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Properties

Label 1-6027-6027.59-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.800 + 0.599i$
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  + (0.946 − 0.323i)2-s + (0.791 − 0.611i)4-s + (−0.163 + 0.986i)5-s + (0.550 − 0.834i)8-s + (0.163 + 0.986i)10-s + (0.873 + 0.486i)11-s + (−0.753 − 0.657i)13-s + (0.251 − 0.967i)16-s + (−0.925 + 0.379i)17-s + (−0.913 + 0.406i)19-s + (0.473 + 0.880i)20-s + (0.983 + 0.178i)22-s + (−0.525 − 0.850i)23-s + (−0.946 − 0.323i)25-s + (−0.925 − 0.379i)26-s + ⋯
L(s)  = 1  + (0.946 − 0.323i)2-s + (0.791 − 0.611i)4-s + (−0.163 + 0.986i)5-s + (0.550 − 0.834i)8-s + (0.163 + 0.986i)10-s + (0.873 + 0.486i)11-s + (−0.753 − 0.657i)13-s + (0.251 − 0.967i)16-s + (−0.925 + 0.379i)17-s + (−0.913 + 0.406i)19-s + (0.473 + 0.880i)20-s + (0.983 + 0.178i)22-s + (−0.525 − 0.850i)23-s + (−0.946 − 0.323i)25-s + (−0.925 − 0.379i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.800 + 0.599i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.800 + 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2195835891 + 0.6592997035i\)
\(L(\frac12)\) \(\approx\) \(0.2195835891 + 0.6592997035i\)
\(L(1)\) \(\approx\) \(1.432404949 - 0.05005304199i\)
\(L(1)\) \(\approx\) \(1.432404949 - 0.05005304199i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.946 - 0.323i)T \)
5 \( 1 + (-0.163 + 0.986i)T \)
11 \( 1 + (0.873 + 0.486i)T \)
13 \( 1 + (-0.753 - 0.657i)T \)
17 \( 1 + (-0.925 + 0.379i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (-0.525 - 0.850i)T \)
29 \( 1 + (0.691 - 0.722i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.280 - 0.959i)T \)
43 \( 1 + (-0.0448 + 0.998i)T \)
47 \( 1 + (-0.946 + 0.323i)T \)
53 \( 1 + (-0.791 + 0.611i)T \)
59 \( 1 + (0.887 - 0.460i)T \)
61 \( 1 + (0.999 + 0.0299i)T \)
67 \( 1 + (-0.978 + 0.207i)T \)
71 \( 1 + (0.691 + 0.722i)T \)
73 \( 1 + (-0.955 + 0.294i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.193 + 0.981i)T \)
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

−17.29463709323353845987777220391, −16.602665780759317539421016867468, −16.17671743113416849952857731701, −15.46549682928955070758690757817, −14.79147844988796361894817709675, −14.15586522988439067576309297760, −13.41186185102011770557421051177, −13.04454139305027949027164238651, −12.15453691863645391235212581747, −11.70165628749592942509620321560, −11.23714114490659977590535772441, −10.18913188663869833902012493765, −9.22696552152171261864169783173, −8.73109980236133181172799356268, −8.05241499634106607280156060487, −7.11970455869732257497175811148, −6.63745417334608330102668772650, −5.84841697932160048112560219732, −5.06194877391742773470184565445, −4.48864057059655375209534393135, −3.956562656208457392674188535533, −3.12724803933727243916923970475, −2.06717817640150680082997803186, −1.54797681646578247019999569754, −0.10375694327491163448523042281, 1.36172428408239752544348988552, 2.30478787070334674758088090161, 2.63351405382624272197987755613, 3.71711796887243875063093992065, 4.14557824947633124627599040722, 4.88154998465188094358248719840, 5.860915463284776449726665673, 6.52205993025337448168031470348, 6.87999887488849968564668994066, 7.73586012223497976952317912682, 8.544629855164074852228279439725, 9.7110601247675325617939515500, 10.12489463845602591577268770844, 10.90483492575024428388067464602, 11.29351882608434551261004852904, 12.29351402389981858429677242147, 12.519395840024916197335555561915, 13.37271010521781476900160439086, 14.28492968161067744661923115512, 14.59792978933949892374702789833, 15.06129194360889254811234417905, 15.798270376479592727491359710043, 16.46509344343004494953327715242, 17.52333840269906648852609916324, 17.808745297893040168419819408525

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