In this paper, we obtain joint simultaneous approximation of analytic functions defined on the strip \(\{s\in \mathbb{C}: 1/2< \operatorname{Re} s<1\}\) by shifts \( { (L(s+i\tau, \chi_1), \dots, L(s+i\tau, \chi_r)) } \) of Dirichlet \(L\) -functions with non-equivalent Dirichlet characters in short intervals, i.e., intervals \([T,T+H]\) with \(T^{27/82} \leq H \leq T^{1/2}\) . It is proved that the set of such approximating shifts has a positive lower density, and even positive density for all but at most countably many approximation accuracies. For the proof, the probabilistic approach is used.
