Distribution of Dirichlet $L$-functions

In this article, we study the distribution of values of Dirichlet $L$-functions, the distribution of values of the random models for Dirichlet $L$-functions, and the discrepancy between these two kinds of distributions. For each question, we consider the cases of $\frac{1}{2}<\text{RE}s<1$ and $\text{RE}s=1$ separately.