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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.

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