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Properties

Label 1-569-569.141-r0-0-0
Degree $1$
Conductor $569$
Sign $0.453 - 0.891i$
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  + (0.903 − 0.428i)2-s + (0.154 + 0.988i)3-s + (0.633 − 0.773i)4-s + (−0.283 − 0.958i)5-s + (0.562 + 0.826i)6-s + (0.984 + 0.176i)7-s + (0.240 − 0.970i)8-s + (−0.952 + 0.304i)9-s + (−0.666 − 0.745i)10-s + (0.325 − 0.945i)11-s + (0.862 + 0.506i)12-s + (−0.921 − 0.387i)13-s + (0.964 − 0.262i)14-s + (0.903 − 0.428i)15-s + (−0.197 − 0.980i)16-s + (0.0663 − 0.997i)17-s + ⋯
L(s)  = 1  + (0.903 − 0.428i)2-s + (0.154 + 0.988i)3-s + (0.633 − 0.773i)4-s + (−0.283 − 0.958i)5-s + (0.562 + 0.826i)6-s + (0.984 + 0.176i)7-s + (0.240 − 0.970i)8-s + (−0.952 + 0.304i)9-s + (−0.666 − 0.745i)10-s + (0.325 − 0.945i)11-s + (0.862 + 0.506i)12-s + (−0.921 − 0.387i)13-s + (0.964 − 0.262i)14-s + (0.903 − 0.428i)15-s + (−0.197 − 0.980i)16-s + (0.0663 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.453 - 0.891i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.453 - 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.125499624 - 1.302519269i\)
\(L(\frac12)\) \(\approx\) \(2.125499624 - 1.302519269i\)
\(L(1)\) \(\approx\) \(1.773676722 - 0.5146428453i\)
\(L(1)\) \(\approx\) \(1.773676722 - 0.5146428453i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.903 - 0.428i)T \)
3 \( 1 + (0.154 + 0.988i)T \)
5 \( 1 + (-0.283 - 0.958i)T \)
7 \( 1 + (0.984 + 0.176i)T \)
11 \( 1 + (0.325 - 0.945i)T \)
13 \( 1 + (-0.921 - 0.387i)T \)
17 \( 1 + (0.0663 - 0.997i)T \)
19 \( 1 + (0.633 + 0.773i)T \)
23 \( 1 + (-0.0221 - 0.999i)T \)
29 \( 1 + (0.0663 + 0.997i)T \)
31 \( 1 + (-0.883 + 0.467i)T \)
37 \( 1 + (0.984 + 0.176i)T \)
41 \( 1 + (-0.0221 - 0.999i)T \)
43 \( 1 + (0.903 + 0.428i)T \)
47 \( 1 + (-0.730 - 0.683i)T \)
53 \( 1 + (0.562 + 0.826i)T \)
59 \( 1 + (-0.730 - 0.683i)T \)
61 \( 1 + (0.759 + 0.650i)T \)
67 \( 1 + (0.0663 + 0.997i)T \)
71 \( 1 + (0.240 + 0.970i)T \)
73 \( 1 + (0.633 + 0.773i)T \)
79 \( 1 + (-0.367 - 0.930i)T \)
83 \( 1 + (0.937 + 0.346i)T \)
89 \( 1 + (-0.883 - 0.467i)T \)
97 \( 1 + (-0.839 + 0.544i)T \)
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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

−23.58166364049735346487506224368, −22.71310458571851687071687751295, −21.97523192467243824122299466518, −21.08367629350399472082348691991, −19.90136533556749657697614009396, −19.517650173442999204359624613649, −18.14055154099462441934628556735, −17.57945902092571642857911523606, −16.877160815166180785633881645732, −15.29158786800407437954833839186, −14.8240801414144757476017162024, −14.19979606270493360159747732837, −13.362453552597777862607964492869, −12.35266887735857792898703985173, −11.60296587525555767093226570509, −11.044142588077808080397568519170, −9.50044706048605881632976073189, −7.960204102960062379348431682001, −7.546339914257380491530545746050, −6.83643690156846644589839006952, −5.890790451598677099671884216433, −4.726506503710701374000664832052, −3.71378400679073279615206914363, −2.48939869216517514578953763883, −1.78432361998636946039269945952, 0.97000044925117110740371886318, 2.42077664997130564435709471876, 3.48840334556118468353308120278, 4.44332067667299237537948787433, 5.164165652561199160161856622218, 5.69844989808841466832415802217, 7.42081628248708547450604595195, 8.50026976984044393610344848453, 9.34473909303085844811407339477, 10.35980571055168535997832473468, 11.30274038943861898251969977798, 11.89457228911224052293174938540, 12.79030799205832556636686215024, 14.16131798656354338297473944810, 14.368948262396333877295515625982, 15.443024311485810167772622088827, 16.30196583611996770977788412373, 16.77936058698756957549545780287, 18.17730482216581576007530190673, 19.36137890028990622717870738629, 20.37268405293390277161773802956, 20.48094398302026108732162458127, 21.51884329230614811667403906957, 22.04207191241934120812716855283, 22.954493932564545453050919331816

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