References for this function: $\Psi_{\chi}(s)=\sum_{n=1}^\infty \chi(n){e^{-n^s}}?$

Question

Consider $$\Psi_{\chi}(s)=\sum_{n=1}^\infty \chi(n){e^{-n^s}}$$

where $\chi$ is a Dirichlet character.

Is anything known about it?

I looked around and didn't see any references involving this function.

Are there any references to read about this function? Any online videos?

I'll take references at any level.