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I like Donald Newman's proof of the non-vanishing of Dirichlet series on the real line $\mathfrak{R}(s) = 1$, where for a given modulus $m$ one should not consider each series $\displaystyle \sum_{n=1}^\infty \frac{\chi(n)}{n^s} = \prod_p \left(\frac{1}{1 - \chi(p) p^{-s}} \right)$ separately, but instead consider the product $\displaystyle Z(s) = \prod_\chi \prod_{p} \left(\frac{1}{1 - \chi(p)p^{-s}} \right)$. Then the non-vanishing of the $L$-series follows almost immediately by first supposing the existence of a zero on the line $\mathfrak{R}(s) = 1$, say $1 + i\sigma$, then considering the product $Z(s)^2 Z(1 + i \sigma)Z(1 - i \sigma)$. This product is entire (by hypothesis) and its series has positive coefficients. This implies that the series in fact converges everywhere, which is plainly absurd.