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Properties

Label 1-571-571.24-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.198 + 0.980i$
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 + (0.685 − 0.728i)3-s + (−0.998 + 0.0550i)4-s + (0.224 + 0.974i)5-s + (0.746 + 0.665i)6-s + (0.997 + 0.0660i)7-s + (−0.0825 − 0.996i)8-s + (−0.0605 − 0.998i)9-s + (−0.968 + 0.250i)10-s + (0.266 + 0.963i)11-s + (−0.644 + 0.764i)12-s + (−0.319 + 0.947i)13-s + (−0.0385 + 0.999i)14-s + (0.863 + 0.504i)15-s + (0.993 − 0.110i)16-s + (−0.739 + 0.673i)17-s + ⋯
L(s)  = 1  + (0.0275 + 0.999i)2-s + (0.685 − 0.728i)3-s + (−0.998 + 0.0550i)4-s + (0.224 + 0.974i)5-s + (0.746 + 0.665i)6-s + (0.997 + 0.0660i)7-s + (−0.0825 − 0.996i)8-s + (−0.0605 − 0.998i)9-s + (−0.968 + 0.250i)10-s + (0.266 + 0.963i)11-s + (−0.644 + 0.764i)12-s + (−0.319 + 0.947i)13-s + (−0.0385 + 0.999i)14-s + (0.863 + 0.504i)15-s + (0.993 − 0.110i)16-s + (−0.739 + 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.198 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.109221097 + 1.356919233i\)
\(L(\frac12)\) \(\approx\) \(1.109221097 + 1.356919233i\)
\(L(1)\) \(\approx\) \(1.157703519 + 0.6851517344i\)
\(L(1)\) \(\approx\) \(1.157703519 + 0.6851517344i\)

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 \( 1 + (0.685 - 0.728i)T \)
5 \( 1 + (0.224 + 0.974i)T \)
7 \( 1 + (0.997 + 0.0660i)T \)
11 \( 1 + (0.266 + 0.963i)T \)
13 \( 1 + (-0.319 + 0.947i)T \)
17 \( 1 + (-0.739 + 0.673i)T \)
19 \( 1 + (-0.480 - 0.876i)T \)
23 \( 1 + (0.922 + 0.386i)T \)
29 \( 1 + (0.975 + 0.218i)T \)
31 \( 1 + (-0.401 + 0.915i)T \)
37 \( 1 + (0.802 - 0.596i)T \)
41 \( 1 + (-0.754 - 0.656i)T \)
43 \( 1 + (0.930 + 0.366i)T \)
47 \( 1 + (-0.592 - 0.805i)T \)
53 \( 1 + (0.565 + 0.824i)T \)
59 \( 1 + (0.245 + 0.969i)T \)
61 \( 1 + (-0.234 - 0.972i)T \)
67 \( 1 + (-0.997 + 0.0770i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (-0.917 - 0.396i)T \)
79 \( 1 + (0.988 - 0.153i)T \)
83 \( 1 + (-0.949 + 0.314i)T \)
89 \( 1 + (0.202 - 0.979i)T \)
97 \( 1 + (0.840 + 0.542i)T \)
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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

−22.67131781098974135483057800703, −21.87462774741678447015668844542, −21.05240281599983845884785730356, −20.678002611816878011734138389137, −19.96087828424439887238991614879, −19.16520973098906296756383165574, −18.099780429874558319396556094408, −17.15436704197235219857179855630, −16.45913329292707270580688763218, −15.19811919186715302397945938361, −14.39626502205861399092548603246, −13.57716848705386314651458688725, −12.9117965906253190161507696661, −11.73305950429815242664663341990, −10.95877365203138450350816073321, −10.101258537284864301327939684006, −9.17950616686958505399147846369, −8.44022546042631668277916461240, −7.94033762889230945532101864531, −5.7352072532570609496972193367, −4.84617497567732333189276435332, −4.2708722039256966434400359339, −3.083908568341710506284251939448, −2.10229592915797291024431161769, −0.89849453712278506705960458520, 1.55446150117196836074329431349, 2.53363730588542437619879689611, 3.94963874704737586264626906875, 4.85752550561710493039240945239, 6.26340813880601302083349058831, 7.01803119486865761350537584758, 7.44462641137656517480168095581, 8.66649900949726036409218089110, 9.22035026795213667869917660299, 10.462424729555259617554788694418, 11.63914937924221758227816398004, 12.69673689096230694616813681011, 13.64064198365458182540351113757, 14.36665478228767666164370985172, 14.90490098797731388915872141245, 15.47900257873118842951174179826, 17.10899776839566374387883912301, 17.701577221202840852288241273700, 18.20490237156601729471634041917, 19.18727721011259209639342419834, 19.83931881103282418675792839132, 21.32463769540542367430085440316, 21.76174073613697577211963543858, 23.02847917212650631372476856051, 23.62519652621111848800857144621

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