Are there any reasonable ways to define zeta function of a maximal abelian extension $\mathbb{Q}^{ab}$ of the rationals, or at least zeta functions of some its infinite-dimensional subfields?

In particular, can we use zeroes of Dedekind zeta functions in the following way? Let $N$ be positive integer, let $\zeta_N$ be the Dedekind zeta function of the $N$-th cyclotomic extention of $\mathbb{Q}$, and let $Z_N:= \{s\in\mathbb{C}: \zeta_N(s)=0, 0< \mathrm{Re} s <1\}$ be its non-trivial zeroes. It is known that $Z_N \subset Z_M$ whenever $N$ divides $M$. But is $$Z:=\bigcup_{N\ge 1}{Z_N}$$ a discrete subset of $\mathbb{C}$, so that we could construct a meromorphic function having $Z$ as its zero set?