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Properties

Label 1-6008-6008.2205-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.690 + 0.723i$
Analytic cond. $27.9010$
Root an. cond. $27.9010$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.690 + 0.723i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (2205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.690 + 0.723i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.048134788 + 0.4487233821i\)
\(L(\frac12)\) \(\approx\) \(1.048134788 + 0.4487233821i\)
\(L(1)\) \(\approx\) \(0.8125303224 + 0.3018063779i\)
\(L(1)\) \(\approx\) \(0.8125303224 + 0.3018063779i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.187 + 0.982i)T \)
5 \( 1 + (-0.876 + 0.481i)T \)
7 \( 1 + (-0.187 - 0.982i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.0627 + 0.998i)T \)
17 \( 1 + (0.728 - 0.684i)T \)
19 \( 1 + (-0.968 + 0.248i)T \)
23 \( 1 + (0.968 + 0.248i)T \)
29 \( 1 + (0.929 - 0.368i)T \)
31 \( 1 + (-0.929 + 0.368i)T \)
37 \( 1 + (-0.535 + 0.844i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.876 - 0.481i)T \)
47 \( 1 + (0.876 - 0.481i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.425 - 0.904i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.929 - 0.368i)T \)
71 \( 1 + (-0.637 + 0.770i)T \)
73 \( 1 + T \)
79 \( 1 + (0.968 + 0.248i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (0.876 - 0.481i)T \)
97 \( 1 + (-0.187 + 0.982i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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