Let $\mathbb P$ denote the set of prime numbers and for a subset $T\subset \mathbb P$ let $$ \zeta_T(s)=\prod_{p\in T}\frac1{1-p^{-s}}, $$ where $\mathrm{Re}(s)>1$. Is there any $T$ such that $T$ and $T^c={\mathbb P}\smallsetminus T$ are both infinite and $\zeta_T$ has a meromorphic continuation to $\mathbb C$?