Looking for a specific zeta function.

Question

I am looking for a zeta function

$$ f(s) = \sum \frac{1}{a_n^s}$$

Where $a_n$ is a sequence of distinct positive integers,

such that

1. $f(s)$ is analytic for all $Re(s) > 1$

2. $f(s)$ has a simple pole at $s=1$.

3. $f(s)$ has an analytic continuation to the entire plane.

4. $f(s)$ has all its poles and zero's at $Re(s) = 1$ and they are all simple zero's and simple poles.

5. $f(s)$ has at least $3$ poles ; at $s= 1,s= 1 + A i,s = 1 - A i$.

I had the idea for making the poles by using some sort of greedy algorithm or greedy sieve method. Basically throwing away the integers that go in the undesired direction on the complex plane.

That idea might work to get 1),2),5) but I have no clue how to force 3) and 4).

All ideas are welcome. Satisfying a majority of properties is already nice.

I wonder if such $a_n$ can have a closed form.