There is a simpler way to derive the result from the approach mentioned by K. Conrad. You can find it in the following 1997 note by Paul T. Bateman:

**Bateman, Paul T.:** *A theorem of Ingham implying that Dirichlet's L-functions have no zeros with real part one*. L'Enseignement Mathématique. 43 (1997), 281-284.

It is important to note that the consideration of the function $\zeta^2(s)L(s, \chi)L(s,\overline{\chi})$ can be traced back to Professor Narasimhan's own attack on the non-vanishing of $L$-functions at $\sigma=1$.