¿ Functions of a Quadratic Form

Let Q be a positive definite integral quadratic form in ti variables, with the additional property that the adjoint form Q' is also integral. Using the functional equation of the Epstein zeta function, we obtain a symmetric functional equation of the ¿-function of Q with a primitive character to mod q (additive or multiplicative) defined by £io(2(x))C(x)-ï. Re(s) > nl2> where the summation extends over all x e z", x ¥= 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.