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Properties

Label 1-6017-6017.2229-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.821 + 0.570i$
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  + (0.669 − 0.743i)2-s + (0.809 + 0.587i)3-s + (−0.104 − 0.994i)4-s + (−0.669 − 0.743i)5-s + (0.978 − 0.207i)6-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s − 10-s + (0.5 − 0.866i)12-s + (−0.669 + 0.743i)13-s + (0.913 − 0.406i)14-s + (−0.104 − 0.994i)15-s + (−0.978 + 0.207i)16-s + (−0.978 + 0.207i)17-s + (0.913 + 0.406i)18-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (0.809 + 0.587i)3-s + (−0.104 − 0.994i)4-s + (−0.669 − 0.743i)5-s + (0.978 − 0.207i)6-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s − 10-s + (0.5 − 0.866i)12-s + (−0.669 + 0.743i)13-s + (0.913 − 0.406i)14-s + (−0.104 − 0.994i)15-s + (−0.978 + 0.207i)16-s + (−0.978 + 0.207i)17-s + (0.913 + 0.406i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.821 + 0.570i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (2229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.821 + 0.570i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1134066550 - 0.3618740199i\)
\(L(\frac12)\) \(\approx\) \(-0.1134066550 - 0.3618740199i\)
\(L(1)\) \(\approx\) \(1.293408447 - 0.4806246132i\)
\(L(1)\) \(\approx\) \(1.293408447 - 0.4806246132i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.669 - 0.743i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.669 - 0.743i)T \)
7 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.669 + 0.743i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.913 - 0.406i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (-0.913 - 0.406i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.978 - 0.207i)T \)
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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

−17.75705138250262727226722627712, −17.71482843693403104167455947206, −16.84738548033568461761369967356, −15.65705178772269903431465495089, −15.34898805672578400036158202200, −14.873333882334037761629480116196, −14.11770198450159437548264710041, −13.825859137460523390889794206984, −12.97098438876573696268699350087, −12.346409495624487261949118942533, −11.72272416346610778314173377590, −10.94234468695950406754934009234, −10.287698329778392621554239736922, −8.9621014308846112863549949411, −8.58820329796879868012364157689, −7.846073285362570608366190681234, −7.18851195878529903862021092035, −7.017112421762088164498377955210, −6.120394402588030379304160740302, −5.05833260915787894755783162617, −4.44510860349959897163520771756, −3.73645703569200742777911799068, −2.93149315610226772828431211882, −2.475120328205662355516641544583, −1.380301148982284764297443466765, 0.058087624329690760481379912672, 1.46470821816975598849922419910, 2.12448143859923228338433778098, 2.65021898397660651922232562087, 3.79260088668048231581119236010, 4.295724131380655250179389603388, 4.72641587323110711778243561089, 5.31646880769446110240511950021, 6.38906596324020683950145309347, 7.32089394077447647082457318479, 8.25814123708479015685226141500, 8.74231448452113516520714789529, 9.23494758969077401285378666303, 10.10601478834801456938259451262, 10.84546228412945171077025555474, 11.36428166457286522744797847645, 12.14320705879992021399015707071, 12.61150124483813594458215822715, 13.456412950265108212903399454509, 14.01374786968311593474881678705, 14.779543663551784140697650551424, 15.20009040561856311385915586685, 15.612778551457285821683915644400, 16.6435323821371118914973389294, 17.11624383173094003172129204829

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