Zeroes of a non-zero entire function

Question

Let $f$ be a nonzero entire function such that $|f(z)|\leq e^{|z|}$. Can $f(\sqrt{n})=0$ for infinitely many $n$? We know that the Weierstrass Factor Theorem asserts that given any closed discrete set $A$ in $\mathbb{C}$, we can always find an entire function can assumes zero of any desired multiplicity at each point of $A$. I am trying to use the WFT but am reaching nowhere. Any help will be appreciated.