Let $k$ be an even integer and $p$ a prime number such that $p-1|k$.
Suppose that $p$ does not divide $L(1-k,\chi)$ and $L(1-k,\psi)$, where $\chi$ and $\psi$ are quadratic characters.
Can we deduce that $p$ does not divide $L(1-k,\chi.\psi)$?
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This is trivially false, infinitely many counterexamples exist. A simple one: p = 5, k = 4, $D(\chi)=12$, $D(\psi)=13$ then $L(\chi,-3)=36$, $L(\psi,-3)=58$ but $L(\chi.\psi,-3)=365800$.
answered May 27, 2016 at 16:45
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$\begingroup$ if we fixes one character can we find another one such that the implication in my question is true ? $\endgroup$ Commented May 27, 2016 at 20:41
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$\begingroup$ how did you find those counter-examples ? did you look at a table of the generalized Bernoulli numbers ? $\endgroup$– reunsCommented May 28, 2016 at 2:27