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Properties

Label 1-6001-6001.764-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.414 + 0.910i$
Analytic cond. $27.8685$
Root an. cond. $27.8685$
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  + (0.841 + 0.540i)2-s + (−0.841 + 0.540i)3-s + (0.415 + 0.909i)4-s + (−0.841 + 0.540i)5-s − 6-s − 7-s + (−0.142 + 0.989i)8-s + (0.415 − 0.909i)9-s − 10-s + (0.959 + 0.281i)11-s + (−0.841 − 0.540i)12-s + (−0.654 + 0.755i)13-s + (−0.841 − 0.540i)14-s + (0.415 − 0.909i)15-s + (−0.654 + 0.755i)16-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (−0.841 + 0.540i)3-s + (0.415 + 0.909i)4-s + (−0.841 + 0.540i)5-s − 6-s − 7-s + (−0.142 + 0.989i)8-s + (0.415 − 0.909i)9-s − 10-s + (0.959 + 0.281i)11-s + (−0.841 − 0.540i)12-s + (−0.654 + 0.755i)13-s + (−0.841 − 0.540i)14-s + (0.415 − 0.909i)15-s + (−0.654 + 0.755i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.414 + 0.910i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (764, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.414 + 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8398268767 + 1.305404872i\)
\(L(\frac12)\) \(\approx\) \(0.8398268767 + 1.305404872i\)
\(L(1)\) \(\approx\) \(0.8268159151 + 0.6572117507i\)
\(L(1)\) \(\approx\) \(0.8268159151 + 0.6572117507i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 + (-0.841 + 0.540i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
7 \( 1 - T \)
11 \( 1 + (0.959 + 0.281i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (-0.415 + 0.909i)T \)
31 \( 1 + (0.959 + 0.281i)T \)
37 \( 1 + (0.142 - 0.989i)T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (0.415 + 0.909i)T \)
53 \( 1 + (0.841 - 0.540i)T \)
59 \( 1 + T \)
61 \( 1 + (0.654 - 0.755i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (-0.415 - 0.909i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (-0.142 - 0.989i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−17.17093655896484081532707788061, −16.9141618025556835846955220369, −16.205912002356897628834592437344, −15.430644730185862282249288778630, −15.020274593987661602158380055422, −14.044247938072658561175900080611, −13.13393714751786721041837617840, −12.92687227976585525601785500592, −12.24583530637674366002528090263, −11.667187061023231309953158165723, −11.314145055696957687563158828868, −10.23606543031642131591288258032, −9.91155061528432545033118347764, −8.87304142691652553750820242735, −8.00223279130147459947706655235, −7.07297404324652469800875396778, −6.66891798083603114299386724341, −5.792495493466502896851934143448, −5.370521106202155549811540343265, −4.36254872416645203120497789908, −3.946571993371991157510907825074, −3.052723505836478156078163735717, −2.25097206106227765952175020283, −1.09318147606050690666412602211, −0.643849140000961955694326523850, 0.58933451155154895990390125781, 2.14951123937349829248017256340, 3.11968617034664271806026765405, 3.63580945867516823489840832315, 4.45123235701459129223014352104, 4.72262935726745981397893212832, 5.83820395884599769477774389697, 6.51972396780264532057937610952, 6.9581287755096046614142805006, 7.325774673695080604961879333575, 8.692314241527143726500291852312, 9.16939419367864809904972690765, 10.07892416556171625418175645357, 10.90822840040954239674386143278, 11.4542517780874791026306178226, 12.037484008584972690206378049407, 12.59825380401128840324341201430, 13.16773135226125213294419166423, 14.364272917118298483158125006586, 14.681907434858278440342132352466, 15.34167569590394502557335871854, 15.9793117001659503158355256317, 16.42716198417914700326862226534, 17.07002220889567277198336252036, 17.55859260641522575627341491311

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