  We prove mod-Gaussian convergence for a Dirichlet polynomial which
approximates $\operatorname{Im}\log\zeta(1/2+it)$. This Dirichlet polynomial is
sufficiently long to deduce Selberg's central limit theorem with an explicit
error term. Moreover, assuming the Riemann hypothesis, we apply the theory of
the Riemann zeta-function to extend this mod-Gaussian convergence to the
complex plane. From this we obtain that $\operatorname{Im}\log\zeta(1/2+it)$
satisfies a large deviation principle on the critical line. Results about the
moments of the Riemann zeta-function follow.
