What is the value of the sum of $e^{i(-ak^2 + bk)}$ when $k$ goes through every integer?

Question

I'm doing a physics question sheet and I stumbled upon a series of the kind $$ \sum_{k \in \mathbb{Z}} e^{i(-ak^{2}+bk)} $$ Where $a$ and $b$ are real numbers with $a>0$.

I have no idea how to do this and not even Mathematica will calculate it for me!

Answer 1

This has the structure of a theta function but the series need not converge unless $a$ is restricted to a complex half-plane ( $a$ cannot be purely real).