On the distribution of $\log \mathrm{\mid }L(\sigma ,\chi )\mathrm{\mid }$ and $\log L(\sigma ,{\chi }_D)$ in the modulus aspect

Let $\chi$ be a primitive Dirichlet character whose conductor $q$ is a prime number. For the certain averages of values of $\log \mathrm{\mid }L(s,\chi )\mathrm{\mid }$ in $q$-aspect at a fixed $s=\sigma >1\mathrm{/}2$, under Generalized Riemann Hypothesis (GRH), we explain it can be written as integrals involving the same density function ($M$-function) for the average of values of the difference between the logarithms of two symmetric power $L$-functions in the level aspect. For the distribution of values of $\log L(s,{\chi }_D)$ and $L^{\mathrm{\prime }}\mathrm{/}L(s,{\chi }_D)$ in the $D$-aspect at a fixed $s=\sigma >1\mathrm{/}2$ which $L({\sigma }^{\mathrm{\prime }},\chi )\mathrm{\ne }0$ in $\sigma \le {\sigma }^{\mathrm{\prime }}\le 1$, where ${\chi }_D$ is a real character attached to a fundamental discriminant $D$, we construct a $M$-function unconditionally.