$L$-functions of a quadratic form

Abstract:

Let Q be a positive definite integral quadratic form in n variables, with the additional property that the adjoint form ${Q^\dagger }$ 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.