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Non-vanishing of class group \(L\)-functions at the central point. (English) Zbl 1063.11040

The author considers \(L\)-functions \(L_K(s, \chi)\) attached to class group characters \(\chi\) of an imaginary quadratic field \(K= {\mathbb{Q}}( \sqrt{-D})\) of discriminant \(-D\). He proves that there exists an absolute constant \(c>0\) such that \[ \frac{1}{| C| } \biggl|\biggl\{ \chi \in C;\;L_K \biggl( \frac{1}{2}, \chi\biggr) \neq 0\biggr\}\biggr|\geq c \prod_{p \mid D} \biggl( 1 - \frac{1}{p}\biggr) \] for sufficiently large \(D\). Here, \(C\) denotes the class group of \(K\).

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11M41 Other Dirichlet series and zeta functions
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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