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Properties

Label 1-6027-6027.467-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.989 + 0.147i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (−0.712 − 0.701i)2-s + (0.0149 + 0.999i)4-s + (−0.925 + 0.379i)5-s + (0.691 − 0.722i)8-s + (0.925 + 0.379i)10-s + (−0.998 + 0.0598i)11-s + (0.963 − 0.266i)13-s + (−0.999 + 0.0299i)16-s + (−0.873 + 0.486i)17-s + (−0.913 − 0.406i)19-s + (−0.393 − 0.919i)20-s + (0.753 + 0.657i)22-s + (−0.992 + 0.119i)23-s + (0.712 − 0.701i)25-s + (−0.873 − 0.486i)26-s + ⋯
L(s)  = 1  + (−0.712 − 0.701i)2-s + (0.0149 + 0.999i)4-s + (−0.925 + 0.379i)5-s + (0.691 − 0.722i)8-s + (0.925 + 0.379i)10-s + (−0.998 + 0.0598i)11-s + (0.963 − 0.266i)13-s + (−0.999 + 0.0299i)16-s + (−0.873 + 0.486i)17-s + (−0.913 − 0.406i)19-s + (−0.393 − 0.919i)20-s + (0.753 + 0.657i)22-s + (−0.992 + 0.119i)23-s + (0.712 − 0.701i)25-s + (−0.873 − 0.486i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.989 + 0.147i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.989 + 0.147i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004907338602 - 0.06631143612i\)
\(L(\frac12)\) \(\approx\) \(0.004907338602 - 0.06631143612i\)
\(L(1)\) \(\approx\) \(0.5172528432 - 0.1012685766i\)
\(L(1)\) \(\approx\) \(0.5172528432 - 0.1012685766i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.712 - 0.701i)T \)
5 \( 1 + (-0.925 + 0.379i)T \)
11 \( 1 + (-0.998 + 0.0598i)T \)
13 \( 1 + (0.963 - 0.266i)T \)
17 \( 1 + (-0.873 + 0.486i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (-0.992 + 0.119i)T \)
29 \( 1 + (0.995 - 0.0896i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.575 + 0.817i)T \)
43 \( 1 + (0.983 + 0.178i)T \)
47 \( 1 + (0.712 + 0.701i)T \)
53 \( 1 + (-0.0149 - 0.999i)T \)
59 \( 1 + (-0.646 + 0.762i)T \)
61 \( 1 + (0.599 - 0.800i)T \)
67 \( 1 + (-0.978 - 0.207i)T \)
71 \( 1 + (0.995 + 0.0896i)T \)
73 \( 1 + (0.988 - 0.149i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (0.251 + 0.967i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−18.18552715017460457756167543910, −17.393696800752862740876439863756, −16.557063576304936254033244223646, −15.97521730728226744706165953439, −15.75176861432340672545587679389, −15.06269788790932262504225276543, −14.2053645502379729502310519156, −13.6259105215389563459398205185, −12.79615673583617967603529093584, −12.11882009437084208447412316421, −11.13603254218958767054266210908, −10.80535437743597461072521912669, −10.11472550044419290602830948440, −9.039112576590506536772438413322, −8.69404262821191667201079953740, −8.0340755624668227477405834521, −7.47057861814870183548172980037, −6.70851770411821638214869939295, −5.99542471084736676385415005041, −5.25953582537903616082288570231, −4.43395716123442220801370946270, −3.91359418628022068311486443766, −2.69124327275895240850145497347, −1.873924355628785344196764814905, −0.846710516186761939009096671893, 0.0313343868006984871466297720, 0.96924977310176363201192733218, 2.154929886095651185134098271687, 2.636898041859159651452402714313, 3.55511491529311128990832622807, 4.130350095933865763829633237, 4.77886197206199693108876952522, 6.08802817253030778845585232968, 6.660923365889616618737587568713, 7.61872570647494758208411939137, 8.09143266826033855014642774389, 8.56041001692842743534291977333, 9.37774610872784239652138346173, 10.316274781199396936054229028, 10.80408930519334463978648656852, 11.176865947164243008936410864510, 11.97293931586605136388220858132, 12.68929889208438105077473181352, 13.19274776381922975346924196727, 13.90522972409427750812005306072, 14.99540765531211488191423331445, 15.62302598581833177787248975506, 15.974615904713460235478160097271, 16.761396471531531745997239600146, 17.65478300231863635665305697494

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