Joint universality for dependent $L$-functions

We prove that, for arbitrary Dirichlet $L$-functions $L(s;{\chi }_1),\dots ,L(s;{\chi }_n)$ (including the case when ${\chi }_j$ is equivalent to ${\chi }_l$ for $j\mathrm{\ne }k$), suitable shifts of type $L(s+i{\alpha }_j{t}^{a_j}{\log }^{b_j}t;{\chi }_j)$ can simultaneously approximate any given analytic functions on a simply connected compact subset of the right open half of the critical strip, provided the pairs $(a_j,b_j)$ are distinct and satisfy certain conditions. Moreover, we consider a discrete analogue of this problem where $t$ runs over the set of positive integers.