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For simplicity, I'm gonna call L-functions associated with primitive Dirichlet characters 'primitive' and same with inprimitive.

GRH (Generalized Riemann Hypothesis) says that all non-trivial zeros of primitive Dirichlet L-functions have real part 1/2.

Most of the time (AFAIK) the analytic NT results which assume GRH only use it to get 'good' estimates on functions of primes, while these only use primitive L-functions.

Thus the question arises: What are inprimitive Dirichlet L-functions are for?

Also, I'm assuming its true, however I couldn't find any explicit statement on the matter: Does 'primitive GRH' imply 'inprimitive GRH'?

Arsenniy
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    Imprimitive $L$ functions are of the type $\Pi_{p_k}(1-\chi(p_k)p_k^{-s})L_1$ where $L_1$ is the function generated by the primitive character inducing $L$ and $p_k$ primes that appear in the bigger modulus but not in the small; so imprimitive $L$ are kind of like eta vs zeta (though eta doesn't correspond to an induced character) so generally have extra zeroes though this time on the line $\sigma=0$; the main problem with imprimitive characters is that their Gauss sum is zero so the functional equation which involves it needs to go through the corresponding primitive character inducing it – Conrad Nov 30 '25 at 15:03
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    It's a good exercise to look at the trivial character mod $n$ which is imprimitive for $n\ge 2$ and convince yourself that the corresponding $L$ is indeed $\Pi_{p|n}(1-p^{-s})\zeta(s)$ – Conrad Nov 30 '25 at 15:10
  • I think it's more that if you want to do something where you range over all conductor $N$ Dirichlet characters say, or study level $N$ Eisenstein series, then you're forced to deal with imprimitive characters. – Kimball Nov 30 '25 at 16:36

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