Is it possible to show that for $Re(s)>0$ the following integral converges.
$$\displaystyle\int_1^\infty \left[{x^{\frac{s}{2}-1} + x^{-\frac{s+1}{2}} }\right] \omega \left({x}\right)dx$$
Where $\omega \left({x}\right)=\sum\limits_{n=1}^{\infty}e^{-\pi n^2x}$. I know that it converges for $Re(s)>1$ but I would like to know if there is a way of proving convergence for $Re(s)>0$.