  We study the distribution of the zeroes of the zeta functions of the family
of Artin-Schreier covers of the projective line over $\mathbb{F}_q$ when $q$ is
fixed and the genus goes to infinity. We consider both the global and the
mesoscopic regimes, proving that when the genus goes to infinity, the number of
zeroes with angles in a prescribed non-trivial subinterval of $[-\pi,\pi)$ has
a standard Gaussian distribution (when properly normalized).
