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Properties

Label 1-571-571.124-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.166 + 0.986i$
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  + (0.0275 + 0.999i)2-s + (0.904 + 0.426i)3-s + (−0.998 + 0.0550i)4-s + (−0.754 − 0.656i)5-s + (−0.401 + 0.915i)6-s + (0.245 + 0.969i)7-s + (−0.0825 − 0.996i)8-s + (0.635 + 0.771i)9-s + (0.635 − 0.771i)10-s + (0.350 − 0.936i)11-s + (−0.926 − 0.376i)12-s + (−0.298 − 0.954i)13-s + (−0.962 + 0.272i)14-s + (−0.401 − 0.915i)15-s + (0.993 − 0.110i)16-s + (0.993 − 0.110i)17-s + ⋯
L(s)  = 1  + (0.0275 + 0.999i)2-s + (0.904 + 0.426i)3-s + (−0.998 + 0.0550i)4-s + (−0.754 − 0.656i)5-s + (−0.401 + 0.915i)6-s + (0.245 + 0.969i)7-s + (−0.0825 − 0.996i)8-s + (0.635 + 0.771i)9-s + (0.635 − 0.771i)10-s + (0.350 − 0.936i)11-s + (−0.926 − 0.376i)12-s + (−0.298 − 0.954i)13-s + (−0.962 + 0.272i)14-s + (−0.401 − 0.915i)15-s + (0.993 − 0.110i)16-s + (0.993 − 0.110i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.166 + 0.986i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.166 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.053245355 + 1.245497103i\)
\(L(\frac12)\) \(\approx\) \(1.053245355 + 1.245497103i\)
\(L(1)\) \(\approx\) \(1.044603081 + 0.7123792205i\)
\(L(1)\) \(\approx\) \(1.044603081 + 0.7123792205i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.0275 + 0.999i)T \)
3 \( 1 + (0.904 + 0.426i)T \)
5 \( 1 + (-0.754 - 0.656i)T \)
7 \( 1 + (0.245 + 0.969i)T \)
11 \( 1 + (0.350 - 0.936i)T \)
13 \( 1 + (-0.298 - 0.954i)T \)
17 \( 1 + (0.993 - 0.110i)T \)
19 \( 1 + (0.904 + 0.426i)T \)
23 \( 1 + (-0.0825 + 0.996i)T \)
29 \( 1 + (0.975 + 0.218i)T \)
31 \( 1 + (-0.401 + 0.915i)T \)
37 \( 1 + (-0.298 + 0.954i)T \)
41 \( 1 + (-0.754 - 0.656i)T \)
43 \( 1 + (0.635 - 0.771i)T \)
47 \( 1 + (-0.592 - 0.805i)T \)
53 \( 1 + (0.0275 - 0.999i)T \)
59 \( 1 + (0.245 + 0.969i)T \)
61 \( 1 + (0.851 - 0.523i)T \)
67 \( 1 + (0.851 + 0.523i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.975 - 0.218i)T \)
79 \( 1 + (0.451 + 0.892i)T \)
83 \( 1 + (-0.592 - 0.805i)T \)
89 \( 1 + (0.993 - 0.110i)T \)
97 \( 1 + (-0.998 + 0.0550i)T \)
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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

−23.064537607832138884292340435517, −22.14964531253499387355251042475, −21.05404855789703518619359361534, −20.3647951413191887040878747914, −19.715283395861649534033546800933, −19.12137733985504221647232932562, −18.31000926877830983888236321715, −17.53257153313290532203348750406, −16.33257696731268715273908969333, −14.93429412536967141005703072750, −14.34228812873850667224181843009, −13.827837500805416179984118132, −12.62461488136488045494937147650, −11.97937463298283934512872277420, −11.07684026904181494655766132442, −10.01613852751048617886507653593, −9.418647034145661236258266949437, −8.14028595597195556436571694114, −7.48813049632702754360822275930, −6.62505735259283907654149735938, −4.62393398889857816850058383733, −4.012674819737421244611740739120, −3.10838138837256723648573521834, −2.10219219486848247436620996749, −0.96382212902182069724312888122, 1.20218099243068383784634645676, 3.16226684048440196213003214820, 3.704079962848468735408593220357, 5.21388681242004661069098129505, 5.390877186960085864637335533112, 7.09433184901334520336442125396, 8.139212305613825284289635375208, 8.39357413421810110788334290129, 9.30563860516579027334782800736, 10.19500985781158020819371501287, 11.758825264055508585704249684133, 12.544367342362966535473946127120, 13.588762174579468392681427041389, 14.38433213790385574828867742071, 15.17606700851138966389591677320, 15.875216254977469146226290572936, 16.34222553695652322705921171414, 17.47000328066146178755228662112, 18.60274883088423615374857755817, 19.20096324605342284678136687741, 20.07451217671817988300300867540, 21.12979893844183689023328040723, 21.840703835693418566186368415294, 22.67009850038261795452004239450, 23.78342244110583994031413172572

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