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Reference.org
Automorphic L-function
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an automorphic <i>L</i>-function is a function <i>L</i>(<i>s</i>,π,<i>r</i>) of a complex variable <i>s</i>, associated to an <a href="/facts/Automorphic_representation/H0Pn7MyP">automorphic representation</a> π of a <a href="/facts/Reductive_group/z1zyJvUh">reductive group</a> <i>G</i> over a <a href="/facts/Global_field/na458xWs">global field</a> and a finite-dimensional complex representation <i>r</i> of the <a href="/facts/Langlands_dual_group/RheSlc7W">Langlands dual group</a> <i>L</i><i>G</i> of <i>G</i>, generalizing the <a href="/facts/Dirichlet_L-series/my8hc6Qy">Dirichlet L-series</a> of a <a href="/facts/Dirichlet_character/jDJHEBWa">Dirichlet character</a> and the <a href="/facts/Mellin_transform/M0087pMe">Mellin transform</a> of a <a href="/facts/Modular_form/PcnbYR1f">modular form</a>. They were introduced by <a href="/facts/Robert_Langlands/JWDgePcn">Langlands</a> (1967, 1970, 1971). </p><p>Borel (1979) and Arthur & Gelbart (1991) gave surveys of automorphic L-functions. </p>