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Properties

Label 1-916-916.603-r0-0-0
Degree $1$
Conductor $916$
Sign $0.553 - 0.832i$
Analytic cond. $4.25388$
Root an. cond. $4.25388$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.401 + 0.915i)3-s + (−0.945 + 0.324i)5-s + (−0.969 − 0.245i)7-s + (−0.677 + 0.735i)9-s + (0.546 + 0.837i)11-s + (−0.324 − 0.945i)13-s + (−0.677 − 0.735i)15-s + (0.945 − 0.324i)17-s + (−0.945 − 0.324i)19-s + (−0.164 − 0.986i)21-s + (−0.475 − 0.879i)23-s + (0.789 − 0.614i)25-s + (−0.945 − 0.324i)27-s + (−0.969 − 0.245i)29-s + (−0.837 + 0.546i)31-s + ⋯
L(s)  = 1  + (0.401 + 0.915i)3-s + (−0.945 + 0.324i)5-s + (−0.969 − 0.245i)7-s + (−0.677 + 0.735i)9-s + (0.546 + 0.837i)11-s + (−0.324 − 0.945i)13-s + (−0.677 − 0.735i)15-s + (0.945 − 0.324i)17-s + (−0.945 − 0.324i)19-s + (−0.164 − 0.986i)21-s + (−0.475 − 0.879i)23-s + (0.789 − 0.614i)25-s + (−0.945 − 0.324i)27-s + (−0.969 − 0.245i)29-s + (−0.837 + 0.546i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(916\)    =    \(2^{2} \cdot 229\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(4.25388\)
Root analytic conductor: \(4.25388\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{916} (603, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 916,\ (0:\ ),\ 0.553 - 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5131167754 - 0.2751652659i\)
\(L(\frac12)\) \(\approx\) \(0.5131167754 - 0.2751652659i\)
\(L(1)\) \(\approx\) \(0.7578449355 + 0.1642459280i\)
\(L(1)\) \(\approx\) \(0.7578449355 + 0.1642459280i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (0.401 + 0.915i)T \)
5 \( 1 + (-0.945 + 0.324i)T \)
7 \( 1 + (-0.969 - 0.245i)T \)
11 \( 1 + (0.546 + 0.837i)T \)
13 \( 1 + (-0.324 - 0.945i)T \)
17 \( 1 + (0.945 - 0.324i)T \)
19 \( 1 + (-0.945 - 0.324i)T \)
23 \( 1 + (-0.475 - 0.879i)T \)
29 \( 1 + (-0.969 - 0.245i)T \)
31 \( 1 + (-0.837 + 0.546i)T \)
37 \( 1 + (-0.986 + 0.164i)T \)
41 \( 1 + (0.735 - 0.677i)T \)
43 \( 1 + (0.986 - 0.164i)T \)
47 \( 1 + (0.735 + 0.677i)T \)
53 \( 1 + (-0.401 - 0.915i)T \)
59 \( 1 + (0.164 - 0.986i)T \)
61 \( 1 + (-0.677 + 0.735i)T \)
67 \( 1 + (0.735 + 0.677i)T \)
71 \( 1 + (0.546 - 0.837i)T \)
73 \( 1 + (-0.614 - 0.789i)T \)
79 \( 1 + (0.969 - 0.245i)T \)
83 \( 1 + (0.986 + 0.164i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.789 - 0.614i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.060215663714962520770531264228, −21.18884290935185900521956809405, −20.1427407172788846047004445593, −19.39716952595121837353870352025, −19.07456945260900557461561022193, −18.48122115180916668247048598134, −16.99106858795367692143115575579, −16.64859779820364752325814017947, −15.65014603696443638414557772010, −14.72367748887319143588835905044, −14.01847618696874359943340805439, −13.038539507233929059944814211421, −12.36234837564238322348114709344, −11.80405889534926363878441320451, −10.88342355898288732403717720158, −9.40249047395849665913081495675, −8.9481678905634347808376563339, −7.956912544098569819985922924712, −7.2575446754570338134396599412, −6.33656067557660854480656041322, −5.58658804813360647329779472648, −3.887151690568356816646598271072, −3.53892281414390163418555715069, −2.27259160367695016872062679497, −1.112301080656450070869707385579, 0.26801228259650639329360518833, 2.31104910253704622742909949860, 3.29172047191840226906318445822, 3.895085808785155518201056883369, 4.75122869164475117569415598316, 5.89445070222739628223627955881, 7.06548946344449444457281148547, 7.73788510822049564840232244407, 8.76974762270550093462271246620, 9.6041591114250172043488971638, 10.38768516613846006169484063971, 10.97138687179214180089695034831, 12.257645467768754721484656045, 12.69794327490535670758314781884, 14.04541428285530082065747828515, 14.765292368119281487215995759856, 15.362009310019455577184952915832, 16.11478139792565801564949079025, 16.77316805225928123534253355253, 17.69670768737200726423745162581, 19.04308259076876031114413665719, 19.39164926577387067332491171788, 20.30808450401807237718138582932, 20.672009467795072469480296512877, 22.02606913259985279200660790422

Graph of the $Z$-function along the critical line