $\theta(x,\chi) \neq 0$ for any
$x>0$. We give similar results for odd Dirichlet characters mod $p$.
