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Properties

Label 1-6027-6027.353-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.999 + 0.0170i$
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  + (0.599 + 0.800i)2-s + (−0.280 + 0.959i)4-s + (−0.447 − 0.894i)5-s + (−0.936 + 0.351i)8-s + (0.447 − 0.894i)10-s + (0.420 + 0.907i)11-s + (−0.393 + 0.919i)13-s + (−0.842 − 0.538i)16-s + (−0.971 − 0.237i)17-s + (−0.104 − 0.994i)19-s + (0.983 − 0.178i)20-s + (−0.473 + 0.880i)22-s + (0.646 − 0.762i)23-s + (−0.599 + 0.800i)25-s + (−0.971 + 0.237i)26-s + ⋯
L(s)  = 1  + (0.599 + 0.800i)2-s + (−0.280 + 0.959i)4-s + (−0.447 − 0.894i)5-s + (−0.936 + 0.351i)8-s + (0.447 − 0.894i)10-s + (0.420 + 0.907i)11-s + (−0.393 + 0.919i)13-s + (−0.842 − 0.538i)16-s + (−0.971 − 0.237i)17-s + (−0.104 − 0.994i)19-s + (0.983 − 0.178i)20-s + (−0.473 + 0.880i)22-s + (0.646 − 0.762i)23-s + (−0.599 + 0.800i)25-s + (−0.971 + 0.237i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.999 + 0.0170i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.999 + 0.0170i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008506372020 + 0.9963588577i\)
\(L(\frac12)\) \(\approx\) \(0.008506372020 + 0.9963588577i\)
\(L(1)\) \(\approx\) \(0.9623642253 + 0.5120332561i\)
\(L(1)\) \(\approx\) \(0.9623642253 + 0.5120332561i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.599 + 0.800i)T \)
5 \( 1 + (-0.447 - 0.894i)T \)
11 \( 1 + (0.420 + 0.907i)T \)
13 \( 1 + (-0.393 + 0.919i)T \)
17 \( 1 + (-0.971 - 0.237i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.646 - 0.762i)T \)
29 \( 1 + (0.134 - 0.990i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.791 + 0.611i)T \)
43 \( 1 + (-0.963 + 0.266i)T \)
47 \( 1 + (0.599 + 0.800i)T \)
53 \( 1 + (-0.280 + 0.959i)T \)
59 \( 1 + (0.712 + 0.701i)T \)
61 \( 1 + (0.337 - 0.941i)T \)
67 \( 1 + (-0.669 + 0.743i)T \)
71 \( 1 + (0.134 + 0.990i)T \)
73 \( 1 + (-0.955 + 0.294i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.992 - 0.119i)T \)
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

−17.673316904525188602528580813642, −16.67595423481552200466986043835, −15.82850571901971388764916587178, −15.189581922133035672386116008, −14.713959991135455944204437429636, −14.08642895269119023273572744711, −13.40230039343110800822166780518, −12.80215155584966646416137559509, −11.94446524110211818441312411491, −11.51004719317867633013457225746, −10.755073649871818931837279991008, −10.40910504218209793897271141457, −9.625042408792858082126307927093, −8.731964474014305402600952878956, −8.10254682223713430244169897364, −7.11058486008735610547885947681, −6.453362535483901560342744572146, −5.78398910637983964861394422998, −5.06467618240647211908005851646, −4.13835731969178603795860442059, −3.473795352557966911451399904555, −3.021133568261296781775752375588, −2.207048272044569537888805400627, −1.27098143041717089218122021686, −0.22370117377820550182992222920, 1.01669184493477670828332760158, 2.23762588714094790364912015626, 2.864421168502831756234730540189, 4.21970513924428768782490377584, 4.40783337408948952607021357432, 4.8529290849232116857977573416, 5.869191793666156803114491441071, 6.78432352914924049710420959798, 7.014777201002139619551233231667, 7.983020417535254427286811754284, 8.58490776337808522420419566956, 9.26411640821177942342321431187, 9.7073394491038240583685233347, 11.0592482585603706546152567209, 11.77442529401776975412283980840, 12.10176617031954461414008265778, 13.01884014476244797381672802830, 13.33495803290525716878151026043, 14.13736697351824940307073919128, 14.97823170840227013334609199498, 15.34594551208163990081759062148, 16.018660635345479587973824245486, 16.672400887609509344728615357609, 17.33066574667204272099788595965, 17.56606180578430485418172242603

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