The principal Dirichlet L-function factors through zeta

By [L2], $L(s,{\chi }_0)={\prod }_p(1-{\chi }_0(p)p^{-s}{)}^{-1}$. Using [L1], the Euler factors are $1$ at primes dividing $q$ and $(1-p^{-s}{)}^{-1}$ at all other primes, so $L(s,{\chi }_0)={\prod }_{p\nmid q}(1-p^{-s}{)}^{-1}=({\prod }_p(1-p^{-s}{)}^{-1}){\prod }_{p\mid q}(1-p^{-s})=\zeta (s){\prod }_{p\mid q}(1-p^{-s})$.

The right-hand side of step 1.1, using [L3], defines a meromorphic continuation of $L(s,{\chi }_0)$ to $\mathrm{Re}s>0$, which contains a neighborhood of $1$. Any further meromorphic continuation agrees with this germ. The finite factor ${\prod }_{p\mid q}(1-p^{-s})$ is holomorphic at $s=1$ and has value ${\prod }_{p\mid q}(1-p^{-1})$. Multiplying this with the residue-one pole from [L3] gives a simple pole of $L(s,{\chi }_0)$ at $1$ with residue ${\prod }_{p\mid q}(1-p^{-1})$. Finally, $\phi (q)=q{\prod }_{p\mid q}(1-p^{-1})$ by inclusion-exclusion on the finite set of primes dividing $q$ (with empty product $1$ when $q=1$), so the residue is $\phi (q)\mathrm{/}q$.