The function I have is $f(z)=\zeta(z)\Gamma(z − 1)\sin(\pi z)$ and I need to find all singularities and their residues so I can evaluate a clockwise contour integral for the contour $\left\lvert z+\frac{1}{2}\right\rvert=4.$
So far I've simplified $f(z)$ using the identity $\Gamma(z)=(z-1)\Gamma(z − 1)$ and then with Euler's reflection formula $\Gamma(z)\sin(\pi z)=\frac{\pi}{\Gamma(1-z)}$ to get $f(z)=\zeta(z)\pi /(\Gamma(1-z))(z-1)).$
$\zeta(z)$ has a pole at $z=1$, so combined with the $\frac{1}{z-1}$, $f(z)$ has a double pole at $z=1$. Is my logic correct so far? I'm not sure how to calculate the residue now. I know $\zeta(z)$ has a residue of $1$ at $z=1,$ unsure where to go from here.