Discrepancy bounds for the distribution of $L$-functions near the critical line

We investigate the joint distribution of $L$-functions on the line $\sigma =\frac{1}{2}+\frac{1}{G(T)}$ and $t\in [T,2T]$, where $\log \log T\le G(T)\le \frac{\log T}{(\log \log T{)}^2}$. We obtain an upper bound on the discrepancy between the joint distribution of $L$-functions and that of their random models. As an application we prove an asymptotic expansion of a multi-dimensional version of Selberg’s central limit theorem for $L$-functions on $\sigma =\frac{1}{2}+\frac{1}{G(T)}$ and $t\in [T,2T]$, where $(\log T{)}^{\epsilon }\le G(T)\le \frac{\log T}{(\log \log T{)}^{2+\epsilon }}$ for $\epsilon >0$.