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Properties

Label 1-569-569.31-r1-0-0
Degree $1$
Conductor $569$
Sign $0.00303 - 0.999i$
Analytic cond. $61.1475$
Root an. cond. $61.1475$
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  + (0.219 + 0.975i)2-s + (−0.0773 − 0.997i)3-s + (−0.903 + 0.428i)4-s + (−0.598 + 0.801i)5-s + (0.955 − 0.294i)6-s + (0.0883 − 0.996i)7-s + (−0.616 − 0.787i)8-s + (−0.988 + 0.154i)9-s + (−0.912 − 0.408i)10-s + (−0.165 + 0.986i)11-s + (0.496 + 0.867i)12-s + (0.980 + 0.197i)13-s + (0.991 − 0.132i)14-s + (0.845 + 0.534i)15-s + (0.633 − 0.773i)16-s + (0.683 + 0.730i)17-s + ⋯
L(s)  = 1  + (0.219 + 0.975i)2-s + (−0.0773 − 0.997i)3-s + (−0.903 + 0.428i)4-s + (−0.598 + 0.801i)5-s + (0.955 − 0.294i)6-s + (0.0883 − 0.996i)7-s + (−0.616 − 0.787i)8-s + (−0.988 + 0.154i)9-s + (−0.912 − 0.408i)10-s + (−0.165 + 0.986i)11-s + (0.496 + 0.867i)12-s + (0.980 + 0.197i)13-s + (0.991 − 0.132i)14-s + (0.845 + 0.534i)15-s + (0.633 − 0.773i)16-s + (0.683 + 0.730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00303 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00303 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.00303 - 0.999i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ 0.00303 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3439867227 - 0.3450321797i\)
\(L(\frac12)\) \(\approx\) \(0.3439867227 - 0.3450321797i\)
\(L(1)\) \(\approx\) \(0.7860991307 + 0.2073760753i\)
\(L(1)\) \(\approx\) \(0.7860991307 + 0.2073760753i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.219 + 0.975i)T \)
3 \( 1 + (-0.0773 - 0.997i)T \)
5 \( 1 + (-0.598 + 0.801i)T \)
7 \( 1 + (0.0883 - 0.996i)T \)
11 \( 1 + (-0.165 + 0.986i)T \)
13 \( 1 + (0.980 + 0.197i)T \)
17 \( 1 + (0.683 + 0.730i)T \)
19 \( 1 + (-0.336 - 0.941i)T \)
23 \( 1 + (0.0110 + 0.999i)T \)
29 \( 1 + (-0.999 + 0.0331i)T \)
31 \( 1 + (0.856 + 0.515i)T \)
37 \( 1 + (-0.766 + 0.641i)T \)
41 \( 1 + (-0.699 + 0.714i)T \)
43 \( 1 + (0.975 + 0.219i)T \)
47 \( 1 + (-0.397 - 0.917i)T \)
53 \( 1 + (-0.294 - 0.955i)T \)
59 \( 1 + (0.917 - 0.397i)T \)
61 \( 1 + (0.346 - 0.937i)T \)
67 \( 1 + (0.683 - 0.730i)T \)
71 \( 1 + (0.616 - 0.787i)T \)
73 \( 1 + (-0.941 + 0.336i)T \)
79 \( 1 + (-0.826 - 0.562i)T \)
83 \( 1 + (-0.820 + 0.571i)T \)
89 \( 1 + (-0.515 - 0.856i)T \)
97 \( 1 + (0.878 + 0.477i)T \)
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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

−22.86120195364477386417499617578, −22.435208568297852372023800878544, −21.23998624809103232453833046255, −20.90342463817903359947292342806, −20.3313228851498714387915322596, −18.99156484370291040156052691425, −18.72938177756974591979903525897, −17.38269273911632386778736317699, −16.33355314409149763316476918477, −15.76479344946321643773561547451, −14.771540122707438084407621408437, −13.91699283212064963964820426321, −12.7972371487596482861298053805, −11.97095947885401798402707539672, −11.32895771918403756322092475772, −10.50913548079327856356913065506, −9.435999158220646974221211453213, −8.65285331092053756680226775697, −8.21596433574168587940454949565, −5.79074616836614536179881201702, −5.4837199308668796120434533779, −4.2942395678279655601469461667, −3.56113663721406351178256237622, −2.601287896491456912804534429889, −1.02292219406375500250976567083, 0.13608941141365783957775958572, 1.52008944998153625643000124155, 3.19411570733517036546817137144, 4.02421890332926999143086156192, 5.24396405720831061479813867281, 6.5373781652135510273689207405, 6.92182671292799760012352316406, 7.76904375586950185470284514354, 8.383525426637986734904484911739, 9.81484930470843486097654552902, 10.93249763912286053920367076829, 11.85088269272198358245435050543, 12.934757886782661070266218439699, 13.5479617313538045791891808733, 14.39778871268617932036906028122, 15.13397578585491819965061364221, 16.04521300425323142076506335731, 17.2154388792840465942184679301, 17.618988228850560713296618286594, 18.558467371398397386104991613158, 19.25376167178516027129221967939, 20.16364993063104055859391873609, 21.3474768292724853171276321609, 22.5790244880857450634950732447, 23.13004054310919029131906345426

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