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