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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

$ L$-functions of a quadratic form


Authors: T. Callahan and R. A. Smith
Journal: Trans. Amer. Math. Soc. 217 (1976), 297-309
MSC: Primary 10H10
DOI: https://doi.org/10.1090/S0002-9947-1976-0404164-6
MathSciNet review: 0404164
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Abstract: Let Q be a positive definite integral quadratic form in n variables, with the additional property that the adjoint form $ {Q^\dag }$ is also integral. Using the functional equation of the Epstein zeta function, we obtain a symmetric functional equation of the L-function of Q with a primitive character $ \omega \bmod q$ (additive or multiplicative) defined by $ \Sigma \omega (Q({\text{x}}))Q{({\text{x}})^{ - s}},\operatorname{Re} (s) > n/2$, where the summation extends over all $ {\text{x}} \in {Z^n},{\text{x}} \ne 0$; our result does not depend upon the usual restriction that q be relatively prime to the discriminant of Q, but rather on a much milder restriction.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0404164-6
Keywords: L-functions, quadratic forms, functional equation, Gaussian sums
Article copyright: © Copyright 1976 American Mathematical Society