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Properties

Label 1-6004-6004.503-r0-0-0
Degree $1$
Conductor $6004$
Sign $-0.763 + 0.645i$
Analytic cond. $27.8824$
Root an. cond. $27.8824$
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  + (−0.872 + 0.488i)3-s + (0.611 − 0.791i)5-s + (−0.632 − 0.774i)7-s + (0.523 − 0.852i)9-s + (−0.885 + 0.464i)11-s + (0.994 + 0.107i)13-s + (−0.147 + 0.989i)15-s + (−0.830 + 0.556i)17-s + (0.930 + 0.367i)21-s + (0.939 + 0.342i)23-s + (−0.252 − 0.967i)25-s + (−0.0402 + 0.999i)27-s + (0.783 + 0.621i)29-s + (0.919 + 0.391i)31-s + (0.545 − 0.837i)33-s + ⋯
L(s)  = 1  + (−0.872 + 0.488i)3-s + (0.611 − 0.791i)5-s + (−0.632 − 0.774i)7-s + (0.523 − 0.852i)9-s + (−0.885 + 0.464i)11-s + (0.994 + 0.107i)13-s + (−0.147 + 0.989i)15-s + (−0.830 + 0.556i)17-s + (0.930 + 0.367i)21-s + (0.939 + 0.342i)23-s + (−0.252 − 0.967i)25-s + (−0.0402 + 0.999i)27-s + (0.783 + 0.621i)29-s + (0.919 + 0.391i)31-s + (0.545 − 0.837i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
Sign: $-0.763 + 0.645i$
Analytic conductor: \(27.8824\)
Root analytic conductor: \(27.8824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6004} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6004,\ (0:\ ),\ -0.763 + 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09534670411 + 0.2604513533i\)
\(L(\frac12)\) \(\approx\) \(0.09534670411 + 0.2604513533i\)
\(L(1)\) \(\approx\) \(0.7261690427 + 0.007118057149i\)
\(L(1)\) \(\approx\) \(0.7261690427 + 0.007118057149i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
79 \( 1 \)
good3 \( 1 + (-0.872 + 0.488i)T \)
5 \( 1 + (0.611 - 0.791i)T \)
7 \( 1 + (-0.632 - 0.774i)T \)
11 \( 1 + (-0.885 + 0.464i)T \)
13 \( 1 + (0.994 + 0.107i)T \)
17 \( 1 + (-0.830 + 0.556i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.783 + 0.621i)T \)
31 \( 1 + (0.919 + 0.391i)T \)
37 \( 1 + (-0.428 - 0.903i)T \)
41 \( 1 + (-0.611 + 0.791i)T \)
43 \( 1 + (-0.991 - 0.133i)T \)
47 \( 1 + (-0.711 + 0.702i)T \)
53 \( 1 + (-0.0134 - 0.999i)T \)
59 \( 1 + (-0.404 + 0.914i)T \)
61 \( 1 + (0.0938 + 0.995i)T \)
67 \( 1 + (0.452 - 0.891i)T \)
71 \( 1 + (0.329 - 0.944i)T \)
73 \( 1 + (0.994 - 0.107i)T \)
83 \( 1 + (-0.278 + 0.960i)T \)
89 \( 1 + (-0.982 - 0.186i)T \)
97 \( 1 + (-0.252 + 0.967i)T \)
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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

−17.467230173196837081933673877949, −17.01562870482756958404538907814, −15.9792816762543738949214782020, −15.67858845446236155527572728454, −15.05556446072158348106577234303, −13.8046555209896885103222529033, −13.50709150863065469763037098759, −12.983114013255814841044226528251, −12.167779090129011465987324862912, −11.369299917781702140865072374977, −10.99927481058726527008708131464, −10.20472290148578783263528076621, −9.72392811883509807612182859548, −8.64082800175662737907584691522, −8.15756066592624454510738097865, −7.009891271168170713116275643223, −6.56777929803072465131936980214, −6.08655974158356852787216221027, −5.35228358753168771271612638379, −4.792725021980187063015906361357, −3.508193670423977500934748165235, −2.740860298221879102630084550435, −2.2417331175030327771393322352, −1.20297200538371204813454878346, −0.08946195799238220265703765424, 1.01363312642067764939612466039, 1.61812158561653471335982013841, 2.864405265525059180270167609757, 3.70510812590310196800705071073, 4.49113584679613859984436344964, 4.99611708712130277922054940130, 5.69729014810885548322563625925, 6.59707918821803027048443059458, 6.78989691684030039733752185967, 8.038711207281109747525306061408, 8.73519927045099551773154261786, 9.50164946526601032006252266467, 10.053467604408494646825229693006, 10.69915277877399650917980657328, 11.11471365783365811143628506680, 12.20835872113311065139094600989, 12.69838319697135400843475403949, 13.38148361088590385493629916393, 13.68604491011056172207450518863, 14.913437028465825123765962945407, 15.58405399933951081718356351786, 16.15190098109746308844373763804, 16.58723845757680364931956188946, 17.302704382871100940279748514225, 17.83011854770786446221698912118

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