Identity for $L(s,\chi)L(s,\bar\chi)$

Question

I was told recently that there is an identity roughly of the form

$$L(s,\chi)L(s,\bar\chi)=\zeta(s)^2$$

If true, it seems like there should be a short proof of this. Could someone supply a reference, and/or the proof of this?

Answer 1

I am not sure whether this identity is the one you were told recently. It cannot be true, because for $s=1$ the LHS is finite, but the RHS is not. However we have a functional equation for $L$-series, of the form $$ L(s,\chi)=\epsilon(\chi)L(1-s,\overline{\chi}), $$ which one can find in several books on analytic number theory. Also studied is the product over all Dirichlet characters, the Dedekind zeta function $$ \prod_{\chi \bmod N}L(s,\chi). $$