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We prove an explicit upper bound of the function $S(t,\chi)$, defined by the argument 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 an idea of Fujii's result on $S_1(t)$. The constant part of the explicit upper bound of $S(t,\chi)$ in this paper does not depend on a primitive Dirichlet character $\chi\pmod {q>1}$.

An Explicit upper Bound of the Argument of Dirichlet <i>L</i>-functions on the Generalized Riemann Hypothesis