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Properties

Label 1-569-569.40-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.879 + 0.474i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
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  + (−0.964 − 0.262i)2-s + (0.219 + 0.975i)3-s + (0.862 + 0.506i)4-s + (−0.883 − 0.467i)5-s + (0.0442 − 0.999i)6-s + (−0.408 − 0.912i)7-s + (−0.699 − 0.714i)8-s + (−0.903 + 0.428i)9-s + (0.730 + 0.683i)10-s + (0.980 − 0.197i)11-s + (−0.304 + 0.952i)12-s + (0.525 + 0.850i)13-s + (0.154 + 0.988i)14-s + (0.262 − 0.964i)15-s + (0.487 + 0.873i)16-s + (−0.937 + 0.346i)17-s + ⋯
L(s)  = 1  + (−0.964 − 0.262i)2-s + (0.219 + 0.975i)3-s + (0.862 + 0.506i)4-s + (−0.883 − 0.467i)5-s + (0.0442 − 0.999i)6-s + (−0.408 − 0.912i)7-s + (−0.699 − 0.714i)8-s + (−0.903 + 0.428i)9-s + (0.730 + 0.683i)10-s + (0.980 − 0.197i)11-s + (−0.304 + 0.952i)12-s + (0.525 + 0.850i)13-s + (0.154 + 0.988i)14-s + (0.262 − 0.964i)15-s + (0.487 + 0.873i)16-s + (−0.937 + 0.346i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.879 + 0.474i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.879 + 0.474i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05979954363 + 0.2366845893i\)
\(L(\frac12)\) \(\approx\) \(0.05979954363 + 0.2366845893i\)
\(L(1)\) \(\approx\) \(0.5054358255 + 0.08622483460i\)
\(L(1)\) \(\approx\) \(0.5054358255 + 0.08622483460i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.964 - 0.262i)T \)
3 \( 1 + (0.219 + 0.975i)T \)
5 \( 1 + (-0.883 - 0.467i)T \)
7 \( 1 + (-0.408 - 0.912i)T \)
11 \( 1 + (0.980 - 0.197i)T \)
13 \( 1 + (0.525 + 0.850i)T \)
17 \( 1 + (-0.937 + 0.346i)T \)
19 \( 1 + (-0.506 - 0.862i)T \)
23 \( 1 + (-0.801 - 0.598i)T \)
29 \( 1 + (-0.346 + 0.937i)T \)
31 \( 1 + (0.999 - 0.0221i)T \)
37 \( 1 + (-0.912 + 0.408i)T \)
41 \( 1 + (-0.598 + 0.801i)T \)
43 \( 1 + (0.964 - 0.262i)T \)
47 \( 1 + (-0.176 + 0.984i)T \)
53 \( 1 + (-0.0442 + 0.999i)T \)
59 \( 1 + (0.176 - 0.984i)T \)
61 \( 1 + (0.110 + 0.993i)T \)
67 \( 1 + (-0.937 - 0.346i)T \)
71 \( 1 + (-0.699 + 0.714i)T \)
73 \( 1 + (0.506 + 0.862i)T \)
79 \( 1 + (-0.996 + 0.0883i)T \)
83 \( 1 + (-0.745 - 0.666i)T \)
89 \( 1 + (-0.999 - 0.0221i)T \)
97 \( 1 + (-0.826 + 0.562i)T \)
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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

−22.90646875572482957447284647154, −22.49201733120091847522736501912, −20.88739871306898895517024246861, −19.934693284843432753915089438340, −19.37236917239027297300312810382, −18.80598771825736262407522797088, −17.996116141477032650858684538126, −17.36727143996396390215233093204, −16.122437834273212937047810434833, −15.357107205846683954291607542425, −14.75292776366912049992884035813, −13.63219965691902136479612518568, −12.25528259594568724080175440407, −11.875405479316094542055213486132, −10.95562273091656891734463060253, −9.75754444003813126548559406750, −8.67019613359513963419146753870, −8.20746616439392231902503542136, −7.196834403466223304594279595436, −6.42992207987042838598253012672, −5.72688166539661661509015866432, −3.743361117820523691178080386482, −2.67006836047378692880744926504, −1.70538679801036188738890519255, −0.17425461062089193756808999099, 1.34790142606013120757621404022, 2.929952731387840813778725954430, 4.01575343636281132532238706212, 4.37388178312070447854789943481, 6.31419953171493992103427558391, 7.09733115266237947757409102228, 8.404793294337257870130514064278, 8.83946189967446646539342492845, 9.69820966952960034656886312157, 10.7852221116488571162697623804, 11.251915089855303527276661314382, 12.19176907027304887602913784338, 13.40895000394450507266417614866, 14.51550859350630841092587055939, 15.61830031307557488446333433628, 16.12005158662657251432831983880, 16.87652948772347648775725410077, 17.43212977072234753769629643248, 18.95354838340864244884854454667, 19.62008064271220644540607339239, 20.11514558692371369276021089938, 20.79845815272721588671808819305, 21.81589879065640412717201962086, 22.56514499745737370791088806922, 23.76890285598278415958478575327

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