What is the most simple entire function vanishing at $n+in$ for $n \in \mathbb{N}$? I want to construct an entire function which vanishes at points $n+in$ for all $n$ integers. 
I'm looking for the most simple entire function which satisfies this condition, in the sense that the integers $p_n$ in weierstrass elementary factor should be as small as possible.
If i enumerate non-zero points of the form $n+in$ as $\{z_n\}$ i get
$$f(z)=z\prod_{n=1}^{+\infty}E(\frac{z}{z_n},p_n)$$
which is entire with zeros where i want, for suitable integers $p_n$.
But i don't know hot to determine $p_n$'s. Moreover i don't know which is the right way to enumerate $n+in$ in this case. I need integers $p_n$ such that
$$\sum_{n=1}^{+\infty}\frac{1}{|z_n|^{p_n+1}}<+\infty$$
How can i find them?

If you indeed want $\sum_{n}|z_n|^{-p_n-1}<\infty$, then enumeration makes no difference. Just set $z_n=n+in$. Now you see that $p_n\equiv 0$ does not work, while $p_n\equiv 1$ does. What is the "minimal" sequence of $p_n$? Well, there isn't one. Whatever sequence you come up with can be made "smaller" by replacing one of the nonzero numbers $p_n$ with a $0$. 
However, it is not necessary to have $\sum_{n}|z_n|^{-p_n-1}<\infty$ in order for the product to converge. This is what the example of Marvis and Jonas demonstrates: in it $p_n\equiv 0$, yet the product converges conditionally with the enumeration $1,-1,2,-2,3,-3\dots $ (times $1+i$).
Moral: the Weierstrass theorem does not exhaust all possibilities for constructing entire functions with prescribed zeros. 
