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XI an even primitive character and A(q,I,x1,Y,s) are holomorphic e2 - al ~ O C(q,I,x1,Y,s) is holomorphic for s E C2 PROOF: From q > 1 and (104) we deduce F (q,Xl,~) = g(X1,W). It follows from Landau [22], § 128 that g(X,~) has no zeros for Re ~ ~ ~ . Hence the first part of the theorem follows from (108) (109). The last part is a consequence of (107). Theorem 14 is proved. THEOREM 15: Let q > I and XI be an even primitive character mod q. Then ~ (q,l,xI,Y,w) is holomorphic in the half-plane Re ~ ~ ~ .

IY[P(1)],(s2,sl)) to I (135) ~(q,l,×l,Y,~) = G(×I)~- ~(q,l,~l,Y[p(1)~,l - ~) Let q = I . Then Y = O(2) and W,Q(1), P(1) E ~(2). (1,1,1,Y,(s2,sl)) k(1,1,1,Y,-s)= (137) § 3. &(I,I,I,Y,~) = X(I,I,I,Y,I-~) ELEMENTARY EISTF~STEIN X(I,I,I,y-I,s), = ~(1,I,I,y-I,w) SERIES § 3. contains the definition, analytic continuation equation of elementary Eisenstein series. Let z ~ 3(1). Like in (41) put (138) Y = Y x Then (139) Det Y = I From (46) we deduce (14o) = I Y[h] = ~lhlZ + h212 and functional 25 Let I ' ( I ) = Sp(1, 7Z) (141) For and F1(1) = {M = (±~ ~ ) M = (~ ~) £ Sp(1, ~ ) (142) let, 6 F(t)} l i k e in (38), M - aZcz + db = XM + iy M with real xM,Y M.

The function ~(q,l,xI,Y,~) is holomorphic for all w E C PROOF: Apply theorem 14. 1 Set (130) P(1) = q2 WQ-I(1) = Then (131) abs P(1) = I and (132) p2(1) = E . From (11), (80) we get V (133) Y = q-Iy[P(1)] 24 By theorem 11 and (106) k(q,l,x1,Y,s ) is homogeneous degree O. ,~IY[P(1)],(s2,sl)) to I (135) ~(q,l,×l,Y,~) = G(×I)~- ~(q,l,~l,Y[p(1)~,l - ~) Let q = I . Then Y = O(2) and W,Q(1), P(1) E ~(2). (1,1,1,Y,(s2,sl)) k(1,1,1,Y,-s)= (137) § 3. &(I,I,I,Y,~) = X(I,I,I,Y,I-~) ELEMENTARY EISTF~STEIN X(I,I,I,y-I,s), = ~(1,I,I,y-I,w) SERIES § 3.

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