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Properties

Label 1-571-571.136-r0-0-0
Degree $1$
Conductor $571$
Sign $0.521 + 0.853i$
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  + (0.451 + 0.892i)2-s + (−0.834 − 0.551i)3-s + (−0.592 + 0.805i)4-s + (0.0715 + 0.997i)5-s + (0.115 − 0.993i)6-s + (0.991 − 0.131i)7-s + (−0.986 − 0.164i)8-s + (0.391 + 0.920i)9-s + (−0.857 + 0.514i)10-s + (0.874 − 0.485i)11-s + (0.938 − 0.345i)12-s + (−0.126 − 0.991i)13-s + (0.565 + 0.824i)14-s + (0.490 − 0.871i)15-s + (−0.298 − 0.954i)16-s + (0.815 − 0.578i)17-s + ⋯
L(s)  = 1  + (0.451 + 0.892i)2-s + (−0.834 − 0.551i)3-s + (−0.592 + 0.805i)4-s + (0.0715 + 0.997i)5-s + (0.115 − 0.993i)6-s + (0.991 − 0.131i)7-s + (−0.986 − 0.164i)8-s + (0.391 + 0.920i)9-s + (−0.857 + 0.514i)10-s + (0.874 − 0.485i)11-s + (0.938 − 0.345i)12-s + (−0.126 − 0.991i)13-s + (0.565 + 0.824i)14-s + (0.490 − 0.871i)15-s + (−0.298 − 0.954i)16-s + (0.815 − 0.578i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.521 + 0.853i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.521 + 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.302352671 + 0.7302738822i\)
\(L(\frac12)\) \(\approx\) \(1.302352671 + 0.7302738822i\)
\(L(1)\) \(\approx\) \(1.070701063 + 0.4682823300i\)
\(L(1)\) \(\approx\) \(1.070701063 + 0.4682823300i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.451 + 0.892i)T \)
3 \( 1 + (-0.834 - 0.551i)T \)
5 \( 1 + (0.0715 + 0.997i)T \)
7 \( 1 + (0.991 - 0.131i)T \)
11 \( 1 + (0.874 - 0.485i)T \)
13 \( 1 + (-0.126 - 0.991i)T \)
17 \( 1 + (0.815 - 0.578i)T \)
19 \( 1 + (0.999 - 0.0440i)T \)
23 \( 1 + (0.701 - 0.712i)T \)
29 \( 1 + (-0.821 - 0.569i)T \)
31 \( 1 + (-0.677 + 0.735i)T \)
37 \( 1 + (0.685 - 0.728i)T \)
41 \( 1 + (-0.926 + 0.376i)T \)
43 \( 1 + (0.224 + 0.974i)T \)
47 \( 1 + (0.975 + 0.218i)T \)
53 \( 1 + (0.988 + 0.153i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (0.840 - 0.542i)T \)
67 \( 1 + (-0.360 - 0.932i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (0.287 + 0.957i)T \)
79 \( 1 + (-0.739 + 0.673i)T \)
83 \( 1 + (-0.917 + 0.396i)T \)
89 \( 1 + (0.802 + 0.596i)T \)
97 \( 1 + (0.583 + 0.812i)T \)
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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

−23.03686585065031859672987665443, −21.9630694715325187693425204935, −21.5692678252402757971112647814, −20.63400163934463347597841473133, −20.25821010478742500708042904807, −18.99680875736629965316761479716, −18.10799799588235919319967172599, −17.1323203243026374755804020626, −16.696151514803346406580067172035, −15.33067104355037225924872213445, −14.65285103081364804494668189775, −13.672850666759174506199043614953, −12.565445393963003546052139386667, −11.74771889070414591102689742906, −11.53590766856808839571996203576, −10.29370020570717903182478176884, −9.379185876160463898651990695822, −8.8613902469473035941008643520, −7.26439175652264767419888876023, −5.77641255301791255693122325032, −5.22345512265435025382573021552, −4.35635555863864862453966073312, −3.69190561515230902931042069713, −1.76718954708860627734265678935, −1.14421975660538226699279563813, 1.00155508465350173514790596964, 2.689376359283093913047046603432, 3.83194035399379278235008998751, 5.14187466222785338972078848942, 5.70287170978360500690167500181, 6.72639768831339304863381169823, 7.46234351256155464312972816199, 8.085074204756142114434396350692, 9.514621496444293672156534801311, 10.78843550393893134564387722375, 11.50569888184519395211683753216, 12.29348053494467053079049480761, 13.39393574703778352931724607195, 14.222832438197819875827138808339, 14.74979669250179236804543688097, 15.83016661603850287899308614573, 16.829259479887981971877776819477, 17.41429977885367796232445972312, 18.30768671629663207441041654304, 18.62361143753462201982555455110, 20.07718243696619690121858489644, 21.35073464197238820327622884314, 22.022778514790077620550982552225, 22.82556826752806753778518941729, 23.17656167122428606352157988701

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