Contents
Preface vii
Advice to the Reader ix
Introduction xiii
Chapter I. HECKE L-FUNCTIONS 1
1. Abstract Harmonic Analysis: A Survey 1
2. Zeta Distributions 18
3. Appendix: Principal L-functions on GL(n) 62
Notes 71
Chapter II. ARTIN-HECKE L-FUNCTIONS 73
1. Variations on Brauer's Theorem 73
2. Characteristic Polynomials and Integral Formulas 80
3. Representations of Weil Groups 84
4. Relative Local Weil Groups 87
5. Relative Global Weil Groups 88
6. Non-Archimedean Local L-factors 88
7. Conductors of Representations (The Herbrand Distribution) 89
8. Archimedean Local L-Factors 93
9. Decomposition Groups at Infinity 94
10. The Gamma Factors (Non-Abelian Case) 95
11. Local Properties of L-Functions 97
12. Global Properties of L-Functions 99
13. Artin-Hecke L-Functions (Functional Equation) 100
14. The Root Number 102
15. The Local Langlands Correspondence for GL(n) 103
16. The Principle of Functoriality 109
Notes 113
Chapter III. ANALYTIC PROPERTIES OF L-FUNCTIONS 115
1. Jensen's Formula 116
2. The Classical Convexity Estimate 119
3. Riemann's Product Formula 120
4. Phragmen-Lindelof Theorems 125
5. Bounds for Gamma Functions 131
6. Bounds for L-Functions on Vertical Strips 132
Notes 141
Chapter IV. THE EXPLICIT FORMULAS 143
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