Let $(z_i)$ be a square-summable sequence which is even summable but not absolute summable, i.e. $\sum_{i=1}^{\infty} \vert z_i \vert = \infty$,$\sum_{i=1}^{\infty} \vert z_i \vert^2 < \infty$ and $\sum_{i=1}^{\infty} z_i$ exists. I would like to ask if the following function $$f(\mu):=\prod_{i=1}^{\infty}(1+\mu^2 \vert z_i \vert^2 - 2 \mu \Re(z_i))$$ is necessarily entire?

I must say that I do not even know if this product exists away from the real axis. On the real axis it is clear that it exists by using that $(1+x) \le e^x$ such that $\vert f(\mu) \vert \le e^{\mu^2 \sum_i \vert z_i \vert^2 -2\mu \Re \sum_i z_i }.$

Assuming it was entire, does there exist a similar growth bound on $\vert f(\mu) \vert$ as the one I obtained on the real axis?