Number of zeros of an entire function

Question

I want to find the zeros of the function \begin{equation} e^{\gamma z}\prod_{n=1}^{\infty}\left(1+\frac{z}{n}\right)e^{-\frac{z}{n}}-1 \end{equation} for some constant $\gamma$. The function is clearly equal to zero at $z=0$. But I am unable to decide whether it has zero at other points in the complex plane as well. I can not use the Casorati Weierstrass theorem since the function does not have any essential singularity in the finite plane. It is actually an entire function having oder $1$. Any entire function having infinite number of zeros have its convergence exponent $>1$. Can we use it in some way to find the zeros of the given function in any way? Please help.