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Properties

Label 1-571-571.509-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.121 - 0.992i$
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  + (−0.592 − 0.805i)2-s + (−0.754 − 0.656i)3-s + (−0.298 + 0.954i)4-s + (0.716 + 0.697i)5-s + (−0.0825 + 0.996i)6-s + (0.546 − 0.837i)7-s + (0.945 − 0.324i)8-s + (0.137 + 0.990i)9-s + (0.137 − 0.990i)10-s + (−0.926 − 0.376i)11-s + (0.851 − 0.523i)12-s + (0.635 + 0.771i)13-s + (−0.998 + 0.0550i)14-s + (−0.0825 − 0.996i)15-s + (−0.821 − 0.569i)16-s + (−0.821 − 0.569i)17-s + ⋯
L(s)  = 1  + (−0.592 − 0.805i)2-s + (−0.754 − 0.656i)3-s + (−0.298 + 0.954i)4-s + (0.716 + 0.697i)5-s + (−0.0825 + 0.996i)6-s + (0.546 − 0.837i)7-s + (0.945 − 0.324i)8-s + (0.137 + 0.990i)9-s + (0.137 − 0.990i)10-s + (−0.926 − 0.376i)11-s + (0.851 − 0.523i)12-s + (0.635 + 0.771i)13-s + (−0.998 + 0.0550i)14-s + (−0.0825 − 0.996i)15-s + (−0.821 − 0.569i)16-s + (−0.821 − 0.569i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.121 - 0.992i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.121 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5554966760 - 0.6277480633i\)
\(L(\frac12)\) \(\approx\) \(0.5554966760 - 0.6277480633i\)
\(L(1)\) \(\approx\) \(0.6320405464 - 0.3578493024i\)
\(L(1)\) \(\approx\) \(0.6320405464 - 0.3578493024i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.592 - 0.805i)T \)
3 \( 1 + (-0.754 - 0.656i)T \)
5 \( 1 + (0.716 + 0.697i)T \)
7 \( 1 + (0.546 - 0.837i)T \)
11 \( 1 + (-0.926 - 0.376i)T \)
13 \( 1 + (0.635 + 0.771i)T \)
17 \( 1 + (-0.821 - 0.569i)T \)
19 \( 1 + (-0.754 - 0.656i)T \)
23 \( 1 + (0.945 + 0.324i)T \)
29 \( 1 + (0.350 - 0.936i)T \)
31 \( 1 + (-0.0825 + 0.996i)T \)
37 \( 1 + (0.635 - 0.771i)T \)
41 \( 1 + (0.716 + 0.697i)T \)
43 \( 1 + (0.137 - 0.990i)T \)
47 \( 1 + (0.904 - 0.426i)T \)
53 \( 1 + (-0.592 + 0.805i)T \)
59 \( 1 + (0.546 - 0.837i)T \)
61 \( 1 + (0.993 - 0.110i)T \)
67 \( 1 + (0.993 + 0.110i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.350 + 0.936i)T \)
79 \( 1 + (0.975 + 0.218i)T \)
83 \( 1 + (0.904 - 0.426i)T \)
89 \( 1 + (-0.821 - 0.569i)T \)
97 \( 1 + (-0.298 + 0.954i)T \)
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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

−23.6811712351785773429394625037, −22.73694680332277810595361029318, −21.87895758102460656061939791128, −20.901649035371951459764395781236, −20.44571672554697381081801566888, −18.942596998658790515185817265270, −17.963767686741531616321228038465, −17.696860240560765964761948708325, −16.78312757450544774789945503373, −15.984188134359870373873875623213, −15.2516336206774095687195492, −14.636043795030334718611408379505, −13.173380051949575411483844232669, −12.556640816191598116404874926438, −11.068022209719390355512674420478, −10.50834074756433238146700243417, −9.55313881929978091717910657291, −8.72461911747556036354357716341, −8.053204769442923334433488398173, −6.5088652768979694963002810131, −5.76013140777492820592580886658, −5.15133386379138243426609904332, −4.343525284333702296752140506143, −2.32320654648652634746272140245, −1.04163000167253732763296363598, 0.73294207636815550023594815416, 1.91507780812168931670830747577, 2.698734809699854198535232794149, 4.19539863778110667444102919404, 5.23003831704137676536220882818, 6.61199719495751092884552096599, 7.194471970252359029769698420788, 8.215255996579371494571179407179, 9.31193021337663565669051829565, 10.541775132500655424426221960177, 10.94992932066162500185796385586, 11.48582704612050595773595570205, 12.8761714950469996146728759947, 13.48965306551509098094731949342, 14.04944314484255124946682633206, 15.72945069504333856398003765022, 16.74342215267691473761408795832, 17.51211312776994888289246659340, 17.97042647096107202610995481280, 18.77235590360546162944154971863, 19.41763438451679288920680679492, 20.62238295892816808654801998510, 21.3968954831251294227460955654, 21.94601461904387143488359812996, 23.14989741410917777691584234776

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