↑

Properties

Label 1-6017-6017.314-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.984 - 0.173i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.966 + 0.254i)2-s + (−0.309 − 0.951i)3-s + (0.870 + 0.493i)4-s + (0.554 − 0.832i)5-s + (−0.0563 − 0.998i)6-s + (0.457 − 0.889i)7-s + (0.715 + 0.698i)8-s + (−0.809 + 0.587i)9-s + (0.748 − 0.663i)10-s + (0.200 − 0.979i)12-s + (0.104 + 0.994i)13-s + (0.669 − 0.743i)14-s + (−0.962 − 0.270i)15-s + (0.513 + 0.857i)16-s + (−0.789 + 0.613i)17-s + (−0.932 + 0.362i)18-s + ⋯
L(s)  = 1  + (0.966 + 0.254i)2-s + (−0.309 − 0.951i)3-s + (0.870 + 0.493i)4-s + (0.554 − 0.832i)5-s + (−0.0563 − 0.998i)6-s + (0.457 − 0.889i)7-s + (0.715 + 0.698i)8-s + (−0.809 + 0.587i)9-s + (0.748 − 0.663i)10-s + (0.200 − 0.979i)12-s + (0.104 + 0.994i)13-s + (0.669 − 0.743i)14-s + (−0.962 − 0.270i)15-s + (0.513 + 0.857i)16-s + (−0.789 + 0.613i)17-s + (−0.932 + 0.362i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (314, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.782632021 - 0.3308372737i\)
\(L(\frac12)\) \(\approx\) \(3.782632021 - 0.3308372737i\)
\(L(1)\) \(\approx\) \(1.991202150 - 0.3048447943i\)
\(L(1)\) \(\approx\) \(1.991202150 - 0.3048447943i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.966 + 0.254i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.554 - 0.832i)T \)
7 \( 1 + (0.457 - 0.889i)T \)
13 \( 1 + (0.104 + 0.994i)T \)
17 \( 1 + (-0.789 + 0.613i)T \)
19 \( 1 + (0.657 + 0.753i)T \)
23 \( 1 + (-0.428 + 0.903i)T \)
29 \( 1 + (0.607 + 0.794i)T \)
31 \( 1 + (0.681 + 0.732i)T \)
37 \( 1 + (-0.991 + 0.128i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (0.948 - 0.316i)T \)
47 \( 1 + (-0.231 - 0.972i)T \)
53 \( 1 + (0.369 - 0.929i)T \)
59 \( 1 + (0.657 - 0.753i)T \)
61 \( 1 + (0.0563 + 0.998i)T \)
67 \( 1 + (0.948 - 0.316i)T \)
71 \( 1 + (0.293 + 0.955i)T \)
73 \( 1 + (-0.937 - 0.347i)T \)
79 \( 1 + (0.995 + 0.0965i)T \)
83 \( 1 + (0.997 + 0.0643i)T \)
89 \( 1 + (0.354 + 0.935i)T \)
97 \( 1 + (-0.984 - 0.176i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.708962833838607449283592520516, −17.129954824588822952714308932461, −15.94024606458616425736172035762, −15.6503041738413909587188171353, −15.1226783161105500303773154907, −14.52755318325771795209795737885, −13.87189430839625544368498350091, −13.307709875111359230942659840018, −12.28849384683503445872132750286, −11.75814558856036749252375108911, −11.1005004206648361234944683851, −10.65500828192411786825839864086, −9.931005104638560600711408495195, −9.36810355110100305646313691405, −8.446325546500499112337581566526, −7.51991459319429762716615278828, −6.503557118093371633458354622773, −6.09915177407574125147876719755, −5.368927910889266188815680483360, −4.85767880946228106255352816118, −4.138281028214274117823079730995, −3.09094791190670595051425624441, −2.726787550700723646162674810300, −2.0891420489443918737471060897, −0.719702491736852695268966931322, 1.01860112884125688454695218894, 1.739116316992600698738891883656, 2.09192059685091300494477543914, 3.40938922312036982266459457093, 4.08515798867781504808729203520, 4.97262721757855195784786662111, 5.348830491428932316621076799042, 6.231715825799246082975064688673, 6.83633887922471259027923317252, 7.3213607924606192792426248937, 8.37580904442692584272219407652, 8.52760883595773739165252188470, 9.87169587273217252113678930796, 10.62348444049649781683709788768, 11.36708836946272866903299295598, 12.02608452524023507515577848102, 12.42744784689148050708725996246, 13.30280872526573557110068465953, 13.797626970292507789190222533884, 13.97645617000974106071148101295, 14.82805269469553599879456723613, 15.95043877919118519807231052638, 16.37649156458674930623133732830, 17.02537280842817449630189906769, 17.58665523223910445819625668336

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