# Riemann hypothesis for $L(s,\chi)$ and $L(s,\chi^\sigma)$

If $$\sigma \in \text{Gal}(\mathbb{Q}(\zeta_{\infty})/\mathbb{Q})$$ do we know or expect that two Dirichlet L-functions $$L(s,\chi)$$ and $$L(s,\chi^\sigma)$$ have more in common, especially in term of their non-trivial zeros, than with the other Dirichlet L-functions of the same modulus ?

Heuristically, can we say $$L(s,\chi)$$ and $$L(s,\chi^\sigma)$$ are related iff there exists some algebraization of the generalized Riemann hypothesis ?

On the modular form/automorphic representation side, it seems that $$f$$ looks completely similar to $$f^\sigma$$, so it seems natural to ask if this similarity can pass through the Mellin transform and stay true on the L-series side.