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Properties

Label 1-6017-6017.4806-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.239 + 0.970i$
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  + (0.826 + 0.563i)2-s + (−0.623 + 0.781i)3-s + (0.365 + 0.930i)4-s + (−0.955 − 0.294i)5-s + (−0.955 + 0.294i)6-s + (0.365 − 0.930i)7-s + (−0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.623 − 0.781i)10-s + (−0.955 − 0.294i)12-s + (−0.0747 − 0.997i)13-s + (0.826 − 0.563i)14-s + (0.826 − 0.563i)15-s + (−0.733 + 0.680i)16-s + (−0.733 + 0.680i)17-s + (0.365 − 0.930i)18-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)2-s + (−0.623 + 0.781i)3-s + (0.365 + 0.930i)4-s + (−0.955 − 0.294i)5-s + (−0.955 + 0.294i)6-s + (0.365 − 0.930i)7-s + (−0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.623 − 0.781i)10-s + (−0.955 − 0.294i)12-s + (−0.0747 − 0.997i)13-s + (0.826 − 0.563i)14-s + (0.826 − 0.563i)15-s + (−0.733 + 0.680i)16-s + (−0.733 + 0.680i)17-s + (0.365 − 0.930i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.239 + 0.970i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (4806, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.239 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.020406355 + 0.7993019206i\)
\(L(\frac12)\) \(\approx\) \(1.020406355 + 0.7993019206i\)
\(L(1)\) \(\approx\) \(0.9159617824 + 0.4610486873i\)
\(L(1)\) \(\approx\) \(0.9159617824 + 0.4610486873i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.826 + 0.563i)T \)
3 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (-0.955 - 0.294i)T \)
7 \( 1 + (0.365 - 0.930i)T \)
13 \( 1 + (-0.0747 - 0.997i)T \)
17 \( 1 + (-0.733 + 0.680i)T \)
19 \( 1 + (-0.826 + 0.563i)T \)
23 \( 1 + (-0.826 + 0.563i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + (-0.623 + 0.781i)T \)
37 \( 1 + (0.733 - 0.680i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.365 - 0.930i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.0747 + 0.997i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.733 + 0.680i)T \)
67 \( 1 + (-0.733 + 0.680i)T \)
71 \( 1 + (0.733 - 0.680i)T \)
73 \( 1 + (-0.826 - 0.563i)T \)
79 \( 1 + (0.623 + 0.781i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.222 - 0.974i)T \)
97 \( 1 + (0.826 - 0.563i)T \)
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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

−17.94385588538665798926671910067, −16.74262018467994546653605772672, −16.157688714036416812034405317483, −15.56236902091946994815319503923, −14.702468627126900477340167292548, −14.40950606464312031838218643295, −13.42350319860796203023373424317, −12.853943675399599726948395803055, −12.20997434662079588536157891005, −11.60243580482212537047711642118, −11.33352503768611314517640048470, −10.765320597144310723656381780000, −9.73166104966648426212711450129, −8.84000862318401511377590166379, −8.137323533658510128764907515017, −7.265776204676245815903818417760, −6.41584253142558546967330463898, −6.35350969712887244498740730373, −5.01377773299805598266742056728, −4.78285911851164622116170723358, −3.95570783636344359748664723776, −2.88685178325040991084875923330, −2.26799692102330156276577858814, −1.67902556837455901163063122845, −0.48288926827530223688755125391, 0.51069094578279659552654037909, 1.82380981999685171833562550626, 3.10761355596089658388709694232, 3.78007959094435635686226287103, 4.2792706643065367507287748862, 4.674196232735797543865133827254, 5.68515192423719960313802664672, 6.082664528651950660681512855436, 7.130321737786575667121381820240, 7.62525467941217843416794748964, 8.375699942730729212789174905, 8.94751344566415049368242141960, 10.234899726710485706822073831487, 10.67584028341838934835000375296, 11.29788819841828949257809606454, 11.99008318367037854524481528990, 12.58310646806565421889191679946, 13.18280193144678636996318089842, 14.09453207058047360408157507913, 14.75094080594860223255761373369, 15.39157937886474112341045311037, 15.697669145895143151079197712459, 16.51928300368580926639957716130, 16.94470077849522525040575845449, 17.578890223820506870022811695967

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