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Properties

Label 1-6017-6017.576-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.213 + 0.976i$
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  + (0.202 − 0.979i)2-s + (−0.963 + 0.266i)3-s + (−0.918 − 0.396i)4-s + (0.945 + 0.325i)5-s + (0.0655 + 0.997i)6-s + (−0.766 + 0.642i)7-s + (−0.573 + 0.819i)8-s + (0.858 − 0.512i)9-s + (0.509 − 0.860i)10-s + (0.990 + 0.137i)12-s + (0.134 − 0.990i)13-s + (0.473 + 0.880i)14-s + (−0.998 − 0.0621i)15-s + (0.686 + 0.727i)16-s + (−0.0310 − 0.999i)17-s + (−0.328 − 0.944i)18-s + ⋯
L(s)  = 1  + (0.202 − 0.979i)2-s + (−0.963 + 0.266i)3-s + (−0.918 − 0.396i)4-s + (0.945 + 0.325i)5-s + (0.0655 + 0.997i)6-s + (−0.766 + 0.642i)7-s + (−0.573 + 0.819i)8-s + (0.858 − 0.512i)9-s + (0.509 − 0.860i)10-s + (0.990 + 0.137i)12-s + (0.134 − 0.990i)13-s + (0.473 + 0.880i)14-s + (−0.998 − 0.0621i)15-s + (0.686 + 0.727i)16-s + (−0.0310 − 0.999i)17-s + (−0.328 − 0.944i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.213 + 0.976i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (576, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.213 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08825171823 + 0.07104829883i\)
\(L(\frac12)\) \(\approx\) \(0.08825171823 + 0.07104829883i\)
\(L(1)\) \(\approx\) \(0.6267402078 - 0.3056557171i\)
\(L(1)\) \(\approx\) \(0.6267402078 - 0.3056557171i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.202 - 0.979i)T \)
3 \( 1 + (-0.963 + 0.266i)T \)
5 \( 1 + (0.945 + 0.325i)T \)
7 \( 1 + (-0.766 + 0.642i)T \)
13 \( 1 + (0.134 - 0.990i)T \)
17 \( 1 + (-0.0310 - 0.999i)T \)
19 \( 1 + (-0.938 - 0.344i)T \)
23 \( 1 + (-0.596 - 0.802i)T \)
29 \( 1 + (-0.775 - 0.631i)T \)
31 \( 1 + (-0.467 - 0.883i)T \)
37 \( 1 + (0.972 + 0.232i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.952 + 0.305i)T \)
47 \( 1 + (-0.989 - 0.144i)T \)
53 \( 1 + (-0.367 + 0.930i)T \)
59 \( 1 + (0.995 + 0.0965i)T \)
61 \( 1 + (-0.492 - 0.870i)T \)
67 \( 1 + (0.990 + 0.137i)T \)
71 \( 1 + (0.804 - 0.593i)T \)
73 \( 1 + (-0.249 + 0.968i)T \)
79 \( 1 + (-0.584 - 0.811i)T \)
83 \( 1 + (-0.527 + 0.849i)T \)
89 \( 1 + (0.675 + 0.736i)T \)
97 \( 1 + (0.999 + 0.0138i)T \)
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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

−17.323211367333482852632726961560, −16.83536955097347313748637326402, −16.447018001772308704306175074836, −15.953798582073851021842004193770, −14.917010734945076974332722559836, −14.28144557321640424453442214053, −13.53032558160648737572915646205, −12.95316167299746359587409421205, −12.73565215829729980527725956446, −11.78380894540558222099316107813, −10.879788130347809550541661177287, −10.04722666371745040980200772060, −9.7169236234937689204370312149, −8.821318526511021408973380006206, −8.103399189222099448245773066901, −7.12166885357096839488464822159, −6.63185364690570723828520389614, −6.15746876702418648489209210221, −5.56420807220311813275023879097, −4.79174546584701944847435642572, −4.09065129180045688709051510586, −3.41987008475478775641318786541, −1.913908437743507191794493642161, −1.3468096209408490055571514375, −0.03910037448715280700259995124, 0.78453175867953477394304387440, 1.96413587534068849061259802759, 2.525418252746485529428997873475, 3.30796981432888903384948672497, 4.124312926658147583777536893255, 5.07353128363165963626442339006, 5.4862848761564351050427783670, 6.24194558925033814593554369713, 6.62531605250052940949221613357, 7.905430766627150750766638617264, 8.90138450828720189308543460656, 9.58129455108493525641999633800, 9.95145706273781346278296815665, 10.58011805146248600654097027496, 11.245099672771154027440777185094, 11.83078887819621760536517127941, 12.68272460585715567878801132965, 13.037964842832589584101457508294, 13.56340177976249012331179820997, 14.65058768971216793086944959729, 15.1175111383583487902512314875, 15.88018484093794148460380735095, 16.795562723315161177865611918263, 17.266461192116687533683930576090, 18.05090460546170539600404098555

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