I can show, it is bounded by $p^4\log p$. But I think the bound should be smaller.

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    $\begingroup$ Undoubtedly there is an interesting question underlying this, but your presentation needs quite a bit of work. Please check out our guide for new askers. Note that: 1) the only math here is in the title, but most of that should be in the question body. 2) Your question is lacking in context such as A) Why is this interesting? Is it a homework problem or something you encountered working on something else? B) If homework you definitely should at least outline how you derive your bound. $\endgroup$ – Jyrki Lahtonen Jul 13 at 12:11
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    $\begingroup$ I have no idea what such a weighted sum of $|L(1,\chi)|$ would be useful for, are you sure you didn't mean $\log |L(1,\chi)|$ ? You mean $\chi$ the characters $\bmod p$ ? What bound do you assume for $|L(1,\chi)|$ ? $\endgroup$ – reuns Jul 13 at 16:36
  • $\begingroup$ $\chi$ is a dirichlet character modulo p. I encountered this problem while solving one moment weighted by L function.. $\endgroup$ – Nilanjan Bag Jul 13 at 23:53

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