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Properties

Label 1-6017-6017.8-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.791 - 0.610i$
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  + (−0.341 + 0.939i)2-s + (0.0448 − 0.998i)3-s + (−0.766 − 0.642i)4-s + (0.405 + 0.914i)5-s + (0.923 + 0.383i)6-s + (−0.503 − 0.863i)7-s + (0.865 − 0.500i)8-s + (−0.995 − 0.0896i)9-s + (−0.997 + 0.0689i)10-s + (−0.675 + 0.736i)12-s + (0.691 − 0.722i)13-s + (0.983 − 0.178i)14-s + (0.931 − 0.364i)15-s + (0.175 + 0.984i)16-s + (−0.982 − 0.185i)17-s + (0.424 − 0.905i)18-s + ⋯
L(s)  = 1  + (−0.341 + 0.939i)2-s + (0.0448 − 0.998i)3-s + (−0.766 − 0.642i)4-s + (0.405 + 0.914i)5-s + (0.923 + 0.383i)6-s + (−0.503 − 0.863i)7-s + (0.865 − 0.500i)8-s + (−0.995 − 0.0896i)9-s + (−0.997 + 0.0689i)10-s + (−0.675 + 0.736i)12-s + (0.691 − 0.722i)13-s + (0.983 − 0.178i)14-s + (0.931 − 0.364i)15-s + (0.175 + 0.984i)16-s + (−0.982 − 0.185i)17-s + (0.424 − 0.905i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.791 - 0.610i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.791 - 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9643132106 - 0.3287277998i\)
\(L(\frac12)\) \(\approx\) \(0.9643132106 - 0.3287277998i\)
\(L(1)\) \(\approx\) \(0.7895779063 + 0.05327836549i\)
\(L(1)\) \(\approx\) \(0.7895779063 + 0.05327836549i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.341 + 0.939i)T \)
3 \( 1 + (0.0448 - 0.998i)T \)
5 \( 1 + (0.405 + 0.914i)T \)
7 \( 1 + (-0.503 - 0.863i)T \)
13 \( 1 + (0.691 - 0.722i)T \)
17 \( 1 + (-0.982 - 0.185i)T \)
19 \( 1 + (0.515 + 0.856i)T \)
23 \( 1 + (-0.770 - 0.636i)T \)
29 \( 1 + (0.573 - 0.819i)T \)
31 \( 1 + (-0.975 + 0.219i)T \)
37 \( 1 + (-0.161 + 0.986i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.289 + 0.957i)T \)
47 \( 1 + (0.644 - 0.764i)T \)
53 \( 1 + (0.634 + 0.773i)T \)
59 \( 1 + (-0.836 + 0.548i)T \)
61 \( 1 + (0.998 - 0.0552i)T \)
67 \( 1 + (0.675 - 0.736i)T \)
71 \( 1 + (0.783 + 0.620i)T \)
73 \( 1 + (0.0586 - 0.998i)T \)
79 \( 1 + (0.821 + 0.570i)T \)
83 \( 1 + (0.981 - 0.192i)T \)
89 \( 1 + (-0.256 - 0.966i)T \)
97 \( 1 + (0.996 - 0.0827i)T \)
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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

−17.786502278994589166269031062458, −17.21832139912318319617614717106, −16.39630869807405324260898076798, −15.95632461781301023396242487125, −15.449904968357995396967929671486, −14.31078730376072791920988513352, −13.70855270241609490058185264942, −13.138580197238690615720226943230, −12.35684896476821188630599293680, −11.77105942805825329769190524299, −11.138449706525955518565354061742, −10.48479413645193651945794025886, −9.57344787985780071965466540509, −9.28658359658213669466037234696, −8.722738791060588455152281999598, −8.33581418673666706457845482161, −7.05044246158053774234045007684, −6.013309490824696109845485264876, −5.286641620109818358984454807982, −4.79299007372809964097618672591, −3.850332796919761654378210730417, −3.484173893871621138451789018813, −2.273974705000139538374590746377, −1.99816062720246638993270534724, −0.69345840983842664063111942147, 0.41897372644410233035421944745, 1.32718890058580704012735473635, 2.17118606768238671271130630886, 3.200227332594921910592470330921, 3.8316046157276410879833660266, 4.92735201641089085695416203081, 5.93712570014214466237935990289, 6.27479368015677296251659810641, 6.827460226898926677765214613030, 7.496184886819854276103202581752, 8.0475905913182521307873535297, 8.72996636560429365369737092942, 9.68581542374250338979978488355, 10.269554575963492218113016035776, 10.83223314976496301032193940742, 11.65543323055603260258002683129, 12.70431018925775783677964867433, 13.33212459684330237481391147953, 13.81825098746171426209577175769, 14.18161492246984412891806404587, 15.05735226401919540541937153252, 15.62767759773129028999583885528, 16.5431882575033791841798973514, 17.001998903243205062596267691329, 17.793271558387917613046981663893

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