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oo. §4 Heeke's L-funetions 21 §4 Hecke's L-functions We begin by constructing the analogues of Dirichlet's L-functions. We first need to define the notion of "ideal classes" and then define characters of these classes. Let K be an algebraic number field and f an ideal of OK. A natural starting point is to consider the ideal class group and to define characters of this group.

Then, if A i- {I}, (AA, X) is positive for all Xi-I and = 0 if X = 1. If A = {I} then AA = O. This proves Step 1. To prove the equality of Step 2, it is enough to show that for any irreducible character 1/J of C, both sides have the same inner product with 1/J. Now (ICI(rege -le),1/J) = L(rege -le)(g)1/J(g) = ICI1/J(l) - L 1/J(g) gEe Also, by Frobenius reciprocity, L(Ind~ AA, 1/J) = ~)AA' 1/JIA) A A = LA {¢(IAI)1/J(l) - L 1/J(0")} aEA =A = 1/J(1) L ¢(IAI) - L 1/J(0"). 2. ICI· 32 Chapter 2 Artin L-Functions We illustrate the equality of Step 2 above with an example.

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