The purpose of this book is to give an exposition of the analytic theory of L-
functions following the ideas of harmonic analysis inaugurated by Tate and Weil.
The central theme is the exploitation of the Local Langlands' Correspondence for
GLn to obtain results that apply to both Artin-Hecke L-functions associated to
representations of the Weil group and to automorphic L-functions of principal type
on GLn. In addition to establishing functional equations, explicit formulas and
non-vanishing theorems, essential ingredients in any discussion of generalized prime
number theorems, we also derive lower bound estimates for discriminants and con-
The author's intention has been to make available to a broad mathematical
audience those aspects of the theory of L-functions that are closely related to the
modern interconnections between the analytic theory of numbers and the theory of
group representations.
A noteworthy characteristic property of number fields that distinguishes them
from function fields is the existence of archimedean primes. These primes not only
make their appearance as gamma factors, but also play a crucial role in controlling
the analytic growth of L-functions and in the distribution of zeros and poles. In
this spirit, we have placed a great deal of emphasis in the study of archimedean
A detailed description of the contents of each chapter can be obtained from the
introductory section. A survey of the local theory of root numbers is also included
as an appendix.
This book is based on lectures given by the author over a period of several
years first at the University of Illinois and more recently at the Graduate School
and University Center of the City University of New York. The author acknowledges
the help he has received from many of his colleagues and students. The appendix,
written in collaboration with Aaron Wan, is the result of many fruitful discussions
about root numbers, and for these the author is particularly thankful.
Carlos Julio Moreno
North Salem, December 14, 2004
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