Properties

Degree 1
Conductor $ 2^{3} \cdot 751 $
Sign $0.690 - 0.723i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.187 − 0.982i)3-s + (−0.876 − 0.481i)5-s + (−0.187 + 0.982i)7-s + (−0.929 − 0.368i)9-s + (−0.309 − 0.951i)11-s + (−0.0627 − 0.998i)13-s + (−0.637 + 0.770i)15-s + (0.728 + 0.684i)17-s + (−0.968 − 0.248i)19-s + (0.929 + 0.368i)21-s + (0.968 − 0.248i)23-s + (0.535 + 0.844i)25-s + (−0.535 + 0.844i)27-s + (0.929 + 0.368i)29-s + (−0.929 − 0.368i)31-s + ⋯
L(s,χ)  = 1  + (0.187 − 0.982i)3-s + (−0.876 − 0.481i)5-s + (−0.187 + 0.982i)7-s + (−0.929 − 0.368i)9-s + (−0.309 − 0.951i)11-s + (−0.0627 − 0.998i)13-s + (−0.637 + 0.770i)15-s + (0.728 + 0.684i)17-s + (−0.968 − 0.248i)19-s + (0.929 + 0.368i)21-s + (0.968 − 0.248i)23-s + (0.535 + 0.844i)25-s + (−0.535 + 0.844i)27-s + (0.929 + 0.368i)29-s + (−0.929 − 0.368i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.690 - 0.723i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.690 - 0.723i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6008\)    =    \(2^{3} \cdot 751\)
\( \varepsilon \)  =  $0.690 - 0.723i$
motivic weight  =  \(0\)
character  :  $\chi_{6008} (485, \cdot )$
Sato-Tate  :  $\mu(50)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6008,\ (0:\ ),\ 0.690 - 0.723i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.048134788 - 0.4487233821i$
$L(\frac12,\chi)$  $\approx$  $1.048134788 - 0.4487233821i$
$L(\chi,1)$  $\approx$  0.8125303224 - 0.3018063779i
$L(1,\chi)$  $\approx$  0.8125303224 - 0.3018063779i

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

−17.55351080502270994422581739080, −17.0260200704481996644500406349, −16.40754196527805644813132006944, −15.824839451609500742204473942315, −15.213962480529931510311121534398, −14.56497143865716696882307848341, −14.11430894927634470815686363423, −13.3599102761158628759270790697, −12.33931423271032233692205561636, −11.83074608829158686828251610800, −10.920830117858296063400749384049, −10.62183340725723536025042205017, −9.85628049436717280504646269800, −9.32784661585493677149266088497, −8.380444865986482039061871832143, −7.80440917269834117167262978071, −6.881395664565734051246619056267, −6.70184240120602785389196905059, −5.20961235331555244501138548066, −4.77850184475825880286353895313, −3.94475454401990668797330400512, −3.61311620900883812410267569641, −2.73541207023325082381481493039, −1.85152132756311485047514581958, −0.47235459284306588814709187826, 0.60470239706240729474912029404, 1.32660905173075324816625057391, 2.44995979204909857122214059312, 3.04847589491009594884355150308, 3.641983058266426601733086519254, 4.80998415317257550797346447730, 5.58157549043157083144756914272, 6.04308733880754516045168485217, 6.90601348926108508416451298436, 7.769695760327816950628827709927, 8.23087708278737388514781743627, 8.73310008878277564533258795351, 9.30890955473373888317333973690, 10.66970328432219917929222992772, 11.0038280441073851663062156856, 12.00817640737403829949679152352, 12.36111294352292461604315301524, 12.972246395589351421019017435045, 13.378211710100403471695796866218, 14.57246359376371765062710973216, 14.92403985295800020770889565470, 15.58922080459137238709632235297, 16.403999173688586446032146909762, 16.912144591748729393946947446676, 17.781072259120991717643923513496

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