I was reading this post about the Bell Numbers where users Lucian and Vladimir Reshetnikov give us Dobiński's formula for the Bell numbers

$$ B(x) = \frac{1}{e} \sum_{k=1}^{\infty} \frac{k^x}{k!}. $$

Now I was trying to reason about this function on the complex plane. It's easy to reason that for no value of $x$ such that $|x| < \infty$ will this expression have a singularity so this function definitely doesn't have any poles.

Now the other question I wanted to turn my attention to are, can we characterize the zeroes of this function? Are there only finitely many or countably many? Is there some natural characterization of the zeroes (like they all live on some well known curve(s))?

There also appears to be some Riemann surface structure attached to this wherein $k^x$ is multivalued for complex $x$ as it is $e^{\ln(k)x}$ where the choice of branch of $\ln(k)$ needs to be made.