Properties

Degree 1
Conductor $ 2^{3} \cdot 751 $
Sign $0.927 - 0.372i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.992 + 0.125i)3-s + (0.187 + 0.982i)5-s + (−0.992 − 0.125i)7-s + (0.968 + 0.248i)9-s + (−0.309 + 0.951i)11-s + (−0.535 − 0.844i)13-s + (0.0627 + 0.998i)15-s + (0.876 + 0.481i)17-s + (0.637 − 0.770i)19-s + (−0.968 − 0.248i)21-s + (−0.637 − 0.770i)23-s + (−0.929 + 0.368i)25-s + (0.929 + 0.368i)27-s + (−0.968 − 0.248i)29-s + (0.968 + 0.248i)31-s + ⋯
L(s,χ)  = 1  + (0.992 + 0.125i)3-s + (0.187 + 0.982i)5-s + (−0.992 − 0.125i)7-s + (0.968 + 0.248i)9-s + (−0.309 + 0.951i)11-s + (−0.535 − 0.844i)13-s + (0.0627 + 0.998i)15-s + (0.876 + 0.481i)17-s + (0.637 − 0.770i)19-s + (−0.968 − 0.248i)21-s + (−0.637 − 0.770i)23-s + (−0.929 + 0.368i)25-s + (0.929 + 0.368i)27-s + (−0.968 − 0.248i)29-s + (0.968 + 0.248i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.927 - 0.372i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.927 - 0.372i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6008\)    =    \(2^{3} \cdot 751\)
\( \varepsilon \)  =  $0.927 - 0.372i$
motivic weight  =  \(0\)
character  :  $\chi_{6008} (4205, \cdot )$
Sato-Tate  :  $\mu(50)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6008,\ (0:\ ),\ 0.927 - 0.372i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.112538453 - 0.4086568723i$
$L(\frac12,\chi)$  $\approx$  $2.112538453 - 0.4086568723i$
$L(\chi,1)$  $\approx$  1.346787903 + 0.1335759962i
$L(1,\chi)$  $\approx$  1.346787903 + 0.1335759962i

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

−17.90817659881522866525516516896, −16.6974055086900646385193194730, −16.462083199381768702892550474142, −15.95066247034534025646402143080, −15.15629404122018508121329023826, −14.30169959494291432837001454298, −13.70762975861808831940189065191, −13.35037313111137733235693881340, −12.51597464319162990347535763926, −12.09216286246874195989380080980, −11.2713032428225299265034363461, −10.02434463148509994837641549771, −9.5583260409691040718871284413, −9.321704849520828292877859141009, −8.35340453000838821664656404604, −7.82430314912515087514781779781, −7.20668470257250716840071155876, −6.05934983809503793363764559472, −5.75217826238319517151535916196, −4.62174345358971512932233516011, −4.00895592223673858864212919401, −3.11555718996405324003463289643, −2.694258315333077715754246593645, −1.55463138361821481229017306901, −0.97322032817846514909920040262, 0.498735449151854985438752800220, 1.958173471827425303882727924877, 2.442722653712694119569896738717, 3.21101600703371493998891884226, 3.63073082331605324989155758301, 4.575999190320980124112824210064, 5.47922524179718204539647435748, 6.34303842404458285104208168387, 7.06107271612478864143089697072, 7.58630841040215294678752826798, 8.12020681072519907393226313601, 9.30097453494621087570682631197, 9.67130951448843755091974816011, 10.33701009299505836576634242998, 10.62751798501595401898127870984, 11.91817247443240547834291114936, 12.578607725113908152375687221050, 13.123940970762880864408873559445, 13.84131151525372592971807391965, 14.412898748680657218808207771819, 15.18795117113874404810527897007, 15.3958585184483267284114773918, 16.18471863657619940460203860960, 17.06668191949788641343977571769, 17.79963804154805388007351039781

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