Consider a function $f(z)$ which has an infinite number of zeros (only) along the positive real axis. I will write $f(z_n) = 0$, for $z_n \in \mathbb{R}$, with $z_n \geq 0 $ and labeled by $n \in \{1,2, \ldots\}$. Also, $\lim_{n\to\infty} z_n = \infty$. Importantly, $f(z)$ is not holomorphic.
Is there anything that I can say about this function if I know that $f(z)$ also has branch points somewhere on the complex plane away from the real axis? What would it take for me to determine $f(z)$ uniquely? How do you incorporate branch cuts into these theorems about entire functions?
In other words, one may try to write $f(z)$ in terms of the Weierstrass product, but of course that won't work very well because this product doesn't know anything about the branch points somewhere in the complex plane. So, is there a way to combine the Weierstrass (or Hadamard) product with the information on branch points and branch cuts? Let's say that I also know the asymptotics of $f(z)$ at infinity.
I would be extremely grateful for any examples of any remotely similar reconstructions of a function from an infinite product, where I could see how to determine such an $f(z)$ from its zeros + some other information (the question is what information? and how?). If it helps, think for example of something like (probably not the world's best example)
\begin{equation} \frac{1}{\Gamma(-\sqrt{1+z})}, \end{equation}
or something along those lines. I guess I am looking for something like
\begin{equation} f(z) = \left[\text{something} \right] \prod_{n=1}^\infty E_p\left(z/z_n\right). \end{equation}
Thanks very much!