 Solution

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.