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In mathematics, an automorphic L-function is a function ''L''(''s'',π,''r'') of a complex variable ''s'', associated to an automorphic form π of a reductive group ''G'' over a global field and a finite-dimensional complex representation ''r'' of the Langlands dual group ''L''''G'' of ''G'', generalizing the Dirichlet L-series of a Dirichlet character and the Mellin transform of a modular form. They were introduced by . and gave surveys of automorphic L-functions. ==Properties== Automorphic L-functions should have the following properties (which have been proved in some cases but are still conjectural in other cases). The L-function ''L''(''s'',π,''r'') should be a product over the places ''v'' of ''F'' of local L functions. :''L''(''s'',π,''r'') = Π ''L''(''s'',π''v'',''r''''v'') Here the automorphic representation π=⊗π''v'' is a tensor product of the representations π''v'' of local groups. The L-function is expected to have an analytic continuation as a meromorphic function of all complex ''s'', and satisfy a functional equation :''L''(''s'',π,''r'') = ε(''s'',π,''r'')''L''(1 – ''s'',π,''r''∨) where the factor ε(''s'',π,''r'') is a product of "local constants" :ε(''s'',π,''r'') = Π ε(''s'',π''v'',''r''''v'', ψ''v'') almost all of which are 1. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Automorphic L-function」の詳細全文を読む スポンサード リンク
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