Power series summing $e^{-n + i n^2}$ over positive integers

Question

Consider the series

$\sum_{n=1}^{\infty} e^{- \alpha n +i\beta n^2}$

with $\beta$ a real number, and $\alpha$ a complex number with positive real part, so the series converges absolutely. Is this series related to any special function, or, more open-endedly, can it be simplified to a closed-form expression that is not an infinite series?