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We prove an explicit upper bound of the function $S(t,\chi)$, defined by theargument of Dirichlet $L$-functions. An explicit upper bound of the function$S_1(t)$, defined by the integral of the argument of the Riemann zeta-function,have already been obtained by A. Fujii. Our result is obtained by applying anidea of Fujii's result on $S_1(t)$. The constant part of the explicit upperbound of $S(t,\chi)$ in this paper does not depend on a primitive Dirichletcharacter $\chi\pmod {q>1}$.

An explicit upper bound of the argument of Dirichlet $L$-functions on the generalized Riemann hypothesis