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Properties

Label 1-6027-6027.20-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.978 + 0.207i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (−0.0448 − 0.998i)2-s + (−0.995 + 0.0896i)4-s + (0.691 + 0.722i)5-s + (0.134 + 0.990i)8-s + (0.691 − 0.722i)10-s + (0.351 + 0.936i)11-s + (0.998 − 0.0448i)13-s + (0.983 − 0.178i)16-s + (0.0896 − 0.995i)17-s + (0.587 + 0.809i)19-s + (−0.753 − 0.657i)20-s + (0.919 − 0.393i)22-s + (−0.753 + 0.657i)23-s + (−0.0448 + 0.998i)25-s + (−0.0896 − 0.995i)26-s + ⋯
L(s)  = 1  + (−0.0448 − 0.998i)2-s + (−0.995 + 0.0896i)4-s + (0.691 + 0.722i)5-s + (0.134 + 0.990i)8-s + (0.691 − 0.722i)10-s + (0.351 + 0.936i)11-s + (0.998 − 0.0448i)13-s + (0.983 − 0.178i)16-s + (0.0896 − 0.995i)17-s + (0.587 + 0.809i)19-s + (−0.753 − 0.657i)20-s + (0.919 − 0.393i)22-s + (−0.753 + 0.657i)23-s + (−0.0448 + 0.998i)25-s + (−0.0896 − 0.995i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.978 + 0.207i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.978 + 0.207i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.069237945 + 0.2165813059i\)
\(L(\frac12)\) \(\approx\) \(2.069237945 + 0.2165813059i\)
\(L(1)\) \(\approx\) \(1.180122100 - 0.2122686563i\)
\(L(1)\) \(\approx\) \(1.180122100 - 0.2122686563i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.0448 - 0.998i)T \)
5 \( 1 + (0.691 + 0.722i)T \)
11 \( 1 + (0.351 + 0.936i)T \)
13 \( 1 + (0.998 - 0.0448i)T \)
17 \( 1 + (0.0896 - 0.995i)T \)
19 \( 1 + (0.587 + 0.809i)T \)
23 \( 1 + (-0.753 + 0.657i)T \)
29 \( 1 + (0.512 + 0.858i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.858 - 0.512i)T \)
43 \( 1 + (-0.473 - 0.880i)T \)
47 \( 1 + (0.998 - 0.0448i)T \)
53 \( 1 + (-0.0896 - 0.995i)T \)
59 \( 1 + (0.473 + 0.880i)T \)
61 \( 1 + (0.753 + 0.657i)T \)
67 \( 1 + (-0.951 + 0.309i)T \)
71 \( 1 + (-0.512 + 0.858i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (0.998 + 0.0448i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−17.46709714823739894018769248613, −16.98146152979896900960961488776, −16.3868443134886191970029294562, −15.87812180172301578944927427264, −15.18394062995112891329331707877, −14.3434786810285434222005096791, −13.75116427180367258107656614140, −13.325492082395634583839158805257, −12.7358255761479641966686031930, −11.84573520547577346301024397164, −10.99739034463820079134964573970, −10.1871405370781100872153860822, −9.465735034925736873882308422337, −8.92744762527795720047451239878, −8.2364548669984804040434210330, −7.88330494860809566201817149934, −6.62567661514587225477416617087, −6.09783197045536302591674939349, −5.79510927622901819306824080549, −4.83047418232395241213746484421, −4.17434565811612538823809173253, −3.47899214154557095674920905371, −2.33475600174002490332359721770, −1.21818854406819604474273008904, −0.65553520542520349867829799872, 1.01488640322215398399615134139, 1.719077436755572347375365389320, 2.3536046213259343166820441509, 3.2900921158451473465553512489, 3.70564231348765329741024904074, 4.68971162053593110619518846173, 5.46484789999706754694457719758, 6.067936615825026440075445202854, 7.093150222463196975077535172084, 7.60565579638929352213473264701, 8.718969815242878886692779841677, 9.195413319079928950534371430057, 10.065555102867868502264586645037, 10.255108278914889177821033184936, 11.12357633473337399366382286665, 11.80225704856416572531985065229, 12.2816329050349140663000662586, 13.23511523474120172156680999931, 13.66662507156954167266608112626, 14.3768401929506349223178189442, 14.74175555100144152150154555627, 15.85318300326929254487087824542, 16.45955828619530034084929576276, 17.469176831400648741592634535734, 17.991423337062581255317413503719

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