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Properties

Label 1-571-571.85-r1-0-0
Degree $1$
Conductor $571$
Sign $0.118 + 0.992i$
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  + (−0.716 + 0.697i)2-s + (−0.975 + 0.218i)3-s + (0.0275 − 0.999i)4-s + (0.350 + 0.936i)5-s + (0.546 − 0.837i)6-s + (−0.789 + 0.614i)7-s + (0.677 + 0.735i)8-s + (0.904 − 0.426i)9-s + (−0.904 − 0.426i)10-s + (−0.821 − 0.569i)11-s + (0.191 + 0.981i)12-s + (−0.592 − 0.805i)13-s + (0.137 − 0.990i)14-s + (−0.546 − 0.837i)15-s + (−0.998 − 0.0550i)16-s + (0.998 + 0.0550i)17-s + ⋯
L(s)  = 1  + (−0.716 + 0.697i)2-s + (−0.975 + 0.218i)3-s + (0.0275 − 0.999i)4-s + (0.350 + 0.936i)5-s + (0.546 − 0.837i)6-s + (−0.789 + 0.614i)7-s + (0.677 + 0.735i)8-s + (0.904 − 0.426i)9-s + (−0.904 − 0.426i)10-s + (−0.821 − 0.569i)11-s + (0.191 + 0.981i)12-s + (−0.592 − 0.805i)13-s + (0.137 − 0.990i)14-s + (−0.546 − 0.837i)15-s + (−0.998 − 0.0550i)16-s + (0.998 + 0.0550i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.118 + 0.992i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.118 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4009224648 + 0.3560562565i\)
\(L(\frac12)\) \(\approx\) \(0.4009224648 + 0.3560562565i\)
\(L(1)\) \(\approx\) \(0.4258968452 + 0.2190004001i\)
\(L(1)\) \(\approx\) \(0.4258968452 + 0.2190004001i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.716 + 0.697i)T \)
3 \( 1 + (-0.975 + 0.218i)T \)
5 \( 1 + (0.350 + 0.936i)T \)
7 \( 1 + (-0.789 + 0.614i)T \)
11 \( 1 + (-0.821 - 0.569i)T \)
13 \( 1 + (-0.592 - 0.805i)T \)
17 \( 1 + (0.998 + 0.0550i)T \)
19 \( 1 + (-0.975 + 0.218i)T \)
23 \( 1 + (-0.677 + 0.735i)T \)
29 \( 1 + (0.993 - 0.110i)T \)
31 \( 1 + (0.546 - 0.837i)T \)
37 \( 1 + (-0.592 + 0.805i)T \)
41 \( 1 + (-0.350 - 0.936i)T \)
43 \( 1 + (0.904 + 0.426i)T \)
47 \( 1 + (-0.451 - 0.892i)T \)
53 \( 1 + (-0.716 - 0.697i)T \)
59 \( 1 + (0.789 - 0.614i)T \)
61 \( 1 + (-0.962 - 0.272i)T \)
67 \( 1 + (0.962 - 0.272i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.993 - 0.110i)T \)
79 \( 1 + (-0.851 + 0.523i)T \)
83 \( 1 + (0.451 + 0.892i)T \)
89 \( 1 + (0.998 + 0.0550i)T \)
97 \( 1 + (0.0275 - 0.999i)T \)
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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

−22.91726883770326608849559565458, −21.72733154902117562338794458204, −21.2379560739445782562671409921, −20.32306906642887649748883204693, −19.40649494131714354692596069103, −18.75046339800258072013575571958, −17.60407979990461967356461573272, −17.223746390268257854229804313007, −16.26787785733705694020722578322, −15.99494630062261761574491194424, −14.03196820199457400459412546175, −12.96972928959256861770535524064, −12.50073645653481497489016902735, −11.88733313548480072804104538590, −10.47903122626993729377934659922, −10.16251948468594510740777106858, −9.24086508380527959662821782868, −8.05717925815019493061201959740, −7.134755744925507548762522126894, −6.22289174046372007532778234958, −4.835711608251935220051394728402, −4.19671375641087921347384024189, −2.57198228350593114736094152058, −1.46085715798128875684493124790, −0.43468133853429560432105635309, 0.46778930187130025093564453597, 2.14173084284862624597896274304, 3.36161733706905138398871575186, 5.07447012751306335680512136420, 5.86470502669092725850838452019, 6.3382077998949960133714464617, 7.384885731116437662329642389929, 8.32383551377920989835086960641, 9.859093140909092550831303392793, 10.03580692058679404302230056093, 10.90100792966614609927965204479, 11.960553272143529262905948931982, 13.03111937040035749759900520603, 14.12929908329834173313155654619, 15.30209455942979882806687318478, 15.5907505412263184465460548430, 16.61213944368203295923602902442, 17.39384075298445751817209194407, 18.114061998701282735406733137582, 18.88508675214993611912140964856, 19.33839988274762187060138551624, 20.92790734331444846273891849601, 21.79872496596402687135837091397, 22.583955341903793377390065229272, 23.20322970113925950093130690881

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