Properties

Label 1-6008-6008.469-r0-0-0
Degree $1$
Conductor $6008$
Sign $-0.991 + 0.128i$
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  + (−0.521 + 0.853i)3-s + (0.170 − 0.985i)5-s + (0.793 + 0.607i)7-s + (−0.455 − 0.890i)9-s + (0.348 + 0.937i)11-s + (0.662 − 0.748i)13-s + (0.751 + 0.659i)15-s + (−0.994 + 0.108i)17-s + (0.828 − 0.560i)19-s + (−0.932 + 0.360i)21-s + (−0.970 + 0.240i)23-s + (−0.941 − 0.336i)25-s + (0.997 + 0.0753i)27-s + (−0.630 − 0.775i)29-s + (−0.154 + 0.988i)31-s + ⋯
L(s)  = 1  + (−0.521 + 0.853i)3-s + (0.170 − 0.985i)5-s + (0.793 + 0.607i)7-s + (−0.455 − 0.890i)9-s + (0.348 + 0.937i)11-s + (0.662 − 0.748i)13-s + (0.751 + 0.659i)15-s + (−0.994 + 0.108i)17-s + (0.828 − 0.560i)19-s + (−0.932 + 0.360i)21-s + (−0.970 + 0.240i)23-s + (−0.941 − 0.336i)25-s + (0.997 + 0.0753i)27-s + (−0.630 − 0.775i)29-s + (−0.154 + 0.988i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-0.991 + 0.128i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ -0.991 + 0.128i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02807565293 + 0.4348671055i\)
\(L(\frac12)\) \(\approx\) \(0.02807565293 + 0.4348671055i\)
\(L(1)\) \(\approx\) \(0.8384409846 + 0.1733934409i\)
\(L(1)\) \(\approx\) \(0.8384409846 + 0.1733934409i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.521 + 0.853i)T \)
5 \( 1 + (0.170 - 0.985i)T \)
7 \( 1 + (0.793 + 0.607i)T \)
11 \( 1 + (0.348 + 0.937i)T \)
13 \( 1 + (0.662 - 0.748i)T \)
17 \( 1 + (-0.994 + 0.108i)T \)
19 \( 1 + (0.828 - 0.560i)T \)
23 \( 1 + (-0.970 + 0.240i)T \)
29 \( 1 + (-0.630 - 0.775i)T \)
31 \( 1 + (-0.154 + 0.988i)T \)
37 \( 1 + (-0.0293 + 0.999i)T \)
41 \( 1 + (-0.637 + 0.770i)T \)
43 \( 1 + (0.962 - 0.272i)T \)
47 \( 1 + (-0.440 - 0.897i)T \)
53 \( 1 + (-0.0627 - 0.998i)T \)
59 \( 1 + (-0.964 + 0.264i)T \)
61 \( 1 + (-0.832 + 0.553i)T \)
67 \( 1 + (0.0544 + 0.998i)T \)
71 \( 1 + (0.0125 - 0.999i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.828 - 0.560i)T \)
83 \( 1 + (-0.996 + 0.0836i)T \)
89 \( 1 + (0.244 + 0.969i)T \)
97 \( 1 + (-0.923 - 0.383i)T \)
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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

−17.5275590638266116085247189033, −16.853112306898446041758378957558, −16.24288141786085253775640096887, −15.51093705607655844480301381421, −14.37377569918463316847189383293, −14.04195420111876016217847632196, −13.731055553265066849636707908776, −12.8609696273378984799454063172, −11.92933497722132304870036260467, −11.28826313710369327318907678822, −11.03142371877052301736368290871, −10.43712047792844126586465119536, −9.345783616360956976298244093892, −8.62118034009616644012578024574, −7.67505568426952338290452052678, −7.42701567699597510921255634323, −6.473300673355067416682055813581, −6.09589693674193611506656198530, −5.38930972004939482140137493788, −4.29544635944986042733358108029, −3.69544091138861338400628963655, −2.709333575218245254481516282944, −1.82595864714656543832782973368, −1.351639425924891305684426160, −0.11585743771102291046928498312, 1.172191998751204645280823835783, 1.8267754544384494958092272875, 2.86383709983941103995129094795, 3.89435403311247613804960239480, 4.5191159869555914088553882658, 5.04782925797359745439757877245, 5.65731467164572445713212023236, 6.29382968049393475699156115469, 7.30972791153757243846350439438, 8.29221462536843742494972936523, 8.70040655211409681119338580241, 9.45047209763687983718835989380, 9.951960311774109838179915243429, 10.774058196315089492592868401878, 11.61695563687267565101974277125, 11.86070780594456297453114781454, 12.67059749157500213665452374463, 13.39189579165867474896130757271, 14.13953465100899868418421788095, 15.102022272544298401145455896737, 15.42853988224588059799825533608, 15.96564718149305493902518849169, 16.734294171328609819114864230537, 17.46512265340447350198566650592, 17.862748128822316900145707021024

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