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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.

For a nonprincipal Dirichlet character, the line is zero-free. Euler product positivity for L-function nonvanishing proves this for nonreal characters at every height, and for all characters at nonzero height, where a possibly principal squared-character factor has no pole. The point of height zero is the nonvanishing of nonprincipal Dirichlet L-functions at one. A principal L-function has a pole at one, and no zeros at the other points of this line.

For $\sigma >1$, the Euler product positivity for L-function nonvanishing givesIndeed the logarithm expands into terms proportional to $3+4\cos \theta +\cos 2\theta =2(1+\cos \theta {)}^2\ge 0$. At primes dividing $q$, the character terms vanish and the remaining zeta term is positive. For a nonreal character, ${\chi }^2$ is nonprincipal, so its L-function is entire, even if imprimitive.

If $L(1+it,\chi )$ vanished to order $m\ge 1$, the product would be $O((\sigma -1{)}^{4m-3})$ as $\sigma \downarrow 1$: zeta has a simple pole, the last factor is bounded, and the middle factor has the asserted vanishing. The product would tend to zero, contradicting its lower bound one. This proves nonvanishing for every real $t$, including zero.