The following integral absolutely converges for $Re(z)<0$ and is analytic in this domain: $$F(z):=\int_{0}^{+\infty}\frac{\sin r}{r}e^{\sqrt{r^2+m^2}z}dr ,$$ where $m>0$ is fixed.

Question. To what maximal domain of $\mathbb{C}$ does $F$ extend as an analytic (possibly multi-valued) function of $z$? Does it extend to the whole of $\mathbb{C}$ without a discrete subset? I am particularly interested whether it extends to the whole real line without 0.