The aim of this paper is to provide an analogue of the Ball-Rivoal theorem for $p$-adic $L$-values of Dirichlet characters. More precisely, we prove for a Dirichlet character $\chi$ and a number field $K$ the formula $\dim_{ K}(K+\sum_{ i=2}^{ s+1} L_p(i,\chi\omega^{ 1-i}) K )\geq \frac{ (1-\epsilon)\log (s)}{ 2[K:\mathbb{ Q}](1+\log 2)}$. As a byproduct, we establish an asymptotic linear independence result for the values of the $p$-adic Hurwitz zeta function.
Categories: math.NT math.AG