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Properties

Label 1-6017-6017.3871-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.660 - 0.750i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
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  + (−0.988 + 0.149i)2-s + (0.222 − 0.974i)3-s + (0.955 − 0.294i)4-s + (−0.0747 − 0.997i)5-s + (−0.0747 + 0.997i)6-s + (0.955 + 0.294i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (−0.0747 − 0.997i)12-s + (−0.365 − 0.930i)13-s + (−0.988 − 0.149i)14-s + (−0.988 − 0.149i)15-s + (0.826 − 0.563i)16-s + (0.826 − 0.563i)17-s + (0.955 + 0.294i)18-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)2-s + (0.222 − 0.974i)3-s + (0.955 − 0.294i)4-s + (−0.0747 − 0.997i)5-s + (−0.0747 + 0.997i)6-s + (0.955 + 0.294i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (−0.0747 − 0.997i)12-s + (−0.365 − 0.930i)13-s + (−0.988 − 0.149i)14-s + (−0.988 − 0.149i)15-s + (0.826 − 0.563i)16-s + (0.826 − 0.563i)17-s + (0.955 + 0.294i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.660 - 0.750i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (3871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.660 - 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6271523956 - 1.387383085i\)
\(L(\frac12)\) \(\approx\) \(0.6271523956 - 1.387383085i\)
\(L(1)\) \(\approx\) \(0.7649505857 - 0.4754572144i\)
\(L(1)\) \(\approx\) \(0.7649505857 - 0.4754572144i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.988 + 0.149i)T \)
3 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (-0.0747 - 0.997i)T \)
7 \( 1 + (0.955 + 0.294i)T \)
13 \( 1 + (-0.365 - 0.930i)T \)
17 \( 1 + (0.826 - 0.563i)T \)
19 \( 1 + (0.988 + 0.149i)T \)
23 \( 1 + (0.988 + 0.149i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (0.222 - 0.974i)T \)
37 \( 1 + (-0.826 + 0.563i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.955 + 0.294i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.365 + 0.930i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.826 - 0.563i)T \)
67 \( 1 + (0.826 - 0.563i)T \)
71 \( 1 + (-0.826 + 0.563i)T \)
73 \( 1 + (0.988 - 0.149i)T \)
79 \( 1 + (-0.222 - 0.974i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + (-0.988 - 0.149i)T \)
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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

−17.64727450081650509126016635323, −17.55837237651609662787102919582, −16.65299709238723087549123593421, −16.04891579909973588509264124789, −15.462348459239858275340221104957, −14.527787957976192461954052550386, −14.49171488819296789377603566450, −13.67261624055658298668969734731, −12.255965518994339289665259915284, −11.73836223471977870503434141226, −11.07158714807656784223787879164, −10.585175664792927833198667711709, −10.09238723317245813838104906597, −9.376826076932430109199752363282, −8.64598435683847606091964163522, −8.09593533796452180331023755554, −7.18538111151193970161922628468, −6.901700900237639761724241954615, −5.72182324681237503774292041732, −5.05616443569389973661413594681, −4.05169189950659443160807558276, −3.393026116072377671600286853464, −2.70705695885814964547740471267, −1.94073686441231660202439132625, −1.0022208291928459719934321661, 0.66798013182273382555184369974, 1.05539669596873246112971580465, 1.841163199125878331555177601080, 2.67066922813916655447691034037, 3.36012539662136199261061019052, 4.84548479660662091306457395506, 5.395113183425728721831091851565, 5.95180350316928421377533558704, 7.04546078164434107018082379797, 7.57838590166992259770816389880, 8.27810663322920238255213307651, 8.39547693079932723706934748703, 9.45054692737209377370671698132, 9.84530549262116788292377092328, 10.97018349451794351287352306057, 11.68172762510004659681700266689, 12.039993041696682416739320903201, 12.68450222137704664710606667365, 13.550265027511283597617995302903, 14.22132773682154639781383702180, 15.001236869947322257188297606330, 15.527251886844487658408737112904, 16.36581726723780194728283324220, 17.14276699294544317744091706474, 17.44095448018412686437995574534

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