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In this paper, we give an asymptotic formula for the second moment ofDirichlet twists of an automorphic $L$-function $L(s, \pi)$ on the criticalline averaged over characters and conductors, where $\pi$ denotes anirreducible tempered cuspidal automorphic representation of $\textrm{GL}_{4}$with unitary central character. We give some hybrid bound for the error termwith respect to the size of conductors of Dirichlet characters and that of theautomorphic representation.

The second moment of Dirichlet twists of a $\textrm{GL}_{4}$ automorphic $L$-function