Langlands' analytic continuation and functional equation conjecture for automorphic L-functions

Let FF be a number field, let A\mathbb{A} be its ring of adeles, let GG be a connected linear algebraic group defined over FF, let π\pi be an automorphic representation of G(A)G(\mathbb{A}), and let rr be a finite-dimensional representation of the Langlands dual group LG{}^{\mathrm{L}}G. The associated Langlands LL-function is denoted by L(s,π,r)L(s,\pi,r). Langlands' conjecture. The LL-function L(s,π,r)L(s,\pi,r) has a meromorphic continuation to the whole complex plane with only finitely many poles, and it satisfies a functional equation relating ss to 1s1-s. This is a foundational analytic expectation for automorphic LL-functions and generalizes the continuation and functional equation known for classical zeta and Dirichlet LL-functions. The candidate is not accompanied by evidence of resolution in the supplied text.

Sources & referencesView supporting material

Primary source

Fangyang Tian, “On the Archimedean Local Gamma Factors for Adjoint Representation of GL_3, Part I”, arXiv:1811.02752 (2018).

Additional references

2 papers in this index state this conjecture (2015–2018). The statement above is taken from the most recent of them; the others are arXiv:1506.09128.

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