Vanishing and non-vanishing theta values

For a primitive Dirichlet character $$\chi $$ of conductor $$N$$ set $$\begin{aligned} \Theta (\chi )=\sum _{n\in \mathbb Z } n^\epsilon \chi (n)\,e^{-\pi n^2/N} \end{aligned}$$ (where $$\epsilon =0$$ for $$\chi $$ even, $$\epsilon =1$$ for $$\chi $$ odd), the value of the associated theta series at its point of symmetry under the modular transformation $$\tau \rightarrow -1/\tau $$ . These numbers are related by $$\Theta (\chi )=W(\chi )\Theta (\bar{\chi })$$ to the root number of the $$L$$ -series of $$\chi $$ and hence can be used to calculate the latter quickly if they do not vanish. We describe experiments showing that $$\Theta (\chi )\ne 0$$ for all $$\chi $$ with $$N\le 52{,}100$$ (roughly 500 million primitive characters) except for precisely two characters (up to $$\chi \rightarrow \bar{\chi }$$ ), of conductors $$300$$ and $$600$$ . The proof that $$\Theta (\chi )$$ vanishes in these two cases uses properties of Ramanujan’s modular function of level $$5$$ . We also characterize all $$\chi $$ for which $$W(\chi )$$ is a root of unity and describe some experimental results concerning the algebraic numbers $$\Theta (\chi )/\eta (i)^{1+2\epsilon }$$ when $$N$$ is prime.