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Properties

Label 1-571-571.179-r0-0-0
Degree $1$
Conductor $571$
Sign $0.231 + 0.972i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
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  + (−0.986 − 0.164i)2-s + (0.431 + 0.901i)3-s + (0.945 + 0.324i)4-s + (−0.995 − 0.0990i)5-s + (−0.277 − 0.960i)6-s + (−0.518 + 0.854i)7-s + (−0.879 − 0.475i)8-s + (−0.627 + 0.778i)9-s + (0.965 + 0.261i)10-s + (0.965 − 0.261i)11-s + (0.115 + 0.993i)12-s + (0.997 − 0.0660i)13-s + (0.652 − 0.757i)14-s + (−0.340 − 0.940i)15-s + (0.789 + 0.614i)16-s + (0.828 − 0.560i)17-s + ⋯
L(s)  = 1  + (−0.986 − 0.164i)2-s + (0.431 + 0.901i)3-s + (0.945 + 0.324i)4-s + (−0.995 − 0.0990i)5-s + (−0.277 − 0.960i)6-s + (−0.518 + 0.854i)7-s + (−0.879 − 0.475i)8-s + (−0.627 + 0.778i)9-s + (0.965 + 0.261i)10-s + (0.965 − 0.261i)11-s + (0.115 + 0.993i)12-s + (0.997 − 0.0660i)13-s + (0.652 − 0.757i)14-s + (−0.340 − 0.940i)15-s + (0.789 + 0.614i)16-s + (0.828 − 0.560i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.231 + 0.972i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.231 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6905804585 + 0.5453981315i\)
\(L(\frac12)\) \(\approx\) \(0.6905804585 + 0.5453981315i\)
\(L(1)\) \(\approx\) \(0.7027433832 + 0.2492479499i\)
\(L(1)\) \(\approx\) \(0.7027433832 + 0.2492479499i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.986 - 0.164i)T \)
3 \( 1 + (0.431 + 0.901i)T \)
5 \( 1 + (-0.995 - 0.0990i)T \)
7 \( 1 + (-0.518 + 0.854i)T \)
11 \( 1 + (0.965 - 0.261i)T \)
13 \( 1 + (0.997 - 0.0660i)T \)
17 \( 1 + (0.828 - 0.560i)T \)
19 \( 1 + (0.180 - 0.983i)T \)
23 \( 1 + (0.991 + 0.131i)T \)
29 \( 1 + (0.245 - 0.969i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (-0.846 + 0.533i)T \)
41 \( 1 + (-0.401 + 0.915i)T \)
43 \( 1 + (0.0495 + 0.998i)T \)
47 \( 1 + (0.789 + 0.614i)T \)
53 \( 1 + (-0.148 + 0.988i)T \)
59 \( 1 + (-0.0825 - 0.996i)T \)
61 \( 1 + (0.701 + 0.712i)T \)
67 \( 1 + (-0.148 + 0.988i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.846 + 0.533i)T \)
79 \( 1 + (-0.956 - 0.293i)T \)
83 \( 1 + (-0.277 - 0.960i)T \)
89 \( 1 + (-0.277 - 0.960i)T \)
97 \( 1 + (-0.0165 + 0.999i)T \)
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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

−23.34166845136258300061120600103, −22.65670860557415918119719028990, −20.77084117818511099892844741014, −20.38249187494384241629062717335, −19.31437784902686015573958885141, −19.16330492318737859179951373101, −18.27090298186854645847554163663, −17.1443102962303414594076399355, −16.6161689176529769575279853036, −15.585694882764525199277262943046, −14.6705723601274330260911922413, −13.91619170370677587412689744489, −12.585135219468445113225141323491, −11.9708166952059926224197384645, −10.99702106559398626311249827820, −10.09307148053330158221865475580, −8.87111067806833589876882020807, −8.30741627425331760571392514241, −7.25518388371708605190925174480, −6.88377812896723602858821047420, −5.85527469828067776397270537956, −3.81510349496674140619127300901, −3.24360183910212989099561093429, −1.604490513271087418280283698023, −0.78423443045194113591435092757, 1.08935593028691852503896169548, 2.913089486841231133417227493044, 3.27559131309396932649983834202, 4.532742691993661013054052350989, 5.90400691594642697049153111199, 6.98331647358336865174431820752, 8.14966723523489619412900595176, 8.8263018285519297667814106984, 9.364416209732981000718457639038, 10.40435593324379346947996407150, 11.51898142175049065608087260485, 11.75234603342712761718946518957, 13.11351896335296986224356400522, 14.4714552384143520595036467484, 15.462232236713207266021840056140, 15.80876437059031640484563725511, 16.547347350996170515436148493509, 17.47887928887107478588850257763, 18.8681057169542281002500933669, 19.14496507875078293863818851214, 19.99090326115828206137155154257, 20.77313948794948119360775808052, 21.53820598843909756330950269411, 22.44903639987048263658272122312, 23.32246911175708214235116502051

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