Asymptotics for Sums of Central Values of Canonical Hecke L-Series Academic Article uri icon

abstract

  • Let K = ℚ(√-D) be an imaginary quadratic field of discriminant -D with D > 3 and D = 3 mod 4. Let OK be the ring of integers of K, let ε be the quadratic character of K of conductor √-DO K, and let ψk be a Hecke character of K of conductor √-DOK satisfying ψk(αOK) = ε(α)α2k-1, for (αOK|√-DO K) = 1, k ε ℤ≥1. Let h(-D) be the class number of K and let εk be the set of h(-D) Hecke characters of the form εk. If L(εk, s) denotes the L-series of εk then its central value is L(εk, k). In our main theorem we establish for each even integer k ≥ 2 an asymptotic formula for the average 1/h(-D) σεkεεk L(εk,K)/L((-D),1) as D→∞. We then use this formula to prove that there exists an absolute constant λ ≥ 0 such that the number of nonvanishing central values in the family {L(εk, k) : εk ε εk} is ≫ Dλ as D→∞ © The Author 2007.

author list (cited authors)

  • Masri, R.

citation count

  • 1

publication date

  • July 2010