Selberg's Central Limit Theorem for automorphic $L$-functions in the $t$-aspect

We present a proof of Selberg’s Central Limit Theorem for automorphic $L$-functions of degree 2 using Radziwi\l\l\space and Soundararajan’s method. Additionally, we prove the independence of the automorphic $L$-functions associated with the sequence of primitive holomorphic cusp forms.