Properties

Label 1-571-571.237-r0-0-0
Degree $1$
Conductor $571$
Sign $0.624 - 0.781i$
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  + (−0.821 − 0.569i)2-s + (0.938 + 0.345i)3-s + (0.350 + 0.936i)4-s + (−0.360 − 0.932i)5-s + (−0.574 − 0.818i)6-s + (0.980 − 0.197i)7-s + (0.245 − 0.969i)8-s + (0.761 + 0.648i)9-s + (−0.234 + 0.972i)10-s + (0.959 + 0.282i)11-s + (0.00551 + 0.999i)12-s + (0.0715 − 0.997i)13-s + (−0.917 − 0.396i)14-s + (−0.0165 − 0.999i)15-s + (−0.754 + 0.656i)16-s + (0.391 + 0.920i)17-s + ⋯
L(s)  = 1  + (−0.821 − 0.569i)2-s + (0.938 + 0.345i)3-s + (0.350 + 0.936i)4-s + (−0.360 − 0.932i)5-s + (−0.574 − 0.818i)6-s + (0.980 − 0.197i)7-s + (0.245 − 0.969i)8-s + (0.761 + 0.648i)9-s + (−0.234 + 0.972i)10-s + (0.959 + 0.282i)11-s + (0.00551 + 0.999i)12-s + (0.0715 − 0.997i)13-s + (−0.917 − 0.396i)14-s + (−0.0165 − 0.999i)15-s + (−0.754 + 0.656i)16-s + (0.391 + 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.624 - 0.781i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.624 - 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.341858724 - 0.6455764331i\)
\(L(\frac12)\) \(\approx\) \(1.341858724 - 0.6455764331i\)
\(L(1)\) \(\approx\) \(1.082018828 - 0.3169186248i\)
\(L(1)\) \(\approx\) \(1.082018828 - 0.3169186248i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.821 - 0.569i)T \)
3 \( 1 + (0.938 + 0.345i)T \)
5 \( 1 + (-0.360 - 0.932i)T \)
7 \( 1 + (0.980 - 0.197i)T \)
11 \( 1 + (0.959 + 0.282i)T \)
13 \( 1 + (0.0715 - 0.997i)T \)
17 \( 1 + (0.391 + 0.920i)T \)
19 \( 1 + (-0.556 - 0.831i)T \)
23 \( 1 + (0.371 - 0.928i)T \)
29 \( 1 + (0.137 + 0.990i)T \)
31 \( 1 + (0.945 + 0.324i)T \)
37 \( 1 + (-0.644 - 0.764i)T \)
41 \( 1 + (-0.998 + 0.0550i)T \)
43 \( 1 + (-0.997 + 0.0770i)T \)
47 \( 1 + (-0.191 - 0.981i)T \)
53 \( 1 + (0.287 + 0.957i)T \)
59 \( 1 + (-0.677 + 0.735i)T \)
61 \( 1 + (-0.982 - 0.186i)T \)
67 \( 1 + (0.685 - 0.728i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.984 - 0.175i)T \)
79 \( 1 + (-0.0605 - 0.998i)T \)
83 \( 1 + (-0.421 + 0.906i)T \)
89 \( 1 + (0.996 - 0.0880i)T \)
97 \( 1 + (-0.782 + 0.622i)T \)
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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

−23.65779485262927402952841846469, −22.80904529373584625724331784264, −21.45653840168002606608297640678, −20.74216612286215406579481110163, −19.70581816090939857270449591693, −18.89455665585335842821024950539, −18.678632412641888016604179289490, −17.631878564577406755932272264051, −16.78162146806705443672719172117, −15.538542496345045765182320455803, −14.99094641427659421841633073808, −14.09887407093110378256985203873, −13.88277959137868279176981000369, −11.792297636642819917686734179354, −11.51832927408731359652108950674, −10.16800142029590801432949670838, −9.36991181077236080895817363225, −8.41807773829788698842647649247, −7.790307660923715407711782074732, −6.903500241398269405783340069413, −6.22337530252045100521464302399, −4.6605457526426322603119564305, −3.46260328165007969165874533432, −2.18731149957243937384258079566, −1.3584321550203348452016518276, 1.09650818978546938978090615971, 1.91437179623398484116925547069, 3.248633291786502233673982395916, 4.16877705844567194440474890677, 4.96268148319387964362250924700, 6.86485766077984740043091795459, 7.90786257556830561705716073895, 8.54129176512184175396352956559, 8.99221748151418243231952447837, 10.210603427319660565525627381677, 10.8533824073042779343129540775, 12.05493758374115460105508935288, 12.72222873593632685378957184537, 13.690101452376682147750767571052, 14.90424491305627279845588909168, 15.4834391604252727505354086185, 16.70816878898733271513068519694, 17.18568015518487932883114896339, 18.186734517159642417434347629819, 19.26718891401761403745616688124, 20.03107474168854603516578212194, 20.27852926118508840091849755271, 21.27276285367307264974141163097, 21.74963464413863812506348784232, 23.12601773560857194493810691394

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