Dirichlet $L$-functions on the critical line and multiplicative chaos

In this paper we prove that the Dirichlet $L$-functions $L(1\mathrm{/}2+ix,{\chi }_q)$, where ${\chi }_q$ is uniformly random Dirichlet character modulo $q$ and $x\in \mathbb{R}$, converges to a random Schwartz distribution ${\zeta }_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}$, which is related to (complex) Gaussian multiplicative chaos. This is the same limiting object that appeared in [34], where the authors proved that the random shifts of the Riemann zeta function on the critical line $\zeta (1\mathrm{/}2+ix+i\omega T)$, where $\omega \sim \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}([0,1])$, converge as $T\to \mathrm{\infty }$.