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Properties

Label 1-569-569.56-r0-0-0
Degree $1$
Conductor $569$
Sign $0.134 + 0.990i$
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  + (0.814 + 0.580i)2-s + (0.862 + 0.506i)3-s + (0.325 + 0.945i)4-s + (0.996 + 0.0883i)5-s + (0.408 + 0.912i)6-s + (0.0663 − 0.997i)7-s + (−0.283 + 0.958i)8-s + (0.487 + 0.873i)9-s + (0.759 + 0.650i)10-s + (−0.448 − 0.894i)11-s + (−0.197 + 0.980i)12-s + (0.240 + 0.970i)13-s + (0.633 − 0.773i)14-s + (0.814 + 0.580i)15-s + (−0.787 + 0.616i)16-s + (−0.975 − 0.219i)17-s + ⋯
L(s)  = 1  + (0.814 + 0.580i)2-s + (0.862 + 0.506i)3-s + (0.325 + 0.945i)4-s + (0.996 + 0.0883i)5-s + (0.408 + 0.912i)6-s + (0.0663 − 0.997i)7-s + (−0.283 + 0.958i)8-s + (0.487 + 0.873i)9-s + (0.759 + 0.650i)10-s + (−0.448 − 0.894i)11-s + (−0.197 + 0.980i)12-s + (0.240 + 0.970i)13-s + (0.633 − 0.773i)14-s + (0.814 + 0.580i)15-s + (−0.787 + 0.616i)16-s + (−0.975 − 0.219i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.134 + 0.990i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.134 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.554141634 + 2.231236670i\)
\(L(\frac12)\) \(\approx\) \(2.554141634 + 2.231236670i\)
\(L(1)\) \(\approx\) \(2.074380128 + 1.145567901i\)
\(L(1)\) \(\approx\) \(2.074380128 + 1.145567901i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.814 + 0.580i)T \)
3 \( 1 + (0.862 + 0.506i)T \)
5 \( 1 + (0.996 + 0.0883i)T \)
7 \( 1 + (0.0663 - 0.997i)T \)
11 \( 1 + (-0.448 - 0.894i)T \)
13 \( 1 + (0.240 + 0.970i)T \)
17 \( 1 + (-0.975 - 0.219i)T \)
19 \( 1 + (0.325 - 0.945i)T \)
23 \( 1 + (0.562 + 0.826i)T \)
29 \( 1 + (-0.975 + 0.219i)T \)
31 \( 1 + (-0.839 + 0.544i)T \)
37 \( 1 + (0.0663 - 0.997i)T \)
41 \( 1 + (0.562 + 0.826i)T \)
43 \( 1 + (0.814 - 0.580i)T \)
47 \( 1 + (-0.110 - 0.993i)T \)
53 \( 1 + (0.408 + 0.912i)T \)
59 \( 1 + (-0.110 - 0.993i)T \)
61 \( 1 + (0.964 + 0.262i)T \)
67 \( 1 + (-0.975 + 0.219i)T \)
71 \( 1 + (-0.283 - 0.958i)T \)
73 \( 1 + (0.325 - 0.945i)T \)
79 \( 1 + (-0.666 - 0.745i)T \)
83 \( 1 + (-0.991 - 0.132i)T \)
89 \( 1 + (-0.839 - 0.544i)T \)
97 \( 1 + (0.984 + 0.176i)T \)
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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

−22.74202160972918046699135912242, −22.318350122759491957386421247063, −21.143422907943679687845223172, −20.685883024412106951369984134096, −20.07937426581790323805719411873, −18.86841426782864775085320558233, −18.32829258061034585410915077877, −17.58824384256150659227658741536, −15.94669675104111125518039885919, −14.99007701528753935443572864323, −14.65216820609345431323552089930, −13.526602335494954643338043929477, −12.76347065910391637869536876855, −12.53410619183395693089011097221, −11.134897526175145998690418559655, −10.043215466164537968045836810130, −9.38120189128136225160428480766, −8.41709338964176489128542983730, −7.14036066317465564406454780273, −6.06659573026007224599658850091, −5.39364215126304426296850280907, −4.17873638754888537068795932618, −2.83127776716679724414752595756, −2.30100956122932652194305020483, −1.4206905802572394205857076386, 1.82260579435442242531561219564, 2.87893002847332695408291029863, 3.79425635308167042930010651850, 4.73250021136462220151056674692, 5.63362409014743940682929644297, 6.87509688616320170979993569656, 7.474570513640914500618357628403, 8.821789271938678141691054950823, 9.30027682884059495793336501627, 10.74886265308095158435677417541, 11.22800059362873176365948490485, 13.09841514430707815397412029971, 13.4384293771215466211947668197, 14.04501184476204754800758292252, 14.778035155982441525821132493411, 15.89190969559004957806509939528, 16.46757521122312061801708646279, 17.32646029340653607760614694635, 18.30227103573727886679322008706, 19.58634987322208138589899749569, 20.398611504724815922480343533627, 21.22069850302816167211064936221, 21.65807585823347866028447402655, 22.429133893342514459533561685369, 23.66320206755956954196557311198

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