## A non-vanishing theorem for zeta functions of $$\mathrm{GL}_n$$.(English)Zbl 0349.12006

### MSC:

 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F11 Holomorphic modular forms of integral weight 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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### References:

 [1] Andrianov, A. N.: On the Zeta-functions of the general linear groups. Congrès International des mathematiciens, pp. 273-276. Paris: Gauthier-Villars 1970 [2] Gelfand, I. M., Kajdan, D. A.: Representations of the groupGL (n, K) whereK is a local field. Lie groups and their representations, Summer School of Group Representations, Budapest, pp. 95-118. New York: Halsted-Press 1971 [3] Godement, R.: Analyse spectrale des fonctions modulaires. Séminaire Bourbaki, (1964/65), No. 278, 26 p. [4] Godement, R.: Introduction à la théorie de Langlands. Séminaire Bourbaki, (1966/67), No. 321, 30 p. [5] Godement, R., Jacquet, H.: Zeta functions of simple algebras. Springer Lecture Notes 260 (1972) · Zbl 0244.12011 [6] Harish-Chandra: Automorphic forms on Semi-Simple Lie Groups. Springer Lecture Notes 62 (1968) · Zbl 0186.04702 [7] Jacquet, H.: Automorphic forms onGL(2). II. Springer Lecture Notes 278 (1972) · Zbl 0243.12005 [8] Jacquet, H., Pyatetskii-Shapiro, I. I., Shalika, J.: Construction de formes automorphes pour le groupeGL(3). Comptes Rendus, Acad. Sci., Paris, Vol. 282, (1976), pp. 91-94 · Zbl 0356.12017 [9] Langlands, R. P.: On the functional equations satisfied by Eisenstein Series, preprint · Zbl 0332.10018 [10] Langlands, R. P.: Problems in the theory of automorphic forms. Modern Analysis and its applications. Springer Lecture Notes 170 (1970), pp. 18-61 [11] Langlands, R. P.: Euler products. Lecture notes, Yale University (1967) [12] Ogg, A. P.: On a convolution ofL-series. Inventiones Math.,7, 297-312 (1969) · Zbl 0205.50902 [13] Pyatetskii-Shapiro, I.: Euler subgroups, Lie groups and their representations. Summer School of Group Representations, Budapest, pp. 597-620. New York: Halsted-Press 1971 [14] Rankin, R.: Contributions to the theory of Ramanujan’s functions ?(n) and similar arithmetical functions. I. Proc. Camb. Phil. Soc.35, 351-356 (1939) · Zbl 0021.39201 [15] Shalika, J.: The multiplicity one theorem forGL n . Annals of math., Vol. 100, No. 1, 171-193 (1974) · Zbl 0316.12010
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