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Properties

Label 1-6027-6027.1781-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: $\chi_{6027} (1781, \cdot )$
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

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

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