Voronin's Universality Theorem for $\zeta(s)$ is that the zeta function can uniformly approximate any non-vanishing holomorphic function to any degree of accuracy in the right-half of the critical strip. Precisely, suppose $h(s)$ is a non-vanishing holomorphic function defined on a open set $U$ in the right-half of the critical strip such that the complement of $U$ is connected. Then for any $\epsilon > 0$ there exists a $t \ge 0$ such that
$$\max_{s \in U}|\zeta(s+it)-h(s)| < \epsilon.$$
It is also known that the lower density for the set of such values of $t$ is positive. There is also an analogous universality theorem for Dirichlet $L$-series $L(s,\chi)$, some Hecke $L$-series (also joint universality), and for some Dekekind zeta functions. The Davenport Heilbronn function
$$f(s) = 1+\frac{\xi}{2^{s}}+\frac{\xi}{3^{s}}-\frac{1}{4^{s}}+\frac{0}{5^{s}}+\cdots$$
which is a Dirichlet series with period $5$ also satisfies universality (but in the strong sense by approximating functions with zeros). In all of these cases, the underlying function is defined by a Dirichlet series in the right-half plane. I'm curious if there has been any investigation towards a converse theorem for universality. Say if $f(z)$ is a meromorphic function on $\mathbb{C}$ satisfying universality in the right-half of the critical strip then is it defined by a Dirichlet series in the right-half plane? Do we need to also assume $f(z)$ has at most a simple pole at $s = 1$, an Euler product, etc? On the other hand, if this result is false are there known counterexamples and are they interesting functions?
I suppose a looser form of what I'm asking is "how strong of a condition is universality for holomorphic functions"?
