Properties

Degree 1
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $0.878 + 0.477i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.900 + 0.433i)5-s + (−0.900 − 0.433i)8-s + (0.900 − 0.433i)10-s + (−0.781 − 0.623i)11-s + (−0.781 − 0.623i)13-s + (−0.900 + 0.433i)16-s + (0.974 + 0.222i)17-s + i·19-s + (0.222 − 0.974i)20-s + (−0.974 + 0.222i)22-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.974 + 0.222i)26-s + ⋯
L(s,χ)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.900 + 0.433i)5-s + (−0.900 − 0.433i)8-s + (0.900 − 0.433i)10-s + (−0.781 − 0.623i)11-s + (−0.781 − 0.623i)13-s + (−0.900 + 0.433i)16-s + (0.974 + 0.222i)17-s + i·19-s + (0.222 − 0.974i)20-s + (−0.974 + 0.222i)22-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.974 + 0.222i)26-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.878 + 0.477i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.878 + 0.477i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $0.878 + 0.477i$
motivic weight  =  \(0\)
character  :  $\chi_{6027} (419, \cdot )$
Sato-Tate  :  $\mu(28)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6027,\ (0:\ ),\ 0.878 + 0.477i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.720799804 + 0.4372445262i$
$L(\frac12,\chi)$  $\approx$  $1.720799804 + 0.4372445262i$
$L(\chi,1)$  $\approx$  1.306610012 - 0.4369553950i
$L(1,\chi)$  $\approx$  1.306610012 - 0.4369553950i

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

−17.48307804987471607569445582795, −17.0156488558097118903848086904, −16.15233449186872856357361617119, −15.93279635605537852419291807474, −14.80301279701115880209048174191, −14.4474539835393927894460390684, −13.8419196407744905359383938874, −13.05354532735305702281796819852, −12.57455260656643043505930913513, −12.12230316721149650919879677303, −11.0914839083588245414872281818, −10.2715306802765860612000424394, −9.419290476190380477769215020378, −9.0931071750495420207864702608, −8.1058233993618398939702839915, −7.48641991117477456189452943766, −6.7995314591750870756400985730, −6.136100164321883293762656253870, −5.33567903913774167733676196075, −4.79637678866900469606004436724, −4.37358633769943786595725075918, −3.07602877398450758062615458385, −2.5451325514380650895101172305, −1.731542235194873911338308453879, −0.335124710053468834559933676845, 1.10156995334638886342880936541, 1.72901444904850093760675743043, 2.72709808834176633505282181489, 3.085740406421463111992262273866, 3.84988888562957497608017477481, 5.02044650682108377399730480819, 5.47393858591203948192601884048, 5.92443283518465268199593201546, 6.81548301288310660718963867526, 7.70200247612439205997983163281, 8.46381639660971768874843483388, 9.58443348355044962657316313347, 9.821196603930596247367151634256, 10.693366096412204154302116595974, 10.867117276114693426718705768575, 12.0844192208805133819062599074, 12.396615091274642487725309275309, 13.275512608998447250168519680523, 13.69642543332618601993551417855, 14.401551857584684458808539727899, 14.87611593539066798782431410414, 15.59573041465088474671810781265, 16.46959491605346763171406792037, 17.21103040433985152623788586362, 17.96655250338374392374104326282

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