= Solution
For $\sigma>1$, the <Euler product positivity for L-function nonvanishing> gives
$$
\zeta(\sigma)^3|L(\sigma+it,\chi)|^4|L(\sigma+2it,\chi^2)|\ge1.
$$
Indeed the logarithm expands into terms proportional to $3+4\cos\theta+\cos2\theta=2(1+\cos\theta)^2\ge0$. 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 $m\ge1$, the product would be $O((\sigma-1)^{4m-3})$ as $\sigma\downarrow1$: 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.
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