When does the "Zetor function" converge?

Question

Let $p_n$ be the n'th non-trivial zero of the Riemann zeta function. We define the Zetor function (acronym of 'zeta' and 'zero') as follows: $$\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}. $$

For which values of $s$ does $\zeta \rho (s)$ converge?

Thanks,

Max