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Properties

Label 1-571-571.285-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.595 + 0.803i$
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 + (−0.962 − 0.272i)3-s + (0.350 − 0.936i)4-s + (−0.998 − 0.0550i)5-s + (0.945 − 0.324i)6-s + (−0.677 − 0.735i)7-s + (0.245 + 0.969i)8-s + (0.851 + 0.523i)9-s + (0.851 − 0.523i)10-s + (0.0275 − 0.999i)11-s + (−0.592 + 0.805i)12-s + (−0.926 + 0.376i)13-s + (0.975 + 0.218i)14-s + (0.945 + 0.324i)15-s + (−0.754 − 0.656i)16-s + (−0.754 − 0.656i)17-s + ⋯
L(s)  = 1  + (−0.821 + 0.569i)2-s + (−0.962 − 0.272i)3-s + (0.350 − 0.936i)4-s + (−0.998 − 0.0550i)5-s + (0.945 − 0.324i)6-s + (−0.677 − 0.735i)7-s + (0.245 + 0.969i)8-s + (0.851 + 0.523i)9-s + (0.851 − 0.523i)10-s + (0.0275 − 0.999i)11-s + (−0.592 + 0.805i)12-s + (−0.926 + 0.376i)13-s + (0.975 + 0.218i)14-s + (0.945 + 0.324i)15-s + (−0.754 − 0.656i)16-s + (−0.754 − 0.656i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.595 + 0.803i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.595 + 0.803i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007789743865 + 0.01547589984i\)
\(L(\frac12)\) \(\approx\) \(0.007789743865 + 0.01547589984i\)
\(L(1)\) \(\approx\) \(0.3261084512 - 0.04842535781i\)
\(L(1)\) \(\approx\) \(0.3261084512 - 0.04842535781i\)

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 \( 1 + (-0.962 - 0.272i)T \)
5 \( 1 + (-0.998 - 0.0550i)T \)
7 \( 1 + (-0.677 - 0.735i)T \)
11 \( 1 + (0.0275 - 0.999i)T \)
13 \( 1 + (-0.926 + 0.376i)T \)
17 \( 1 + (-0.754 - 0.656i)T \)
19 \( 1 + (-0.962 - 0.272i)T \)
23 \( 1 + (0.245 - 0.969i)T \)
29 \( 1 + (0.137 - 0.990i)T \)
31 \( 1 + (0.945 - 0.324i)T \)
37 \( 1 + (-0.926 - 0.376i)T \)
41 \( 1 + (-0.998 - 0.0550i)T \)
43 \( 1 + (0.851 - 0.523i)T \)
47 \( 1 + (-0.191 + 0.981i)T \)
53 \( 1 + (-0.821 - 0.569i)T \)
59 \( 1 + (-0.677 - 0.735i)T \)
61 \( 1 + (0.904 + 0.426i)T \)
67 \( 1 + (0.904 - 0.426i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.137 + 0.990i)T \)
79 \( 1 + (0.635 - 0.771i)T \)
83 \( 1 + (-0.191 + 0.981i)T \)
89 \( 1 + (-0.754 - 0.656i)T \)
97 \( 1 + (0.350 - 0.936i)T \)
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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

−23.61383468909867402872295787918, −22.7386481208085812620518328441, −22.111672700845470367397459003477, −21.39701464334054997160332621186, −20.26683941444594973738784484242, −19.4723681698896065068990383096, −18.90682928379980693921235529956, −17.854394515305183448767679951852, −17.25752473658864059659218633521, −16.382568962449871965941462302706, −15.3907101373272753385773929613, −15.18516016287550605134228389176, −12.95238821383590278150071842618, −12.34165684989407894273948656177, −11.90875358379424455631768686442, −10.84055837350142907188701813891, −10.15713892315594831439917373230, −9.274501478854970869486948373645, −8.2752994636863300828341673333, −7.13295652178209878678671219836, −6.53715801908738824427583089053, −5.02080177786812899861483544913, −4.07689834906685499914394562474, −3.02228886261960112889151619071, −1.70868380501958332484076819208, 0.02040685517276748546862973166, 0.755327035363002182636006944979, 2.49220477184246025368477722565, 4.1809462787033747787542106248, 5.00129499270749132376279209137, 6.404885675542531664823149596462, 6.80309089146529520933279487753, 7.71081208557463738970617937007, 8.64931092349029630250894190707, 9.803123739302571478529343510725, 10.74057764112601811746873674537, 11.30942693994133491725582904799, 12.25010044063479310712909601356, 13.34640535646273269521654601936, 14.37621275702316137018289630401, 15.68938584828654903483312708271, 16.00598993822118932065199299115, 17.0477494308533071262121644389, 17.26380962063893071232718211131, 18.76963069704773406733824770969, 19.07554076824686794969601620772, 19.787181586428894117859979728039, 20.89912735690071611641126137074, 22.34646032013619754524713658671, 22.82847361724938110873705900422

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