This came up when reading about elliptic functions, where $\frac{\omega_1}{\omega_2}\notin\mathbb{R}$, and $\alpha>2$ for $$S(\omega_1,\omega_2,\alpha)=\sum_{\begin{matrix} n,m=-\infty\\ (n,m)\ne (0,0) \end{matrix}}^\infty \frac{1}{(n\omega_1+m \omega_2)^\alpha}$$ to converge. My question is what is the relationship between $\omega_1,\omega_2$ and $\alpha$ so that $S\in \mathbb{R}$ is true?
For instance, if $\alpha$ is odd, then we have
$$\frac{1}{(n\omega_1+m\omega_2)^\alpha}+\frac{1}{(-n\omega_1-m\omega_2)^\alpha}=0$$ and $S=0\in \mathbb{R}$.