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Properties

Label 1-571-571.18-r1-0-0
Degree $1$
Conductor $571$
Sign $0.829 - 0.559i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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  + (0.998 + 0.0550i)2-s + (0.968 − 0.250i)3-s + (0.993 + 0.110i)4-s + (0.984 − 0.175i)5-s + (0.980 − 0.197i)6-s + (−0.431 − 0.901i)7-s + (0.986 + 0.164i)8-s + (0.874 − 0.485i)9-s + (0.992 − 0.120i)10-s + (0.391 + 0.920i)11-s + (0.989 − 0.142i)12-s + (0.287 − 0.957i)13-s + (−0.381 − 0.924i)14-s + (0.909 − 0.416i)15-s + (0.975 + 0.218i)16-s + (−0.509 + 0.860i)17-s + ⋯
L(s)  = 1  + (0.998 + 0.0550i)2-s + (0.968 − 0.250i)3-s + (0.993 + 0.110i)4-s + (0.984 − 0.175i)5-s + (0.980 − 0.197i)6-s + (−0.431 − 0.901i)7-s + (0.986 + 0.164i)8-s + (0.874 − 0.485i)9-s + (0.992 − 0.120i)10-s + (0.391 + 0.920i)11-s + (0.989 − 0.142i)12-s + (0.287 − 0.957i)13-s + (−0.381 − 0.924i)14-s + (0.909 − 0.416i)15-s + (0.975 + 0.218i)16-s + (−0.509 + 0.860i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.829 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.829 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.829 - 0.559i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.829 - 0.559i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(7.212110232 - 2.204641962i\)
\(L(\frac12)\) \(\approx\) \(7.212110232 - 2.204641962i\)
\(L(1)\) \(\approx\) \(3.253810891 - 0.5342538393i\)
\(L(1)\) \(\approx\) \(3.253810891 - 0.5342538393i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.998 + 0.0550i)T \)
3 \( 1 + (0.968 - 0.250i)T \)
5 \( 1 + (0.984 - 0.175i)T \)
7 \( 1 + (-0.431 - 0.901i)T \)
11 \( 1 + (0.391 + 0.920i)T \)
13 \( 1 + (0.287 - 0.957i)T \)
17 \( 1 + (-0.509 + 0.860i)T \)
19 \( 1 + (-0.930 - 0.366i)T \)
23 \( 1 + (0.894 + 0.446i)T \)
29 \( 1 + (0.904 - 0.426i)T \)
31 \( 1 + (-0.677 + 0.735i)T \)
37 \( 1 + (-0.795 - 0.605i)T \)
41 \( 1 + (-0.137 + 0.990i)T \)
43 \( 1 + (-0.421 - 0.906i)T \)
47 \( 1 + (0.298 + 0.954i)T \)
53 \( 1 + (0.256 - 0.966i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (0.159 - 0.987i)T \)
67 \( 1 + (0.709 + 0.705i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (0.126 - 0.991i)T \)
79 \( 1 + (-0.00551 - 0.999i)T \)
83 \( 1 + (-0.319 + 0.947i)T \)
89 \( 1 + (0.660 + 0.750i)T \)
97 \( 1 + (0.202 + 0.979i)T \)
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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

−22.8937188384035599024561205846, −21.99056217877883303471584468730, −21.49244684171590787459929744771, −20.987565831960217612079700256597, −19.98934488597992615959174063203, −18.968282079649536921371137821734, −18.567569024188335171258952215348, −16.87406993515973046932151139757, −16.20050079509240871551862167691, −15.26793172157554837516255273798, −14.50341929650841644607473922367, −13.792056646551114356856323298090, −13.2354657170482143691197333797, −12.26595320422960653226101426244, −11.155439859012524722016543248749, −10.2384515369568986966145339561, −9.13182027416924432892393748372, −8.6240518691899789775909780266, −6.97603126344599581044522774159, −6.35280844235569593419725025475, −5.338557167123632707537521877906, −4.28276719313692674243274314307, −3.14483086883188465080442996375, −2.478636760195414348326883379339, −1.56392530689244193533802299557, 1.23374028275483926892559012220, 2.1123750865400269043887250400, 3.16359039668108316517350707642, 4.07491202213014836770143652818, 5.03778983942721951173084810943, 6.42750547323726331999585836586, 6.87376608002785498697938126887, 7.9809693422357960325609561919, 9.09767234087349099651700996659, 10.18501802609906723329140067541, 10.74582899275111005623842937385, 12.45541528870141188680006249889, 12.98946254738142383670059315420, 13.4968570722431507528613516809, 14.40933705150476357677118462588, 15.068795628564801828299758231, 15.937398058161816172609421343550, 17.21664336832749775369500462863, 17.64143082683890469148086476062, 19.21672597093474431824683498430, 19.92298509607555587586312856734, 20.49227247539968406871558751634, 21.27430488132541853655884710548, 22.03135192856847121130703431293, 23.085049045832778469533321065986

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