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Properties

Label 1-6017-6017.1682-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.512 + 0.858i$
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  + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s − 10-s + (0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s − 10-s + (0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.512 + 0.858i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1682, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.512 + 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09272272492 - 0.05265601037i\)
\(L(\frac12)\) \(\approx\) \(-0.09272272492 - 0.05265601037i\)
\(L(1)\) \(\approx\) \(0.4106353128 - 0.3448340938i\)
\(L(1)\) \(\approx\) \(0.4106353128 - 0.3448340938i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 - T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−18.04946067472486715907440470932, −17.83835450291172766640183491579, −16.8450988299459026132606448765, −16.4650479404960378724932598455, −15.63530329550931318533092377039, −15.08231697339301456362561058022, −14.76795109382425497203718140217, −13.59516414192169070377622287523, −13.086651527942366549206039639976, −12.479197991792292699329132339955, −11.28297555049287943411028594253, −11.01576555830973181893418592156, −10.16315048973151711514202485142, −9.69052132259008129963872641693, −9.056243997928850534026711598795, −8.05124389279055720116587222484, −7.420930252215171186599610202872, −6.652636478129344882823404933064, −6.06379283898569685296544529831, −5.655707010281018327059621352588, −5.17047888770901062358118328029, −3.93203673557801742035049734972, −3.21704389167922736314001926407, −1.93651326951883940456556413919, −1.377493532592216260261135611933, 0.04873592928925056438074863653, 0.80028655378390732910588863648, 1.49945910538819076017522205822, 2.29517107605440987672315076431, 3.43116221536946057223152779346, 4.1913801066188745298519351669, 4.77459419133976061228444426886, 5.36824438456097254678229886902, 6.55231842071356452389338447636, 6.947909423027686059492744413029, 7.77136787842963173827566556866, 8.83424160260861953729021265913, 9.34044148937807962799773216475, 9.8289465713511979858427426843, 10.64126874651788392378366897218, 11.20882002254043664728112876083, 11.71698368063134816955023164058, 12.535699443862490669101306578910, 13.12955828163303165654171893004, 13.48113760411832186494304632853, 14.192419744661279386091393585741, 15.62110582640924625294691354865, 16.22890378938079911451590356741, 16.7691429594422388466758831512, 17.05418574741639423854899184603

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