I need to find, in fact, need to show that $I(s)=\int_{1}^{\infty}\frac{S(x)-x+½}{x^{s+1}}dx$
where $S(x)=\sum_{n\leq x} \chi (n)$
is analytic in the region Re($s$)>-q and $\chi(n)$ is non principal character mod q
There could be error in the question as well. Please notify the same with solution. Thank you