  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 $\chi\pmod {q>1}$.
