Properties

Degree 1
Conductor $ 2^{3} \cdot 751 $
Sign $-0.136 + 0.990i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.309 + 0.951i)3-s + (−0.309 + 0.951i)5-s + (0.309 − 0.951i)7-s + (−0.809 − 0.587i)9-s − 11-s + (0.809 − 0.587i)13-s + (−0.809 − 0.587i)15-s + (−0.309 − 0.951i)17-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)21-s + (0.809 + 0.587i)23-s + (−0.809 − 0.587i)25-s + (0.809 − 0.587i)27-s + (−0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + ⋯
L(s,χ)  = 1  + (−0.309 + 0.951i)3-s + (−0.309 + 0.951i)5-s + (0.309 − 0.951i)7-s + (−0.809 − 0.587i)9-s − 11-s + (0.809 − 0.587i)13-s + (−0.809 − 0.587i)15-s + (−0.309 − 0.951i)17-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)21-s + (0.809 + 0.587i)23-s + (−0.809 − 0.587i)25-s + (0.809 − 0.587i)27-s + (−0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.136 + 0.990i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.136 + 0.990i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6008\)    =    \(2^{3} \cdot 751\)
\( \varepsilon \)  =  $-0.136 + 0.990i$
motivic weight  =  \(0\)
character  :  $\chi_{6008} (291, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6008,\ (0:\ ),\ -0.136 + 0.990i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5514893005 + 0.6329515605i$
$L(\frac12,\chi)$  $\approx$  $0.5514893005 + 0.6329515605i$
$L(\chi,1)$  $\approx$  0.7365034707 + 0.2386824003i
$L(1,\chi)$  $\approx$  0.7365034707 + 0.2386824003i

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

−17.582752412252561668775378512156, −16.86034120919596631643500417437, −16.40012822103793772763524144498, −15.53225281453137929182557397996, −15.0626571781550735204984790729, −14.16957626984779627689876958988, −13.28603667789562447025864236637, −12.753408274948103769852338428438, −12.59501014220015282033043080466, −11.59443934952405484552645568523, −11.11506484465655017518795827078, −10.51260991173925711822776177834, −9.160810365266824872987737354666, −8.65683535025718045953540299018, −8.35943485046721924153995330990, −7.497283063856634837108111336771, −6.73251865879775041889170916370, −5.9505754229793094847164842819, −5.35019643133976893704505905505, −4.79679795121110871830373335197, −3.855562381283359854380334933384, −2.80533873639216285455471906633, −1.94389701201541241725057870200, −1.52941640988543948993758011511, −0.33585914582912431349151260438, 0.61604752770154287299305370282, 1.9155269894412425822975016219, 3.05565875149655459624811284601, 3.34154841750624912724697210791, 4.26355202780928044550661319875, 4.78623374744155224176070392050, 5.73038594002007113274862662789, 6.2834330207698922148182922791, 7.22312480833348500552503744930, 7.81419524346657092588540533038, 8.43555140638812119409672749430, 9.539790045566131970319658369322, 10.02199134203693803195722308854, 10.730626277784108562781278312277, 11.23329034027818117622382960884, 11.41455702110767728840711900108, 12.76796965004724990152875518073, 13.34105276396874283351781997195, 14.101178906520016525994468582555, 14.76721866688661253731343348201, 15.31018138113685755087162862993, 15.85957392660708227830380358220, 16.50555247234929740338962724391, 17.23711734583835423412902738493, 17.815922687641632999961148542069

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