Let $χ$ range over the $(p-1)/2$ even Dirichlet characters modulo a prime $p$ and denote by $θ(x,χ)$ 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 $θ(x,χ) \neq 0$ for any $x>0$. We give similar results for odd Dirichlet characters mod $p$.