= Dirichlet character theta function
{c}
{title2=$\theta_\chi(x)=\sum_{n\in\mathbb Z}n^a\chi(n)e^{-\pi n^2x/q}$}
For a <primitive Dirichlet character> whose <conductor of a Dirichlet character> is $q>1$ and whose <character parity> is $a$, <Poisson summation> gives $\theta_\chi(x)=\varepsilon_\chi x^{-a-1/2}\theta_{\overline\chi}(1/x)$, where $\varepsilon_\chi=\tau(\chi)/(i^a\sqrt q)$. Its rapid decay at both ends yields the entire Mellin representation of the <completed Dirichlet L-function>. The case of <conductor of a Dirichlet character> equal to one uses the ordinary <Jacobi theta function> and a subtracted constant term.
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