Representation exponential type functions

Question

I am looking for a representation for the entire functions of $\pi$-exponential type. Let us suppose that $f$ is of $\pi$-exponential type and that it is equal to zero on the integers. It would mean that $$ f(z)=\sin(\pi z)h(z) $$ where $h$ is another entire function of exponential type. If I add some hypothesis to $h$, for example it is bounded on $\mathbb{Z}+1/2$, I mean $1/2, 3/2, -1/2$ etc..., is $h$ forced to be a constant? Any references would be extremely appreciate.