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Results are known in many different cases to bound powers of L-functions on average over a wide enough family. I am interested in results for general number fields, not only for the rationals, for powers of the quadratic twists. More precisely let $\pi$ be an automorphic representation of $GL_n$ and $\chi_d$ the quadratic character associated to $\mathbb{Q}(\sqrt d)/\mathbb{Q}$. Are there any bound known of the form $$\sum_{d < D} |L(1/2, \pi \otimes \chi_d)|^2 \ll D^{\theta + \varepsilon}?$$

All the results I find deal with the rational field, I would like to know if there is any problem in generalizing to general fields.

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  • $\begingroup$ I am confused. How do you wish to formulate this for more general number fields? Given an automorphic representation $\pi$ of $\mathrm{GL}_n$ over a number field $F$, the twist $\pi \otimes \omega$ by a Hecke character $\omega$ only makes sense if $\omega$ is a Hecke character of $F$. So your question only makes sense as it is written for $\pi$ a rep over $\mathbb{Q}$. $\endgroup$
    – Peter Humphries
    Commented Oct 17, 2018 at 19:20

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