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Properties

Label 1-569-569.8-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.867 - 0.496i$
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  + (0.110 − 0.993i)2-s + (−0.346 − 0.937i)3-s + (−0.975 − 0.219i)4-s + (−0.448 + 0.894i)5-s + (−0.970 + 0.240i)6-s + (0.999 − 0.0442i)7-s + (−0.325 + 0.945i)8-s + (−0.759 + 0.650i)9-s + (0.839 + 0.544i)10-s + (0.304 − 0.952i)11-s + (0.132 + 0.991i)12-s + (−0.633 + 0.773i)13-s + (0.0663 − 0.997i)14-s + (0.993 + 0.110i)15-s + (0.903 + 0.428i)16-s + (0.367 − 0.930i)17-s + ⋯
L(s)  = 1  + (0.110 − 0.993i)2-s + (−0.346 − 0.937i)3-s + (−0.975 − 0.219i)4-s + (−0.448 + 0.894i)5-s + (−0.970 + 0.240i)6-s + (0.999 − 0.0442i)7-s + (−0.325 + 0.945i)8-s + (−0.759 + 0.650i)9-s + (0.839 + 0.544i)10-s + (0.304 − 0.952i)11-s + (0.132 + 0.991i)12-s + (−0.633 + 0.773i)13-s + (0.0663 − 0.997i)14-s + (0.993 + 0.110i)15-s + (0.903 + 0.428i)16-s + (0.367 − 0.930i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.867 - 0.496i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.867 - 0.496i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2588869234 - 0.9730757251i\)
\(L(\frac12)\) \(\approx\) \(0.2588869234 - 0.9730757251i\)
\(L(1)\) \(\approx\) \(0.6474580872 - 0.6142949900i\)
\(L(1)\) \(\approx\) \(0.6474580872 - 0.6142949900i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.110 - 0.993i)T \)
3 \( 1 + (-0.346 - 0.937i)T \)
5 \( 1 + (-0.448 + 0.894i)T \)
7 \( 1 + (0.999 - 0.0442i)T \)
11 \( 1 + (0.304 - 0.952i)T \)
13 \( 1 + (-0.633 + 0.773i)T \)
17 \( 1 + (0.367 - 0.930i)T \)
19 \( 1 + (0.219 + 0.975i)T \)
23 \( 1 + (0.387 - 0.921i)T \)
29 \( 1 + (0.930 - 0.367i)T \)
31 \( 1 + (-0.616 - 0.787i)T \)
37 \( 1 + (-0.0442 - 0.999i)T \)
41 \( 1 + (-0.921 - 0.387i)T \)
43 \( 1 + (-0.110 - 0.993i)T \)
47 \( 1 + (0.826 + 0.562i)T \)
53 \( 1 + (0.970 - 0.240i)T \)
59 \( 1 + (-0.826 - 0.562i)T \)
61 \( 1 + (-0.984 + 0.176i)T \)
67 \( 1 + (0.367 + 0.930i)T \)
71 \( 1 + (-0.325 - 0.945i)T \)
73 \( 1 + (-0.219 - 0.975i)T \)
79 \( 1 + (0.883 + 0.467i)T \)
83 \( 1 + (0.0883 + 0.996i)T \)
89 \( 1 + (0.616 - 0.787i)T \)
97 \( 1 + (0.801 - 0.598i)T \)
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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

−23.48259908373987110174383511955, −23.11858133248837386116963742052, −21.82285845072152873078756405260, −21.488894593379642761910587185386, −20.28077976630997681237905362951, −19.711440151882520280416166292930, −18.094303774604888724592081772, −17.30110889571656407086541050436, −17.04922953412708514476103803394, −15.92727021226939278086781149856, −15.09103269088195765272551450224, −14.89374802290191268708370986423, −13.59861016784852735068458580809, −12.444241667145120874279131472560, −11.84532184914871249579058324201, −10.558135276443541214916219698575, −9.58504489669314684824017191346, −8.75851655898777596744409092136, −7.99470824781328130228952612341, −7.03102076952679484989210726736, −5.6195140088842135833949193960, −4.90631958385014215835834640290, −4.482816644593638409811493283488, −3.31850076975053824463939228417, −1.188000798947228310694085294156, 0.62859925074710932045256148137, 1.91391433693673944389602146661, 2.74000434064437969111563736571, 3.91066213460434974664383435450, 5.05481310281079261355721840389, 6.067676603195873280075453045928, 7.25379440249557814343916729874, 8.048053240753646745190441180466, 9.001521713002005598955838443332, 10.41433693672910246455688768903, 11.08197867586466213660342645626, 11.873764385864426305071610845726, 12.19280930601596978582011789079, 13.75705740996751252624511547160, 14.09776808644518469909964147533, 14.82086142918592043233963991525, 16.49208815650162800791001002383, 17.32022265963385388870233596618, 18.39351333389345710265813535840, 18.66065809814204213535714275706, 19.38463131621995675242945017980, 20.33724369808427073769609842307, 21.29317541606288517489466127041, 22.164218218103541851938181997774, 22.80736751204119174423263711470

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