  We prove a functional limit theorem in a space of analytic functions for the
random Dirichlet series $D(\alpha;z)=\sum_{n\geq 2}(\log
n)^{\alpha}(\eta_n+{\rm i} \theta_n)/n^z$, properly scaled and normalized,
where $(\eta_n,\theta_n)_{n\in\mathbb{N}}$ is a sequence of independent copies
of a centered $\mathbb{R}^2$-valued random vector $(\eta,\theta)$ with a finite
second moment and $\alpha>-1/2$ is a fixed real parameter. As a consequence, we
show that the point processes of complex and real zeros of $D(\alpha;z)$
converge vaguely, thereby obtaining a universality result. In the real case,
that is, when $\mathbb{P}\{\theta=0\}=1$, we also prove a law of the iterated
logarithm for $D(\alpha;z)$, properly normalized, as $z\to (1/2)+$.
