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Properties

Label 1-6017-6017.1115-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.996 - 0.0796i$
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  + (0.457 + 0.889i)2-s + (−0.809 − 0.587i)3-s + (−0.581 + 0.813i)4-s + (0.247 + 0.968i)5-s + (0.152 − 0.988i)6-s + (0.0563 − 0.998i)7-s + (−0.989 − 0.144i)8-s + (0.309 + 0.951i)9-s + (−0.748 + 0.663i)10-s + (0.948 − 0.316i)12-s + (0.669 − 0.743i)13-s + (0.913 − 0.406i)14-s + (0.369 − 0.929i)15-s + (−0.324 − 0.945i)16-s + (0.899 − 0.435i)17-s + (−0.704 + 0.709i)18-s + ⋯
L(s)  = 1  + (0.457 + 0.889i)2-s + (−0.809 − 0.587i)3-s + (−0.581 + 0.813i)4-s + (0.247 + 0.968i)5-s + (0.152 − 0.988i)6-s + (0.0563 − 0.998i)7-s + (−0.989 − 0.144i)8-s + (0.309 + 0.951i)9-s + (−0.748 + 0.663i)10-s + (0.948 − 0.316i)12-s + (0.669 − 0.743i)13-s + (0.913 − 0.406i)14-s + (0.369 − 0.929i)15-s + (−0.324 − 0.945i)16-s + (0.899 − 0.435i)17-s + (−0.704 + 0.709i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.996 - 0.0796i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.996 - 0.0796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.310031564 - 0.05228607770i\)
\(L(\frac12)\) \(\approx\) \(1.310031564 - 0.05228607770i\)
\(L(1)\) \(\approx\) \(0.9317368815 + 0.2939764540i\)
\(L(1)\) \(\approx\) \(0.9317368815 + 0.2939764540i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.457 + 0.889i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.247 + 0.968i)T \)
7 \( 1 + (0.0563 - 0.998i)T \)
13 \( 1 + (0.669 - 0.743i)T \)
17 \( 1 + (0.899 - 0.435i)T \)
19 \( 1 + (-0.293 - 0.955i)T \)
23 \( 1 + (-0.996 + 0.0804i)T \)
29 \( 1 + (0.0241 + 0.999i)T \)
31 \( 1 + (-0.906 + 0.421i)T \)
37 \( 1 + (0.853 - 0.520i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.200 + 0.979i)T \)
47 \( 1 + (0.184 - 0.982i)T \)
53 \( 1 + (0.937 - 0.347i)T \)
59 \( 1 + (-0.293 + 0.955i)T \)
61 \( 1 + (0.152 - 0.988i)T \)
67 \( 1 + (-0.200 + 0.979i)T \)
71 \( 1 + (0.0884 + 0.996i)T \)
73 \( 1 + (0.997 - 0.0643i)T \)
79 \( 1 + (0.399 - 0.916i)T \)
83 \( 1 + (-0.962 - 0.270i)T \)
89 \( 1 + (-0.354 - 0.935i)T \)
97 \( 1 + (0.999 - 0.0322i)T \)
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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

−17.84065698365053160714282453558, −16.91645602063555162284944165332, −16.53730030470845912816083672424, −15.67029570486742572101171701745, −15.19588419621218494967711822344, −14.34092994210463207202323733331, −13.72184076975867187553736071104, −12.7763460878827511323305094469, −12.31818312391219954423545573275, −11.89806219789514864097329623173, −11.296369453323320318339157910330, −10.457769690179754879852394306120, −9.80628773059491630202840643005, −9.31084257205805282249367876533, −8.66294717435794048051409090702, −7.933178348375025694472700520443, −6.3704021342555094129896673910, −5.850588029549632818244343677747, −5.55298398082739005361381155135, −4.69349319171554325666685561979, −4.00576653563037534929713764801, −3.56404747865605719705888489392, −2.23411381997791029205851528088, −1.68488542720920546452510813605, −0.8057399889347864700777660376, 0.41224253834709440909837560498, 1.40960929413683011256636480689, 2.59969648706470363989574375175, 3.37846239062673446755345356190, 4.080720796092038767564079434873, 5.02172260893728900423356503091, 5.61357678718750282411788830411, 6.32757125601208540925344176321, 6.82007482141773756347840267861, 7.52948669667240077216507234040, 7.81417973545466093160137229918, 8.832543592727618960346989705209, 9.96044215909392126137785780122, 10.41014097749760066966366914843, 11.27205582536139651792408784057, 11.673713512840827277763288655898, 12.86765883608542362329352848944, 13.040468391862264618683771587644, 13.87910232633743066137233060723, 14.309420038989511716833769318065, 15.00933052517985538380934143858, 15.91625533858164248471120893920, 16.3952809803587103946809465091, 17.013612930814033229927428934378, 17.73558532931073200382645257042

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