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# Orbital $L$-functions for the Space of Binary Cubic Forms

Published:2013-10-09
Printed: Dec 2013
• Takashi Taniguchi,
Department of Mathematics, Graduate School of Science, Kobe University, Kobe 657-8501, Japan
• Frank Thorne,
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
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## Abstract

We introduce the notion of orbital $L$-functions for the space of binary cubic forms and investigate their analytic properties. We study their functional equations and residue formulas in some detail. Aside from their intrinsic interest, the results from this paper are used to prove the existence of secondary terms in counting functions for cubic fields. This is worked out in a companion paper.
 Keywords: binary cubic forms, prehomogeneous vector spaces, Shintani zeta functions, $L$-functions, cubic rings and fields
 MSC Classifications: 11M41 - Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11E76 - Forms of degree higher than two