Properties

Degree 1
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $-0.800 + 0.599i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $-0.800 + 0.599i$
motivic weight  =  \(0\)
character  :  $\chi_{6027} (59, \cdot )$
Sato-Tate  :  $\mu(210)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6027,\ (0:\ ),\ -0.800 + 0.599i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2195835891 + 0.6592997035i$
$L(\frac12,\chi)$  $\approx$  $0.2195835891 + 0.6592997035i$
$L(\chi,1)$  $\approx$  1.432404949 - 0.05005304199i
$L(1,\chi)$  $\approx$  1.432404949 - 0.05005304199i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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