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Properties

Label 1-6017-6017.224-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.947 + 0.320i$
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  + (−0.800 + 0.598i)2-s + (−0.963 + 0.266i)3-s + (0.282 − 0.959i)4-s + (0.269 + 0.963i)5-s + (0.612 − 0.790i)6-s + (0.147 − 0.989i)7-s + (0.348 + 0.937i)8-s + (0.858 − 0.512i)9-s + (−0.792 − 0.609i)10-s + (−0.0172 + 0.999i)12-s + (0.134 − 0.990i)13-s + (0.473 + 0.880i)14-s + (−0.515 − 0.856i)15-s + (−0.840 − 0.542i)16-s + (−0.492 − 0.870i)17-s + (−0.380 + 0.924i)18-s + ⋯
L(s)  = 1  + (−0.800 + 0.598i)2-s + (−0.963 + 0.266i)3-s + (0.282 − 0.959i)4-s + (0.269 + 0.963i)5-s + (0.612 − 0.790i)6-s + (0.147 − 0.989i)7-s + (0.348 + 0.937i)8-s + (0.858 − 0.512i)9-s + (−0.792 − 0.609i)10-s + (−0.0172 + 0.999i)12-s + (0.134 − 0.990i)13-s + (0.473 + 0.880i)14-s + (−0.515 − 0.856i)15-s + (−0.840 − 0.542i)16-s + (−0.492 − 0.870i)17-s + (−0.380 + 0.924i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.947 + 0.320i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.947 + 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05818552895 + 0.3537694454i\)
\(L(\frac12)\) \(\approx\) \(0.05818552895 + 0.3537694454i\)
\(L(1)\) \(\approx\) \(0.5126647612 + 0.1725460784i\)
\(L(1)\) \(\approx\) \(0.5126647612 + 0.1725460784i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.800 + 0.598i)T \)
3 \( 1 + (-0.963 + 0.266i)T \)
5 \( 1 + (0.269 + 0.963i)T \)
7 \( 1 + (0.147 - 0.989i)T \)
13 \( 1 + (0.134 - 0.990i)T \)
17 \( 1 + (-0.492 - 0.870i)T \)
19 \( 1 + (0.828 + 0.559i)T \)
23 \( 1 + (0.962 - 0.272i)T \)
29 \( 1 + (-0.315 + 0.948i)T \)
31 \( 1 + (0.665 + 0.746i)T \)
37 \( 1 + (-0.127 - 0.991i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.418 - 0.908i)T \)
47 \( 1 + (0.215 + 0.976i)T \)
53 \( 1 + (-0.974 + 0.225i)T \)
59 \( 1 + (-0.989 + 0.144i)T \)
61 \( 1 + (-0.209 + 0.977i)T \)
67 \( 1 + (-0.0172 + 0.999i)T \)
71 \( 1 + (-0.923 + 0.383i)T \)
73 \( 1 + (0.655 + 0.755i)T \)
79 \( 1 + (0.734 - 0.678i)T \)
83 \( 1 + (-0.998 + 0.0483i)T \)
89 \( 1 + (-0.994 - 0.103i)T \)
97 \( 1 + (0.891 - 0.452i)T \)
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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

−17.31227787818573963443566088369, −16.98936479270005259295451384940, −16.45101255904630420865229080940, −15.56208456268190368294978417679, −15.276440749895146218567118739685, −13.67700727397369204667621281190, −13.28265830981795804191419727340, −12.5717109301861925896563809636, −12.00760367484431767097694046467, −11.427677762011909071337918111677, −11.10803042948788248335456906181, −9.893963813404163542427952510542, −9.568242885031163981726165731742, −8.81674910647032675197398130096, −8.21067417180641404506595090156, −7.493309241701815023552049173332, −6.4158766224177851952832092852, −6.16649965925278113673855301740, −4.91016697012047070302521555455, −4.71171007884840179518343364477, −3.615425363899996488945679261225, −2.51698695353432722368897975287, −1.70623023920127814167967470462, −1.32875361026479241247907975468, −0.16706149521073132448405232593, 0.89988941512950640572013199315, 1.541444135844198190939912850995, 2.80165629098036871632175588963, 3.55914409470906100468508431262, 4.65628221999151278605761317979, 5.29876739656171987264089885410, 5.912172286045777331007717984484, 6.74104520072197337586922196816, 7.21098731611250673604855247786, 7.59683778552980437277036180465, 8.729852017687568576476435495129, 9.52413839092204646308339040103, 10.24152491864795173864345448413, 10.570842178334200286218629516200, 11.08465454043643641062851845793, 11.73329913498597279378361660967, 12.742983848105981678935259543658, 13.65242649064626702962517704788, 14.19812658140380768663954280055, 14.89077625724059075854967650544, 15.669168058375366065088322651848, 16.02276054262256785802154199583, 16.88256320942677515664888488721, 17.37170331279001606869940999332, 17.90775389929203473939451196404

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