-functions of a quadratic form

Authors:
T. Callahan and R. A. Smith

Journal:
Trans. Amer. Math. Soc. **217** (1976), 297-309

MSC:
Primary 10H10

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 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 (additive or multiplicative) defined by , where the summation extends over all ; 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