Let $\psi$ be a character with conductor $f_\psi$, that is, $$\psi:(\mathbb{Z}/f_\psi\mathbb{Z})^{\times}\rightarrow\mathbb{C}^{\times}$$ is a group homomorphism. Define $$F_\psi(z)=\begin{cases}\sum_{n=1}^{\infty}\psi(n)e^{2\pi i nz}&\text{ if }& \text{Im}(z)>0\\ -\sum_{n=1}^{\infty}\psi(-n)e^{-2 \pi inz}&\text{ if }&\text{Im}(z)<0\end{cases}.$$ How to compute the following integral? $$\int_{-i\infty}^{i\infty}F_\psi(z)\ dz=\int_{0}^{i\infty}\sum_{n=1}^{\infty}\psi(n)e^{2\pi i nz}\ dz-\int_{-i\infty}^{0}\sum_{n=1}^{\infty}\psi(-n)e^{-2 \pi inz}\ dz.$$ I know from the Fubini's theorem, the summation cannot interchange with the integral here, but I can show that $F_\psi$ has no pole at $s=0$. Is that helpful for this question?
I guess that there are two possible ways to approach. The first one is to rearrange $F_\psi$ into something looks analytic. The other is to make an analogue to the Gamma function.