Duke Mathematical Journal

Low-lying zeros of dihedral L-functions

E. Fouvry and H. Iwaniec

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Assuming the grand Riemann hypothesis, we investigate the distribution of the lowlying zeros of the $L$-functions $L(s,\psi)$, where $\psi$ is a character of the ideal class group of the imaginary quadratic field $\mathbb {Q}(\sqrt{-D}) (D\text {squarefree},D>3,D\equiv 3(\mod 4))$. We prove that, in the vicinity of the central point $s = 1/2$, the average distribution of these zeros (for $D\longrightarrow \infty$) is governed by the symplectic distribution. By averaging over $D$, we go beyond the natural bound of the support of the Fourier transform of the test function. This problem is naturally linked with the question of counting primes $p$ of the form $4p = m\sp 2+Dn\sp 2$, and sieve techniques are applied.

Article information

Duke Math. J., Volume 116, Number 2 (2003), 189-217.

First available in Project Euclid: 26 May 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Secondary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 11N36: Applications of sieve methods 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]


Fouvry, E.; Iwaniec, H. Low-lying zeros of dihedral L -functions. Duke Math. J. 116 (2003), no. 2, 189--217. doi:10.1215/S0012-7094-03-11621-X. https://projecteuclid.org/euclid.dmj/1085598267

Export citation


  • E. Bombieri, The asymptotic sieve, Rend. Accad. Naz. XL (5), 1/2 (1975/76), 243--269.
  • D. A. Cox, Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory, and Complex Multiplication, Wiley Intersci. Publ., Wiley, New York, 1989.
  • W. Duke, The dimension of the space of cusp forms of weight one, Internat. Math. Res. Notices 1995, 99--109.
  • I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1965.
  • E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, I, Math. Z. 1 (1918), 357--376.
  • H. Iwaniec, W. Luo, and P. Sarnak, Low lying zeros of families of $L$-functions, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 55--131.
  • N. M. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy, Amer. Math. Soc. Colloq. Publ. 45, Amer. Math. Soc., Providence, 1999.
  • --. --. --. --., Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 1--26.
  • E. Royer, Petits zéros de fonctions $L$ de formes modulaires, Acta Arith. 99 (2001), 147--172.
  • J.-P. Serre, ``Modular forms of weight one and Galois representations'' in Algebraic Number Fields: $L$-Functions and Galois Properties (Durham, England, 1975), Academic Press, London, 1977, 193--268.