Some remarks on Landau-Siegel zeros
The authors study the possible ``exceptional'' zero of a Dirichlet \(L\)-function \(L(s, \chi _D)\) for a real primitive character \(\chi _D \pmod{|D|}\) under certain hypotheses. Let \(E\) be an elliptic curve over \(\mathbb Q\). ``Hypothesis H'' assumes that all zeros of \(L(s, \chi _d)\) and \(L(s, E\otimes \chi _D)\) are either real or lie on the critical line. It is shown that this hypothesis (for Dirichlet \(L\)-functions only) implies the ineffective lower bound \(L(1, \chi _D) \geq c(\varepsilon)(\log |D|)^{-\varepsilon }\). Further, the full hypothesis H gives the effective lower bound \(L(1, \chi _D) \geq c(\eta)|D|^{-\eta }\) for any \(\eta >2/5\) if \(\chi _D(37)=-1\). Also, if \(L(s,E)\) has a zero of order \(m\) at \(s=1/2\), then the effective estimate \(L(1, \chi _D) \geq c(E, \varepsilon)|D|^{-(2+\varepsilon)/(m+1)}\) holds. The general scheme of the argument is to start from a suitable product of \(L\)-functions and to apply corresponding ``explicit formulae'' connecting arithmetic sums with zeros.
scientific article; zbMATH DE number 1820880
scientific article