Welcome to the LMFDB, the database of L-functions, modular forms, and related objects. These pages are intended to be a modern handbook including tables, formulas, links, references, etc. to very concrete objects, in particular specific L-functions and their sources. L-functions are ubiquitous in number theory and have applications to mathematical physics and cryptography. By an L-function, we generally mean a Dirichlet series with a functional equation and an Euler product, the simplest example being the Riemann zeta function. Two of the seven Clay Mathematics Million Dollar Millennium Problems deal with properties of these functions, namely the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture. L-functions arise from and encode information about a number of mathematical objects. It is necessary to exhibit these objects along with the L-functions themselves, since typically we need these objects to compute L-functions. In these pages you will see examples of L-functions coming from modular forms, elliptic curves, number fields, and Dirichlet characters, as well as more generally from automorphic forms, algebraic varieties, and Artin representations. In addition, the database contains details about these objects themselves. See the Map of LMFDB for descriptions of connections between these objects. For additional information, there is a useful collection of freely available online sources at http://www.numbertheory.org/ntw/lecture_notes.html. The subject of L-functions is very rich, with many interrelationships. Our goal is to describe the data in ways that faithfully exhibit these interconnections, and to offer access to the data as a means of prompting further exploration and discovery. We believe that the creation of this website will lead to the development and understanding of new mathematics.

References in zbMATH (referenced in 121 articles )

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  1. Brock, Bradley W.; Elkies, Noam D.; Jordan, Bruce W.: Periodic continued fractions over (S)-integers in number fields and Skolem’s (p)-adic method (2021)
  2. Dawsey, Madeline Locus; McCarthy, Dermot: Generalized Paley graphs and their complete subgraphs of orders three and four (2021)
  3. Dieulefait, Luis; Soto, Eduardo: Solving (a x^p + b y^p = c z^p) with (abc) containing an arbitrary number of prime factors (2021)
  4. Dujella, Andrej; Kazalicki, Matija: Diophantine (m)-tuples in finite fields and modular forms (2021)
  5. Dwarshuis, Arjan; Roelfszema, Majken; Top, Jaap: Mazur’s rational torsion result for pointless genus one curves: examples (2021)
  6. Esparza-Lozano, Jose A.; Pasten, Hector: A conjecture of Watkins for quadratic twists (2021)
  7. Klüners, Jürgen; Komatsu, Toru: Imaginary multiquadratic number fields with class group of exponent (3) and (5) (2021)
  8. Marseglia, Stefano: Computing square-free polarized abelian varieties over finite fields (2021)
  9. Martin, Kimball: An on-average Maeda-type conjecture in the level aspect (2021)
  10. Verzobio, Matteo: Primitive divisors of elliptic divisibility sequences for elliptic curves with (j=1728) (2021)
  11. Akbary, Amir; Francis, Forrest J.: Euler’s function on products of primes in a fixed arithmetic progression (2020)
  12. Battistoni, Francesco: On small discriminants of number fields of degree 8 and 9 (2020)
  13. Berčič, Katja; Vidali, Janoš: DiscreteZOO: a fingerprint database of discrete objects (2020)
  14. Candelas, Philip; de la Ossa, Xenia; Elmi, Mohamed; van Straten, Duco: A one parameter family of Calabi-Yau manifolds with attractor points of rank two (2020)
  15. Chung, Hee-Joong; Kim, Dohyeong; Kim, Minhyong; Park, Jeehoon; Yoo, Hwajong: Arithmetic Chern-Simons theory. II (2020)
  16. Daniels, Harris B.; González-Jiménez, Enrique: On the torsion of rational elliptic curves over sextic fields (2020)
  17. Demark, David; Hindes, Wade; Jones, Rafe; Misplon, Moses; Stoll, Michael; Stoneman, Michael: Eventually stable quadratic polynomials over (\mathbbQ) (2020)
  18. Deo, Shaunak V.; Wiese, Gabor: Dihedral universal deformations (2020)
  19. Dieulefait, Luis Victor; Soto, Eduardo: Raising the level at your favorite prime (2020)
  20. Dummit, David S.: A note on the equivalence of the parity of class numbers and the signature ranks of units in cyclotomic fields (2020)

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