= Solution
Let $\widehat G$ be the group of all <Dirichlet characters> modulo $q$, where $G=(\mathbb Z/q\mathbb Z)^\times$, and consider
$$
F(s)=\prod_{\chi\in\widehat G}L(s,\chi).
$$
For $p\nmid q$, let $f_p$ be the order of $p$ in $G$. The values $\chi(p)$ run through the $f_p$th roots of unity, each $|G|/f_p$ times, so the local factor is
$$
\prod_{\chi\in\widehat G}(1-\chi(p)p^{-s})^{-1}
=(1-p^{-f_ps})^{-|G|/f_p}.
$$
For $p\mid q$ the local factor is one. Thus the <Dirichlet series> for $F$ has nonnegative coefficients.
The principal-character factor has a simple pole at $s=1$, while every nonprincipal <Dirichlet L-function> is entire. If some nonprincipal $L(1,\chi)$ vanished, its zero would cancel that pole and make $F$ entire. The <Landau theorem for a Dirichlet series with nonnegative coefficients> would then force the Dirichlet series of $F$ to converge for every real $s$. This is impossible: its coefficient at $m^{|G|}$ is at least one for every $m$ coprime to $q$, as is clear from the local factors. Hence
$$
\boxed{L(1,\chi)\ne0\qquad
\text{for every nonprincipal }\chi.}
$$
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