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Properties

Label 1-6017-6017.877-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.254 - 0.967i$
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  + (−0.962 − 0.270i)2-s + (0.809 − 0.587i)3-s + (0.853 + 0.520i)4-s + (0.324 − 0.945i)5-s + (−0.937 + 0.347i)6-s + (0.369 + 0.929i)7-s + (−0.681 − 0.732i)8-s + (0.309 − 0.951i)9-s + (−0.568 + 0.822i)10-s + (0.996 − 0.0804i)12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (−0.293 − 0.955i)15-s + (0.457 + 0.889i)16-s + (−0.993 + 0.112i)17-s + (−0.554 + 0.832i)18-s + ⋯
L(s)  = 1  + (−0.962 − 0.270i)2-s + (0.809 − 0.587i)3-s + (0.853 + 0.520i)4-s + (0.324 − 0.945i)5-s + (−0.937 + 0.347i)6-s + (0.369 + 0.929i)7-s + (−0.681 − 0.732i)8-s + (0.309 − 0.951i)9-s + (−0.568 + 0.822i)10-s + (0.996 − 0.0804i)12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (−0.293 − 0.955i)15-s + (0.457 + 0.889i)16-s + (−0.993 + 0.112i)17-s + (−0.554 + 0.832i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.254 - 0.967i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (877, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.254 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.108052824 - 1.437176338i\)
\(L(\frac12)\) \(\approx\) \(1.108052824 - 1.437176338i\)
\(L(1)\) \(\approx\) \(0.9593714241 - 0.4688142228i\)
\(L(1)\) \(\approx\) \(0.9593714241 - 0.4688142228i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.962 - 0.270i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.324 - 0.945i)T \)
7 \( 1 + (0.369 + 0.929i)T \)
13 \( 1 + (0.978 - 0.207i)T \)
17 \( 1 + (-0.993 + 0.112i)T \)
19 \( 1 + (0.892 - 0.450i)T \)
23 \( 1 + (-0.692 + 0.721i)T \)
29 \( 1 + (-0.926 - 0.377i)T \)
31 \( 1 + (-0.779 - 0.626i)T \)
37 \( 1 + (0.136 + 0.990i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (0.428 - 0.903i)T \)
47 \( 1 + (0.339 + 0.940i)T \)
53 \( 1 + (0.0884 + 0.996i)T \)
59 \( 1 + (0.892 + 0.450i)T \)
61 \( 1 + (0.937 - 0.347i)T \)
67 \( 1 + (0.428 - 0.903i)T \)
71 \( 1 + (0.932 + 0.362i)T \)
73 \( 1 + (0.999 - 0.0161i)T \)
79 \( 1 + (0.958 - 0.285i)T \)
83 \( 1 + (-0.657 - 0.753i)T \)
89 \( 1 + (-0.885 + 0.464i)T \)
97 \( 1 + (-0.00805 - 0.999i)T \)
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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

−18.14564276573734349023128283633, −17.3392430112647077394533453736, −16.32022097072245807897863630596, −16.17096500978560970877286925834, −15.36150989140122957910402241804, −14.50136830381721100105754288948, −14.30752129901317895540611021219, −13.64414545360241961785496302493, −12.80318105174882504194023063971, −11.38591506883620513875132034640, −11.0596682640349577737185163435, −10.53134766026604486027301073540, −9.88372163778447633742431509110, −9.30097859959447801915756155866, −8.56944411374559526668397083521, −7.91989541326953775356670894585, −7.22829592780994063050300600735, −6.77796876456172407362408067493, −5.84836598525872103574832016387, −5.074082280443906831675424674799, −3.80993484791773097761181025496, −3.611834034428151649702903577808, −2.35580661639982498164584375532, −2.0065651086185616178043828662, −0.96597370990579124048941206794, 0.60897827377704788954939899833, 1.45867932466574412260961756775, 2.0049505904562252672076954244, 2.634164324607234266270855610862, 3.57360872136755121103470192175, 4.32086997786909168623498067566, 5.61951334946157312368711336182, 6.01104852082468307412240239781, 6.960735555222576850149652568758, 7.83250197519418309974383527821, 8.21488728684102031523666981756, 8.929276918442003756696856999677, 9.30988394891280771063649693612, 9.808480091504007581445550623000, 11.08259198083106522437623107112, 11.50245964271900316710601733197, 12.28494935616115039035568797229, 12.85748024734050084763278143191, 13.43061422398358385182211762212, 14.105654862329153672262298841880, 15.30981364389345856903074914963, 15.487944524075647866352306973718, 16.18364160591275264845815329357, 17.108740132883818315956619720017, 17.76786594047428355654756586868

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