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Could you please provide a link to the source? $$\sum_{\chi\neq \chi_0}\int_{0}^{T}|L(1/2+it,\chi)|^4dt\ll (qT)^{1+\varepsilon},$$ where $\chi_0$ is the principal character modulo $q$, and $L(s,\chi)$ is the Dirichlet L-function associated to $\chi$

P.S. I found only for primitive characters, but I need for non-principal characters.

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    $\begingroup$ Just separate out the impritive characters and write their $L$-functions as the $L$-function of a primitive character times an imprimitive part that is $\ll_{\varepsilon} (qT)^{\varepsilon}$. $\endgroup$
    – Peter Humphries
    Commented Feb 28, 2022 at 18:55
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    $\begingroup$ Perhaps it's worth adding also that one can quite easily find an asymptotic formula with power savings in the error term by using the classical mean square methods for $\zeta(s)$ applied to the Hurwitz zeta function $\zeta(s,a/q)$. This is done, for example, by Rane in mathscinet.ams.org/mathscinet-getitem?mr=575376 . $\endgroup$
    – Anurag Sahay
    Commented Mar 1, 2022 at 2:32

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