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Transcendental values of class group $$L$$-functions. II. (English) Zbl 1282.11082
Summary: Let $$K$$ be an imaginary quadratic field and $$\mathfrak{f}$$ an integral ideal. Denote by $$Cl(\mathfrak{f})$$ the ray class group of $$\mathfrak{f}$$. For every non-trivial character $$\chi$$ of $$Cl(\mathfrak{f})$$, we show that $$L(1,\chi )/\pi$$ is transcendental. If $$\mathfrak{f}=\overline{\mathfrak{f}}$$, then complex conjugation acts on the character group of $$Cl(\mathfrak{f})$$. Denoting by $$\widehat {Cl(f)}^+$$ the orbits of the group of characters, we show that the values $$L(1,\chi )$$ as $$\chi$$ ranges over elements of $$\widehat {Cl(\mathfrak{f})}^+$$ are linearly independent over $$\overline{\mathbb{Q}}$$. We give applications of this result to the study of transcendental values of Petersson inner products and certain special values of Artin $$L$$-series attached to dihedral extensions.
Part I, cf. Math. Ann. 351, No. 4, 835–855 (2011; Zbl 1281.11071).

##### MSC:
 11J81 Transcendence (general theory) 11M32 Multiple Dirichlet series and zeta functions and multizeta values
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##### References:
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