Is there an analytic function with zeroes only at $-2n$, and zeroes at $\frac12\pm it$, and further, symmetric zeroes within the critical strip?

Question

Is there an analytic function with zeroes only at:

• every $-2n$,
• $\frac12\pm it$, and
• at least one at $\frac12\pm\epsilon\pm it$ where $0<\epsilon<\frac12, t\neq0$ (and these zeroes observing the known reflective symmetries within the critical strip)?

A answer assuming the Riemann hypothesis true would be fine.

To my mind the uniqueness of any analytic continuation of $\zeta(s)$ is suggestive of the existence of such a function being incompatible with the Riemann Hypothesis.

If not, uniqueness of $\zeta$ is of course a nice simple sufficiency for the Riemann hypothesis.

Answer 1

Take $\zeta(z)\left(z-\frac14-i\right)\left(z-\frac14+i\right)\left(z-\frac34-i\right)\left(z-\frac34+i\right)$.

Every condition that you mentioned holds.

Answer 2

I think this is an open question. As long as your function is automorphic, the Grand Riemann Hypothesis asks this exact question.