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Let $\chi$ be a Dirichlet character modulo $4$ such that $\chi(-1) = -1$, and let $\chi'$ be a Dirichlet character modulo $5$ such that $\chi'(-1) = 1$, $\chi'(2) = \chi'(3) = -1$. How do I see the following?

  • $L(\chi, 1) = \pi/2\sqrt{2}$.
  • $L(\chi', 1) = \log \eta/\sqrt{5}$, where$$\eta = {{\sin(2\pi/5)\sin(3\pi/5)}\over{\sin(\pi/5)\sin(4\pi/5)}} = {{1 + \sqrt{5}}\over2}.$$
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    $\begingroup$ One of the rare questions that are answered by their titles. $\endgroup$
    – Franz Lemmermeyer
    Commented Dec 14, 2015 at 14:21
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    $\begingroup$ Are you sure your first identity is right? Isn't it the famous $1-1/3+1/5-\cdots=\pi/4$? Or do you mean some Dirichlet character mod 8? $\endgroup$
    – Fan Zheng
    Commented Dec 14, 2015 at 15:06
  • $\begingroup$ @FranzLemmermeyer Perhaps the OP didn't express himeself very well: I guess s/he really wants to see how the general proof of the CNF works in these specific examples. $\endgroup$
    – Fan Zheng
    Commented Dec 14, 2015 at 15:13
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    $\begingroup$ The first one you can write as $\int_0^1 (1-t^2+t^4-t^6+\ldots) dt = \int_0^1 dt/(1+t^2)$, and compute the integral. The second you can write as $\int_0^1 (1-t-t^2+t^3)/(1-t^5) dt$, and again this is the integral of a rational function which can be computed by partial fractions. (This one is trickier than the first, but still fun -- the substitution $y=t+1/t$ will be useful.) $\endgroup$
    – Lucia
    Commented Dec 14, 2015 at 15:37

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