The behavior of HeckeL-functions of real quadratic fields ats= 0

Abstract

For a family of real quadratic fields {Kn=ℚ(f(n))}n∈ℕ, a Dirichlet character χ modulo q, and prescribed ideals {bn⊂Kn}, we investigate the linear behavior of the special value of the partial Hecke L-function LKn(s,χn:=χ∘NKn,bn) at s=0. We show that for n=qk+r, LKn(0,χn,bn) can be written as 1 1 2 q 2 ( A χ ( r ) + k B χ ( r ) ) , where Aχ(r),Bχ(r)∈ℤ[χ(1),χ(2),…,χ(q)] if a certain condition on bn in terms of its continued fraction is satisfied. Furthermore, we write Aχ(r) and Bχ(r) explicitly using values of the Bernoulli polynomials. We describe how the linearity is used in solving the class number one problem for some families and recover the proofs in some cases.