Ineffectivness of constant in Siegel's theorem

Question

I have a question, the following theorem of Siegel says that the constant is ineffective,i.e. not effectively computable, but I do not understand why it is actually not computable and how to prove something like that, it seems to me that in the proof they didn't proved that is ineffective.

Theorem: Let $ \chi $ be any real primitive Dirichlet character $ \mod q $, then for every $ \epsilon > 0 $ $$ L(1,\chi) > \frac{c(\epsilon)}{q^{\epsilon}} $$

where $ c(\epsilon)$ is an ineffective constant.

The proof is very short and can be found here: https://www.pnas.org/doi/pdf/10.1073/pnas.71.4.1055

Answer 1

This is a detail that commonly puzzles people at first. Look at the paragraph following the statement of Lemma 1:

• This must be true since if there are no real zeros in $[1-\varepsilon,1]$ for any $L(s,\chi)$ then $f(\beta)<0$ if $1-\varepsilon<\beta< 1$, since $\xi(\beta)< 0$. On the other hand, if such real zeros do exist, let $\beta$ be such a zero and $\chi_1$ the corresponding character so that $f(\beta)= 0$ independent of $\chi_2$.

Note that this is a proof in two cases that asserts the existence of a real number $\beta$ with particular properties. Could you actually use this proof to produce an explicit such real number $\beta$? Well, no, not with our current knowledge—you would first need to know whether there exists a $\chi$ such that $L(s,\chi)$ had a zero in the interval $[1-\varepsilon,1]$, because the second case defines $\beta$ in terms of that zero. That's why this proof is ineffective—we don't know which construction of the real number $\beta$ we must use.

This already illustrates the underlying idea of ineffective proofs, but to your exact question, the constant $c(\varepsilon)$ in the main theorem is constructed out of this $\beta$, which makes it ineffective as well.