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Let $\chi$ denote the Legendre symbol of conductor $q$. A Siegel zero for the $ L $ series associated to $ \chi $, which we denote by $ L(s,\chi) $ is a real zero $ \sigma $ satisfying $ 1-\frac{c}{\log |q|} < \sigma < 1$ for some constant $c$.

I have read in many places (see for example the second page of the article by Ajit Bhand and Ram Murty available here) that the non-existence of Siegel zeros for $ L(s,\chi) $ can be used to prove the bound $$ h(d) > c_1 \frac{\sqrt{d}}{\log(d)}, $$ where $ h(d)$ is the class number of the associated imaginary quadratic field and $c_1$ is another effective constant which can be calculated depending on $c$.

My Question : How to explicitly compute $c_1$ from $c$?

If anyone can direct me to a proof of the above statement, I think that would also suffice.

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One place to find this worked out in detail is the paper "On the Siegel-Tatuzawa theorem" by Jeffrey Hoffstein (published in 1980 in Acta Arithmetica). Lemma 1 of that paper states that if $\chi$ is a quadratic Dirichlet character with conductor $d > 10^{6}$ and if $L(s,\chi)$ is nonzero on $(\beta,1)$ and $1-\beta$ is small (specifically $(1 - \beta)^{-1} < 11.657 \log(d)$), then $$ L(1,\chi) > 1.507 (1 - \beta). $$ Also given in Lemma 1 is a lower bound on $L(1,\chi)$ under the assumption that $L(s,\chi)$ doesn't vanish on $(0,1)$.

Combining this with the Dirichlet class number formula $L(1,\chi) = \frac{2 \pi h(d)}{w_{d} \sqrt{d}}$ gives the result you seek with $c_{1} = \frac{1.507}{\pi} c$ provided $c < \frac{1}{11.657}$ and $d > 4$.

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    $\begingroup$ In Lemma 1 of the paper that you mentioned, they require $|d| > 10^6$ . I understand that doesn't affect the nature of the result, but since we are dealing with explicit constants here, I think that will create a problem. You have mentioned the same result for $|d| > 4$. Can you please clarify? $\endgroup$
    – Krishnarjun
    Commented Jun 12, 2022 at 10:40
  • $\begingroup$ Apologies. The assumption $|d| > 10^{6}$ is required in the above paper. If you are interested in bounds on $L(1,\chi)$ for small and medium size $d$ ($|d| < 2^{40}$), you could look at Section 7 of the paper here by Mosunov and Jacobson. $\endgroup$
    – Jeremy Rouse
    Commented Jun 12, 2022 at 15:23
  • $\begingroup$ My question was purely theoretic motivated in part by trying to understand the proof of the claim mentioned therein. The paper of Hoffstein that you mentioned is very helpful and I will take a look if I can tweak the proof of Lemma 1 to get what I want. The second reference that you gave, particularly Section 7 contains numerical data and I don't see how that answers my question. BTW I tried to edit your answer to reflect that $|d|$ should be atleast $10^6$ but I was not able to. Can you please change your ans so that others don't get the wrong info. Thanks. $\endgroup$
    – Krishnarjun
    Commented Jun 12, 2022 at 17:52
  • $\begingroup$ My answer has been edited to include the restriction that $d > 10^{6}$. I referenced the paper in case you were interested or concerned about the cases where $d < 10^{6}$. $\endgroup$
    – Jeremy Rouse
    Commented Jun 12, 2022 at 18:49

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