↑

Properties

Label 1-569-569.257-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

Related objects

Downloads

Learn more

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: $\chi_{569} (257, \cdot )$
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 \( 1 + (0.878 - 0.477i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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