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Properties

Label 1-6017-6017.4336-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.413 - 0.910i$
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  + (0.913 − 0.406i)2-s + (−0.309 + 0.951i)3-s + (0.669 − 0.743i)4-s + (−0.913 − 0.406i)5-s + (0.104 + 0.994i)6-s + (−0.978 − 0.207i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s + (0.5 + 0.866i)12-s + (−0.913 + 0.406i)13-s + (−0.978 + 0.207i)14-s + (0.669 − 0.743i)15-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (−0.978 − 0.207i)18-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (−0.309 + 0.951i)3-s + (0.669 − 0.743i)4-s + (−0.913 − 0.406i)5-s + (0.104 + 0.994i)6-s + (−0.978 − 0.207i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s + (0.5 + 0.866i)12-s + (−0.913 + 0.406i)13-s + (−0.978 + 0.207i)14-s + (0.669 − 0.743i)15-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (−0.978 − 0.207i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.413 - 0.910i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (4336, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.413 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7130952609 - 1.107420259i\)
\(L(\frac12)\) \(\approx\) \(0.7130952609 - 1.107420259i\)
\(L(1)\) \(\approx\) \(1.110683770 - 0.2459580167i\)
\(L(1)\) \(\approx\) \(1.110683770 - 0.2459580167i\)

Euler product

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

−17.98446177031895266364258813689, −16.94989459787658108309841988245, −16.64447549337114007380476851107, −15.91728071948028887800670443469, −15.135316749240898389515980866311, −14.730854917013639161656389668134, −13.96391260974415188761635852348, −13.16297914574230212865795014482, −12.66130167330429581530427931160, −12.18428140487775565828686444455, −11.67790276788190195021314815570, −10.84781993136446067232619762691, −10.24444169964455542595086224058, −9.02024955021941082120386500710, −8.17610250394578315916419103211, −7.66959560450951329349715913051, −6.947729714701180062473554762460, −6.60710219161855489194218088107, −5.76880428721020906919869469692, −5.182658751658705760585427509943, −4.220214125753991420754232252954, −3.442870295234906066737950025853, −2.78101650785414427518122977734, −2.23495964704170300771983259270, −0.87963703956765627373522054514, 0.31703276686059563882677405885, 1.18558729992863028329724495207, 2.85561503143797549104341416706, 2.93043458195770190599933867901, 3.92240471266252537927547002269, 4.437155079208716521292091786382, 5.06525477083131364446848595388, 5.66209038986456448386582869530, 6.66214222776292217913438832227, 7.16168104505744704062531553365, 8.05530225059888085193204351046, 9.3160529465490396473863607845, 9.60515900591226598149410188905, 10.14383591605432360301482733849, 11.29902715661158062740637310223, 11.51687126957816292370006776106, 12.05295919526402030081648140382, 12.90099012309604350036085233778, 13.46924217467131443150668763669, 14.30797003479623028914821174208, 14.965583818299211714978049149219, 15.567882338505928031151896888821, 16.14603002632421586307235242512, 16.41549683708002620965562469424, 17.23004800488544746704445399499

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