On Artin \(L\)-functions for octic quaternion fields. (English) Zbl 1046.11083

Omar, Sami

Exp. Math. 10, No. 2, 237-245 (2001).

Summary: We study the Artin \(L\)-function \(L(s,\chi)\) associated to the unique character \(\chi\) of degree 2 in quaternion fields of degree 8. We first explain how to find examples of quaternion octic fields with not too large a discriminant. We then develop a method yielding a quick computation of the order \(n_\chi\) of the zero of \(L(s,\chi)\) at the point \(s=\tfrac 12\). In all our calculations, we find that \(n_\chi\) only depends on the sign of the root number \(W(\chi)\); indeed \(n_\chi=0\) when \(W(\chi)=+1\) and \(n_\chi=1\) when \(W(\chi)=-1\). Finally we give some estimates on \(n_\chi\) and low zeros of \(L(s,\chi)\) on the critical line in terms of the Artin conductor \({\mathfrak f}_\chi\) of the character \(\chi\).

MSC:

11R42

Zeta functions and \(L\)-functions of number fields

References:

[1]	Barner K., J. Reine Angew. Math. 323 pp 139– (1981)
[2]	Cohen H., A course in computational algebraic number theory (1993) · Zbl 0786.11071
[3]	DOI: 10.1007/978-1-4419-8489-0 · Zbl 0977.11056 · doi:10.1007/978-1-4419-8489-0
[4]	Damey P., J. Reine Angew. Math. 244 pp 37– (1970)
[5]	DOI: 10.1007/BF01418937 · Zbl 0261.12008 · doi:10.1007/BF01418937
[6]	Goss D., Basic structures of function field arithmetic (1996) · Zbl 0874.11004
[7]	Kwon S.-H., Arch. Math. (Basel) 67 (2) pp 119– (1996) · Zbl 0854.11057 · doi:10.1007/BF01268925
[8]	Martinet J., Ann. Sci. École Norm. Sup. (4) 4 pp 399– (1971)
[9]	Martinet J., Algebraic number fields: L-functions and Galois properties (Durham, Durham, 1975) pp 1– (1977)
[10]	Mestre J.-F., Séminaire de thaéorie des nombres (Paris, 1981/1982) pp 179– (1983)
[11]	Mestre J.-F., Compositio Math. 58 (2) pp 209– (1986)
[12]	Murty M. R., Non-vanishing of L-functions and applications (1997) · Zbl 1235.11086 · doi:10.1007/978-3-0348-0274-1
[13]	Poitou G., Séminaire Delange–Pisot–Poitou, 18e année (1976/77) (1977)
[14]	Serre J.-P., Cours d’arithmétique (1970)
[15]	Serre J.-P., Représentations linéaires des groupes finis, (1978)
[16]	Stark H. M., Invent. Math. 23 pp 135– (1974) · Zbl 0278.12005 · doi:10.1007/BF01405166
[17]	DOI: 10.1090/S0025-5718-97-00871-5 · Zbl 0877.11061 · doi:10.1090/S0025-5718-97-00871-5
[18]	Weil A., Izv. Akad. Nauk SSSR Ser. Mat. 36 pp 3– (1972)
[19]	DOI: 10.1515/crll.1936.174.237 · Zbl 0013.19601 · doi:10.1515/crll.1936.174.237