The distribution of the values of Hecke $L$-functions at 1

Let $S_2(q)$ be the set of primitive forms in the space $S_2(\Gamma_0(q))$ of holomorpic $\Gamma_0(q)$-cusp forms of weight $2$. Let $f\in S_2(q)$ and let $L_f(S)$ be the $L$-function of $f(z)$. It is proved that the set $\{\log L_f(1)<x,f\in S_2(q)\}$ has a limit distribution function. The rate of convergence to this limit function is estimated.