# explicit lower bounds on $|L(1,\chi)|$

Does anyone know of an explicit effective lower bound for $|L(1,\chi)|$, where $\chi$ is an odd complex (primitive) Dirichlet character?

I know of Landau's paper Uber Dirichletsche Reihen mit komplexen Charakteren, where he bounds $$|L(1,\chi)|>\frac{1}{c \log(q)},$$

where $q$ is the conductor of $\chi$, but the constant $c$ he gets is on the order of $e^{50}$, and is totally useless for computations.

I know of many papers dealing with quadratic characters but very few that address complex characters (explicitly).

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## 1 Answer

This is discussed on page 47 of Narkiewicz's new book (Rational Number Theory in the 20th Century); see

http://books.google.ca/books?id=3SWNZaDM6iMC&lpg=PP1&dq=rational%20number%20theory%20in%20the&pg=PA47#v=onepage&q&f=false

Reference [4268] is to

Metsankyla, T.: Estimations for L-functions and the class numbers of certain imaginary cyclic fields, Ann. Univ. Turku, Ser. AI 140, 1--11 (1970)

[3995] is

Louboutin, Stéphane(F-CAEN) Minoration au point 1 des fonctions L et détermination des corps sextiques abéliens totalement imaginaires principaux. (French) [Lower bound at the point 1 of L-functions and determination of the principal totally imaginary abelian sextic fields] Acta Arith. 62 (1992), no. 2, 109–124.

and [338] is

Barrucand, Pierre; Louboutin, Stéphane(F-CAEN) Minoration au point 1 des fonctions L attachées à des caractères de Dirichlet. (French) [Lower bound at the point 1 of L-functions associated with Dirichlet characters] Colloq. Math. 65 (1993), no. 2, 301–306.

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