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).

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**Knowl status:**

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

**Referred to by:**

- intro
- intro.features
- lmfdb/lfunctions/main.py (line 498)
- lmfdb/lfunctions/main.py (line 504)
- lmfdb/lfunctions/templates/Degree2.html (line 49)
- lmfdb/lfunctions/templates/Degree2.html (line 72)
- lmfdb/lfunctions/templates/Degree3.html (line 49)
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- lmfdb/lfunctions/templates/Degree4.html (line 53)
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- lmfdb/lfunctions/templates/DegreeNavigateL.html (line 17)
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- lmfdb/lfunctions/templates/DegreeNavigateL.html (line 57)
- lmfdb/lfunctions/templates/DegreeNavigateL.html (line 77)
- lmfdb/lfunctions/templates/LfunctionEulerSearchResults.html (line 17)
- lmfdb/lfunctions/templates/LfunctionNavigate.html (line 35)
- lmfdb/lfunctions/templates/LfunctionSearchResults.html (line 32)
- lmfdb/lfunctions/templates/LfunctionTraceSearchResults.html (line 13)

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