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  • I can let Robin Chapman speak for himself for the illusion, but from the real-complex idea, you are using one extra $\zeta(s)$ when $\chi$ is complex (and so, overkill in pretending it is hard), and taking $(\zeta(s)L(s,\chi))^2$ when $\chi$ is real (and so, using the square of what can be used, and so for no reason I can determine, other than to uniformize the language).
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