By Baruch Z. Moroz

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**Example text**

Of If to the c h a r a c t e r s subgroup, again we may assume P = Pl ~ on the d e g r e e corollary X. X that p isn't of lower I allows to the one i n v o l v i n g after passing, If represent- if n e c e s s a r y , is p r i m i t i v e to to and i r r e - 5, Pl 6 gr(k), (cf. L e m m a 2). 1, 29 P2 = G ( K k ) )~j Zm' 8 Ind G (K ik) _ 7m ~ indG(K j= I (K kj) 4 9 j=m'+1 kj for some finite extensions I < j < m. kj and grossencharacters It follows from (11) that ~j Pl (Op) = XI (p) with in (28) gr(kj), X I = tr Pl' so that ip(p,t) (29) = det(1-P2(dp) (XI (p)t)) .

I and d e f i n i t i o n s [78], (20) >__ c5R(t), - (22) is larger constants p. 2),it and follows that If(s)-f(s I (tl)) ] < 2(M - Re f(s I (tl))) r(t I ) R(tl)_r(tl ) , (23) and f, 2R(t I ) I-f-(s) I < 2 (S-- (R(tl)_r (tl)) where = R(tl). 30) By it follows from < log ~k(1+n) and c7 ~k(1+~) < + log ~ ( t I) is a n u m e r i c a l 3 n+2 ~ D ~ e loss of g e n e r a l i t y , Estimate To p r o v e (19) follows (18) we r e m a r k -I we a s s u m e from (20) - therefore, by and the d e f i n i t i o n -I (27) + c7, constant.

16) On the other hand, (/~) log - -9 = 4/£ It follows from 9/4 94/ ~ (/5) d__uu < ~ u- (16) and V(/~) (17) f ~ (u) o u du. og (18) ~(T+3)). ition 2. Lemma Estimate 3. Let T > O, < Re s < I, O < Im s < T, L(s,x) X 6 gr(k). = N(X,T) from Then = O}. Then + O(log(a(x)b(x)(3+T)n)). (19) follows X 6 gr(k). + ~(/~), (18). for card{sIO Let N(x,T+I) Proof. 30) in view of lemma 2. 56 --(s,x) = ~ (s-P) -1 I - I g(x)(s+s--Z~ -) (20) + O(A(X,t)), It-y1<1 I - ~ < Re where (3+Itl) n) Res< s < and p 2, t:= ranges Im s, over y:= the Im p, A ( x , t ) : = zeros of L(s,x) log(a(x)b(x)" in the s t r i p 0 < I.