Euler product positivity for L-function nonvanishing ($\zeta (\sigma {)}^3|L(\sigma +it,\chi ){|}^4|L(\sigma +2it,{\chi }^2)|\ge 1$)

For $\sigma >1$, logarithmic expansion of the Euler product reduces positivity to $3+4\cos \theta +\cos 2\theta =2(1+\cos \theta {)}^2$. At primes dividing the modulus the zeta contribution is positive by itself. If the last factor is bounded at the line-one point, a zero of the middle factor would outweigh the zeta pole and force this product to zero. This proves the relevant nonvanishing of Dirichlet L-functions on the line one.