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I would like to have an explicit upper bound (that is, one with explicit constants) for a possible real zero $\beta$ for $L(1,\chi)$ for real Dirichlet characters $\chi$. I need such a bound for real characters $\chi$ to any modulus, not just prime modulus. (Ford/Luca/Moree, lemma 3 provides such a bound when the modulus is prime, of the form $$ \beta \le 1 - \frac{25\pi}{4\sqrt q\log^2q} $$ when $q$ is a large prime; I expect that the general result I'm seeking would be of similar quality.)

Essentially equivalently, I would like a lower bound for $|L(1,\chi)|$ for such characters $\chi$. (The proof of Ford/Luca/Moree proceeds in this way, and bounds $|L(1,\chi)|$ using Dirichlet's class number formula.)

I would be happy with either an argument laid out here (the more complete, the better) or a reference to the literature.

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    $\begingroup$ The (standard) argument in Ford/Luca/Moree doesn't really use that $q$ is prime, and works for any fundamental discriminant. $\endgroup$
    – Lucia
    Jul 7, 2015 at 20:32
  • $\begingroup$ I see, so ... that would be enough to deal with all primitive real characters, and then one would just use the Euler product to deal with imprimitive characters? $\endgroup$
    – Greg Martin
    Jul 7, 2015 at 21:53
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    $\begingroup$ Yes; the results should only get better for imprimitive characters. Also the $\log^2$ in the denominator can be removed while still using only the trivial bound for the class number (but maybe no one wrote up an explicit version of that -- maybe useful to do if you're really fighting for constants) -- see Goldfeld and Schinzel archive.numdam.org/ARCHIVE/ASNSP/ASNSP_1975_4_2_4/… $\endgroup$
    – Lucia
    Jul 7, 2015 at 22:05

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