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Properties

Label 1-6017-6017.380-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.964 - 0.262i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
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  + (−0.573 − 0.819i)2-s + (0.691 + 0.722i)3-s + (−0.341 + 0.939i)4-s + (−0.545 + 0.838i)5-s + (0.195 − 0.980i)6-s + (0.498 − 0.867i)7-s + (0.965 − 0.259i)8-s + (−0.0448 + 0.998i)9-s + (0.999 − 0.0345i)10-s + (−0.915 + 0.402i)12-s + (0.393 + 0.919i)13-s + (−0.995 + 0.0896i)14-s + (−0.982 + 0.185i)15-s + (−0.766 − 0.642i)16-s + (0.0930 − 0.995i)17-s + (0.843 − 0.536i)18-s + ⋯
L(s)  = 1  + (−0.573 − 0.819i)2-s + (0.691 + 0.722i)3-s + (−0.341 + 0.939i)4-s + (−0.545 + 0.838i)5-s + (0.195 − 0.980i)6-s + (0.498 − 0.867i)7-s + (0.965 − 0.259i)8-s + (−0.0448 + 0.998i)9-s + (0.999 − 0.0345i)10-s + (−0.915 + 0.402i)12-s + (0.393 + 0.919i)13-s + (−0.995 + 0.0896i)14-s + (−0.982 + 0.185i)15-s + (−0.766 − 0.642i)16-s + (0.0930 − 0.995i)17-s + (0.843 − 0.536i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.964 - 0.262i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (380, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.964 - 0.262i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.359608313 - 0.1818671932i\)
\(L(\frac12)\) \(\approx\) \(1.359608313 - 0.1818671932i\)
\(L(1)\) \(\approx\) \(0.9178433775 - 0.03918962493i\)
\(L(1)\) \(\approx\) \(0.9178433775 - 0.03918962493i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.573 - 0.819i)T \)
3 \( 1 + (0.691 + 0.722i)T \)
5 \( 1 + (-0.545 + 0.838i)T \)
7 \( 1 + (0.498 - 0.867i)T \)
13 \( 1 + (0.393 + 0.919i)T \)
17 \( 1 + (0.0930 - 0.995i)T \)
19 \( 1 + (0.492 - 0.870i)T \)
23 \( 1 + (-0.940 - 0.338i)T \)
29 \( 1 + (-0.461 - 0.887i)T \)
31 \( 1 + (-0.993 + 0.110i)T \)
37 \( 1 + (-0.762 + 0.647i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.596 - 0.802i)T \)
47 \( 1 + (-0.906 + 0.421i)T \)
53 \( 1 + (0.903 + 0.427i)T \)
59 \( 1 + (-0.958 + 0.285i)T \)
61 \( 1 + (0.999 - 0.0276i)T \)
67 \( 1 + (0.915 - 0.402i)T \)
71 \( 1 + (0.328 - 0.944i)T \)
73 \( 1 + (-0.686 - 0.727i)T \)
79 \( 1 + (0.954 + 0.299i)T \)
83 \( 1 + (0.995 - 0.0965i)T \)
89 \( 1 + (0.792 + 0.609i)T \)
97 \( 1 + (0.999 - 0.0414i)T \)
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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

−17.79753457661888036996010100461, −17.23289112248907516366068789122, −16.267054025203661867093446989969, −15.830715086017735088616953072, −15.095362374182991074439151794066, −14.6463545024729376132977657320, −14.02607615128457529616760882606, −13.03089999146105674614291739055, −12.6970560823764027673797740155, −11.926208063860619424508341575389, −11.16803722310649011074089358598, −10.20814399031711124356389292655, −9.44187453144568584120551206727, −8.68504102567401642102420108859, −8.40410081547152510118018935888, −7.80204544017745694590410729964, −7.31964051575254000519197192562, −6.24009127069968744354788490009, −5.59770386702735823514368387864, −5.17967333958403175588014752015, −3.92680891125194833519640890227, −3.43304286704193262101226369088, −1.944913327890723926344340238031, −1.66492987198650772372460614287, −0.69210177416967494536042986745, 0.54452460840034822516289507906, 1.80609454751889479091767720954, 2.36564209533636843072382565272, 3.34757444454251003145124397207, 3.66772738900029896994397482982, 4.47749717886626296735253761901, 4.95285961425387008049525054760, 6.480331187866440410800103114684, 7.32245653255325595330134278862, 7.67665838996106020283331639586, 8.42489980592229065258540331282, 9.15586353758691872737236357764, 9.81001906608839311807591977183, 10.35694440747466504379990582711, 11.117029798539634257447202774081, 11.404154683907929466111788065090, 12.076625793580446947226527164354, 13.36253425573134000142489480377, 13.757365734737339844484516822108, 14.24796069485800902323877772292, 15.04195185188771208160732896908, 15.863165015292850312548813398393, 16.3978532972291664318086725585, 16.96276787136021760340279739623, 17.99438393370515378437882028596

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