 Theta function with characteristics

For real characteristics $\alpha ,\beta$ and $\mathrm{Im}\tau >0$, Gaussian decay gives normal convergence, an entire function of $z$ and holomorphic dependence on $\tau$. Integer shifts of alpha leave the function unchanged; a unit shift of beta multiplies it by $e^{2\pi i\alpha }$. Its spatial quasi-periods are $\theta (z+1)=e^{2\pi i\alpha }\theta (z)$ and $\theta (z+\tau )=e^{-\pi i\tau -2\pi i(z+\beta )}\theta (z)$. Binary characteristic labels often mean alpha and beta divided by two, so the convention must be stated.