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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 $L$-functionsassociated with the sequence of primitive holomorphic cusp forms.

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