L-functions arise from many different sources. Already in degree 2 we have examples of L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions).

Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Stephan Ehlen on 2019-04-30 10:00:55

**Referred to by:**

- intro.features
- lmfdb/lfunctions/templates/Degree2.html (line 49)
- lmfdb/lfunctions/templates/Degree2.html (line 79)
- lmfdb/lfunctions/templates/Degree3.html (line 49)
- lmfdb/lfunctions/templates/Degree3.html (line 68)
- lmfdb/lfunctions/templates/Degree4.html (line 53)
- lmfdb/lfunctions/templates/Degree4.html (line 81)
- lmfdb/lfunctions/templates/DegreeNavigateL.html (line 17)
- lmfdb/lfunctions/templates/DegreeNavigateL.html (line 37)
- lmfdb/lfunctions/templates/DegreeNavigateL.html (line 57)
- lmfdb/lfunctions/templates/DegreeNavigateL.html (line 77)
- lmfdb/lfunctions/templates/LfunctionNavigate.html (line 8)
- lmfdb/lfunctions/templates/yamltotable2.pl (line 49)
- lmfdb/lfunctions/templates/yamltotable3.pl (line 44)
- lmfdb/lfunctions/templates/yamltotable4.pl (line 49)

**History:**(expand/hide all)

**Differences**(show/hide)