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Properties

Label 1-571-571.349-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: $\chi_{571} (349, \cdot )$
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

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

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