Real values of $\frac{\zeta(2 s)}{\zeta(s)}$

Question

If $\frac{\zeta(2 s)}{\zeta(s)}$ is a real number, then must $s$ be real ?

Answer 1

Of course not. For example, let $2s$ be any non-real zero of the zeta function where $s$ is not a zero.
Through such a point, there will be a curve along which $\zeta(2z)/\zeta(z)$ is real.

EDIT: Here is a plot of the curve where $\zeta(2s)/\zeta(s)$ is real in one region of the plane. A dot indicates the one non-real point in this region where $\zeta(2s) = 0$.