= Global partial-fraction expansion of a Dirichlet L-function logarithmic derivative
{title2=$L'/L=B_\chi-\tfrac12\log(q/\pi)-\tfrac12\Gamma'/\Gamma((s+a)/2)+\sum_\rho((s-\rho)^{-1}+\rho^{-1})$}
For a primitive nonprincipal <Dirichlet character> of conductor $q$ and parity $a$, the <completed Dirichlet L-function> is entire of order one. Its genus-one <Hadamard factorization> gives the displayed <logarithmic derivative>, with $B_\chi$ the logarithmic derivative of the completion at zero and zeros counted with multiplicity. Combined summands converge locally normally away from zeros. For an imprimitive character induced by $\chi^*$, add $\sum_{p\mid Q,\ p\nmid q}\chi^*(p)\log p/(p^s-\chi^*(p))$ for the removed Euler factors. The primitive principal case is the <global partial-fraction expansion of the zeta logarithmic derivative>.
// Target: algebraic-number-theory.bigb
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