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L‐Series for genera ats= 1

 

作者: Kenneth H. Rosen,  

 

期刊: Mathematika  (WILEY Available online 1980)
卷期: Volume 27, issue 1  

页码: 10-16

 

ISSN:0025-5793

 

年代: 1980

 

DOI:10.1112/S0025579300009888

 

出版商: London Mathematical Society

 

数据来源: WILEY

 

摘要:

AbstractIt has been conjectured that, ifp≡ 1 (mod 4) is prime, and ifd<0 is a square‐free discriminant with(dk)=−1,thenL(1, χk,G)=8πk√|d|log ε,Whereεka/2εbelongs to the fieldΩ(√k), εkis the fundamental unit ofQ(√k),χk(n)=(nk), a=0 or a=1depending on whether there are an even number or an odd number of classes per genus inQ(√d), and Ω is the genus field ofQ(√d). HereL(s, χk, G)=∑L(s, χk, Q),the summation being over a complete set of inequivalent forms in the genusG, andL(s, χk, Q)=12∑(x, y) ≠ (0, 0)χk(Q(x, y))Q(x, y)s.In this paper it will be shown that this conjecture is true whendis the product of two odd discriminants. An example whendis the product of three prime discriminants is discussed.

 

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