My function is $f(z)$ is analytic in Re$(s)>1$, defined by meromorphic continuation elsewhere. It has a simple pole only at $s=0$. I want to integrate $$\int_{-i\infty}^{i\infty}f(z)dz$$ knowing that $f(z)$ has sufficient decay at infinity. My problem is integrating over the pole: I believe that it should be interpreted as a Cauchy principal value, or equivalently, as a deformed contour $$\lim_{R\to\infty}\lim_{\epsilon\to0}(\int_{-iR}^{-i\epsilon}+\int_{C_\epsilon}+\int_{i\epsilon}^{iR}+\int_{C_R}f(z)dz)$$ where $C_R$ is a contour to the right connecting $-iR$ to $iR$ (using a Jordan lemma type argument), and $C_\epsilon$ is a semicircle around the pole $s=0$.
I am told that the contribution of the pole should be half of the residue, as the angle is $\pi$, but I am confused about the sign the residue should take, as my integral is oriented clockwise.
Question: should the contribution of the pole be $-i\pi\text{Res}(f(z),0)$?