  We look at the values of two Dirichlet $L$-functions at the Riemann zeros (or
a horizontal shift of them). Off the critical line we show that for a positive
proportion of these points the pairs of values of the two $L$-functions are
linearly independent over $\mathbb{R}$, which, in particular, means that their
arguments are different. On the critical line we show that, up to height $T$,
the values are different for $cT$ of the Riemann zeros for some positive $c$.
