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Properties

Label 1-6027-6027.3680-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.474 + 0.880i$
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  + (−0.575 − 0.817i)2-s + (−0.337 + 0.941i)4-s + (0.887 − 0.460i)5-s + (0.963 − 0.266i)8-s + (−0.887 − 0.460i)10-s + (0.193 + 0.981i)11-s + (−0.995 + 0.0896i)13-s + (−0.772 − 0.635i)16-s + (0.646 − 0.762i)17-s + (−0.978 + 0.207i)19-s + (0.134 + 0.990i)20-s + (0.691 − 0.722i)22-s + (0.925 − 0.379i)23-s + (0.575 − 0.817i)25-s + (0.646 + 0.762i)26-s + ⋯
L(s)  = 1  + (−0.575 − 0.817i)2-s + (−0.337 + 0.941i)4-s + (0.887 − 0.460i)5-s + (0.963 − 0.266i)8-s + (−0.887 − 0.460i)10-s + (0.193 + 0.981i)11-s + (−0.995 + 0.0896i)13-s + (−0.772 − 0.635i)16-s + (0.646 − 0.762i)17-s + (−0.978 + 0.207i)19-s + (0.134 + 0.990i)20-s + (0.691 − 0.722i)22-s + (0.925 − 0.379i)23-s + (0.575 − 0.817i)25-s + (0.646 + 0.762i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.474 + 0.880i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (3680, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.474 + 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03805979190 + 0.06374309145i\)
\(L(\frac12)\) \(\approx\) \(0.03805979190 + 0.06374309145i\)
\(L(1)\) \(\approx\) \(0.7016160990 - 0.2566357183i\)
\(L(1)\) \(\approx\) \(0.7016160990 - 0.2566357183i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.575 - 0.817i)T \)
5 \( 1 + (0.887 - 0.460i)T \)
11 \( 1 + (0.193 + 0.981i)T \)
13 \( 1 + (-0.995 + 0.0896i)T \)
17 \( 1 + (0.646 - 0.762i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.925 - 0.379i)T \)
29 \( 1 + (0.473 - 0.880i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (-0.999 + 0.0299i)T \)
43 \( 1 + (-0.550 + 0.834i)T \)
47 \( 1 + (-0.575 - 0.817i)T \)
53 \( 1 + (-0.337 + 0.941i)T \)
59 \( 1 + (-0.447 - 0.894i)T \)
61 \( 1 + (-0.791 + 0.611i)T \)
67 \( 1 + (0.104 + 0.994i)T \)
71 \( 1 + (0.473 + 0.880i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.420 - 0.907i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−17.39178201704562969582300549912, −16.811939911689959052768811479191, −16.54370884182251321893580386651, −15.40363418353199553519943476909, −14.907867145639628573352087868508, −14.35982817349621217372701497205, −13.80315698908416387721810501284, −13.06432670406679675604640123776, −12.38220540366675937825442261148, −11.19852187111374597177505461214, −10.67061797599120741958067742297, −10.178593400646001844120277340914, −9.33445039860505102655384864244, −8.89890975713728229102678589076, −8.118139902653113344560899000753, −7.34895353864322582631525658555, −6.65454175892966331039164955956, −6.16104355853378157630610845982, −5.35453707575991050305499207592, −4.94300601776216243921219895495, −3.72484842906942938358406235658, −2.91050901046731956650209575543, −1.91546744836603073161781486320, −1.29026928917456857435372066902, −0.022193193597226733699435979404, 1.170163792262189491317428632009, 1.859395647505747594616123909013, 2.49305084356074206199477251450, 3.20401096333411293224406442808, 4.39174757047730709897497116362, 4.76108896767736289097428224039, 5.5566562595845941998143753978, 6.720959926866919862056994206989, 7.18473412618697982442410103057, 8.06876698469345757749670676577, 8.81818594397940622496078907830, 9.41290800723913598998266807261, 9.97717907183969221122810779981, 10.38335912330608978904653621501, 11.31193281125488262915594479239, 12.12499148583719586204068390290, 12.54585228878529650967607568723, 13.06446776060621907089744031801, 13.91019920672003755838128181231, 14.50924232627548718160234820967, 15.2696045225294314151658120280, 16.3613237349161512751026356543, 16.81300711649526227732243801719, 17.401237884858590486611089127982, 17.80276039724414289132910482464

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