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For every nonprincipal Dirichlet character, $L(1,\chi )\ne 0$. A nonreal character is covered by Euler product positivity for L-function nonvanishing at height zero, since its squared character is nonprincipal.

For a real character, suppose $L(1,\chi )=0$. The simple zeta pole would cancel, making $F(s)=\zeta (s)L(s,\chi )$ an entire function. Its nonnegative zeta-times-real-L coefficients satisfy $a_n\ge 0$ and $a_{m^2}\ge 1$. Termwise derivatives at two give $(-1{)}^k{F}^{(k)}(2)={\sum }_n{a}_n(\log n{)}^k{n}^{-2}$. The Taylor series of this entire function at two converges at zero. Every term there is nonnegative, and Tonelli theorem identifies its sum asThe last series diverges because all square coefficients are at least one. This contradiction proves the claim. It is the positive-coefficient mechanism behind the more general Landau theorem for a Dirichlet series with nonnegative coefficients.

The Dirichlet-series coefficients areThey are multiplicative. At a prime power, their values are $k+1$ if ${\chi }_D(p)=1$, one for even $k$ and zero for odd $k$ if ${\chi }_D(p)=-1$, and one if ${\chi }_D(p)=0$. Thus every $a_n$ is nonnegative. In particular $a_{m^2}\ge 1$. This is the nonnegative zeta-times-real-L coefficients identity. The same Euler expansion gives $-{\zeta }^{\prime }/\zeta (\sigma )-L^{\prime }/L(\sigma ,{\chi }_D)\ge 0$ for $\sigma >1$, with coefficients $\Lambda (n)(1+{\chi }_D(n))\ge 0$.