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Properties

Label 1-569-569.318-r0-0-0
Degree $1$
Conductor $569$
Sign $0.341 + 0.939i$
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  + (0.633 + 0.773i)2-s + (0.952 + 0.304i)3-s + (−0.197 + 0.980i)4-s + (−0.839 − 0.544i)5-s + (0.367 + 0.930i)6-s + (0.937 − 0.346i)7-s + (−0.883 + 0.467i)8-s + (0.814 + 0.580i)9-s + (−0.110 − 0.993i)10-s + (0.787 − 0.616i)11-s + (−0.487 + 0.873i)12-s + (0.699 − 0.714i)13-s + (0.862 + 0.506i)14-s + (−0.633 − 0.773i)15-s + (−0.921 − 0.387i)16-s + (−0.991 + 0.132i)17-s + ⋯
L(s)  = 1  + (0.633 + 0.773i)2-s + (0.952 + 0.304i)3-s + (−0.197 + 0.980i)4-s + (−0.839 − 0.544i)5-s + (0.367 + 0.930i)6-s + (0.937 − 0.346i)7-s + (−0.883 + 0.467i)8-s + (0.814 + 0.580i)9-s + (−0.110 − 0.993i)10-s + (0.787 − 0.616i)11-s + (−0.487 + 0.873i)12-s + (0.699 − 0.714i)13-s + (0.862 + 0.506i)14-s + (−0.633 − 0.773i)15-s + (−0.921 − 0.387i)16-s + (−0.991 + 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.341 + 0.939i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.341 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.168335731 + 1.518563425i\)
\(L(\frac12)\) \(\approx\) \(2.168335731 + 1.518563425i\)
\(L(1)\) \(\approx\) \(1.715317699 + 0.8408371421i\)
\(L(1)\) \(\approx\) \(1.715317699 + 0.8408371421i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.633 + 0.773i)T \)
3 \( 1 + (0.952 + 0.304i)T \)
5 \( 1 + (-0.839 - 0.544i)T \)
7 \( 1 + (0.937 - 0.346i)T \)
11 \( 1 + (0.787 - 0.616i)T \)
13 \( 1 + (0.699 - 0.714i)T \)
17 \( 1 + (-0.991 + 0.132i)T \)
19 \( 1 + (0.197 + 0.980i)T \)
23 \( 1 + (0.999 + 0.0442i)T \)
29 \( 1 + (0.991 + 0.132i)T \)
31 \( 1 + (-0.562 - 0.826i)T \)
37 \( 1 + (-0.937 + 0.346i)T \)
41 \( 1 + (-0.999 - 0.0442i)T \)
43 \( 1 + (0.633 - 0.773i)T \)
47 \( 1 + (-0.0663 + 0.997i)T \)
53 \( 1 + (0.367 + 0.930i)T \)
59 \( 1 + (-0.0663 + 0.997i)T \)
61 \( 1 + (0.154 - 0.988i)T \)
67 \( 1 + (-0.991 - 0.132i)T \)
71 \( 1 + (-0.883 - 0.467i)T \)
73 \( 1 + (0.197 + 0.980i)T \)
79 \( 1 + (-0.730 - 0.683i)T \)
83 \( 1 + (-0.759 + 0.650i)T \)
89 \( 1 + (-0.562 + 0.826i)T \)
97 \( 1 + (-0.408 - 0.912i)T \)
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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

−23.1358872509353360275244613982, −22.12522343589847897297132465233, −21.34623597399084100855082424137, −20.55898875357442901435842272071, −19.73497552567807038868980015144, −19.290119296365334338977079436629, −18.31029392576877942964793034929, −17.73807464706522074388495194702, −15.81893306701499394630403308276, −15.19179094534332391299215389844, −14.50935742788809332014814024778, −13.87290312373406346006582664904, −12.89937874447129266796256291262, −11.84667889451207013945129969955, −11.38623446984731824771162857117, −10.38564740584792994637199881335, −8.99298441509241417231654063372, −8.6674770548267108122430479254, −7.12665959330374384013755778818, −6.62078910719257966368623885453, −4.83717828739133918297687882118, −4.17999233592722842686876128862, −3.21371697098579309123240573685, −2.23428170853566245338569841413, −1.30541091049805628858250734581, 1.37968829221717326142852018251, 3.063059553770726130711729768508, 3.91756784188649010319147281169, 4.508274365578211130610082869876, 5.566134978726552758066265483274, 6.953178866464136755486551482986, 7.82655037147951387523366561286, 8.52562881694533316209479403939, 8.97602100967542556041038496344, 10.6686777873606450592034472209, 11.5667854916274414974955791335, 12.59287564002219305291000533585, 13.52373460288524728613192068577, 14.14935915843825699875912291583, 15.09077553987538573271476290608, 15.5679859625990356016059106572, 16.50153746800292633994000505341, 17.241254498485493962681423126869, 18.39851229747768609967613668186, 19.42364850904557980039377545429, 20.51164275673574254282597891000, 20.70891513076339361745098319587, 21.78727301884680329425029020595, 22.677297849843746678691848942463, 23.61406520428374287440828894788

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