Skip to main content
SpringerLink
Log in

L-functions of second-order cusp forms

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We discuss equivalent definitions of holomorphic second-order cusp forms and prove bounds on their Fourier coefficients. We also introduce their associated L-functions, prove functional equations for twisted versions of these L-functions and establish a criterion for a Dirichlet series to originate from a second order form. In the last section we investigate the effect of adding an assumption of periodicity to this criterion.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chinta, G., Diamantis, N., O’Sullivan, C.: Second order modular forms. Acta Arithmetica 103, 209–223 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Conrey, B., Farmer, D.: An extension of Hecke’s converse theorem. Internat. Math. Res. Notices 9, 445–463 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Farmer, D.: Converse theorems and second order modular forms. AMS Sectional Meeting Talk, Salt Lake City (2002)

  4. Flood, K.: Two Correspondence Theorems for Modular Integrals. Dissertation (Temple University) (1993)

  5. Iwaniec, H.: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics 17 (1997)

  6. Iwaniec, H.: Spectral methods of automorphic forms, 2nd ed., vol. 53, Graduate studies in mathematics. Amer. Math. Soc. (2002)

  7. Kleban, P., Zagier, D.: Crossing probabilities and modular forms. J. Stat. Phys. 113, 431–454 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ogg, A.: Modular forms and Dirichlet series. W.A. Benjamin, Inc. (1969)

  9. Razar, M.: Modular forms for Г0(N) and Dirichlet series. Trans. AMS 231, 489–495 (1977)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, University Park, NG7 2RD, United Kingdom

    N. Diamantis

  2. Department of Mathematics, Temple University, Philadelphia, Pennsylvania, 19122-2585

    M. Knopp

  3. Mathematics Department, University of California Santa Cruz, 194 Baskin Engineering, Santa Cruz, California, 95064

    G. Mason

  4. Department of Math & Computer Science, Bronx Community College of the City University of New York, University Ave. at West 181 St., Bronx, New York, 10453

    C. O’Sullivan

Authors
  1. N. Diamantis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Knopp
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Mason
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. O’Sullivan
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

2000 Mathematics Subject Classification Primary—11F12, 11F66

Rights and permissions

Reprints and permissions

About this article

Cite this article

Diamantis, N., Knopp, M., Mason, G. et al. L-functions of second-order cusp forms. Ramanujan J 12, 327–347 (2006). https://doi.org/10.1007/s11139-006-0147-2

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-006-0147-2

Keywords

Search

Navigation