Imprimitive Dirichlet L-function Euler correction ($L(s,\chi )=L(s,{\chi }_0){\prod }_{p\mid q,p\nmid q_0}(1-{\chi }_0(p)p^{-s})$)

The inducing primitive Dirichlet character has conductor of a Dirichlet character $q_0$. Increasing the modulus removes Euler factors only at new prime divisors. The correction creates zeros on the imaginary axis; raising the exponents of primes already dividing the conductor of a Dirichlet character changes no Euler factor. Primitive gamma factors continue to use $q_0$. For real-even characters this includes principal primitive conductor of a Dirichlet character one, with its Riemann zeta function pole treated separately.