Properties

Label 1-6017-6017.128-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.0625 + 0.998i$
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  + (−0.231 − 0.972i)2-s + (0.809 + 0.587i)3-s + (−0.892 + 0.450i)4-s + (0.827 + 0.561i)5-s + (0.384 − 0.923i)6-s + (0.184 + 0.982i)7-s + (0.644 + 0.764i)8-s + (0.309 + 0.951i)9-s + (0.354 − 0.935i)10-s + (−0.987 − 0.160i)12-s + (−0.669 + 0.743i)13-s + (0.913 − 0.406i)14-s + (0.339 + 0.940i)15-s + (0.594 − 0.804i)16-s + (0.0884 + 0.996i)17-s + (0.853 − 0.520i)18-s + ⋯
L(s)  = 1  + (−0.231 − 0.972i)2-s + (0.809 + 0.587i)3-s + (−0.892 + 0.450i)4-s + (0.827 + 0.561i)5-s + (0.384 − 0.923i)6-s + (0.184 + 0.982i)7-s + (0.644 + 0.764i)8-s + (0.309 + 0.951i)9-s + (0.354 − 0.935i)10-s + (−0.987 − 0.160i)12-s + (−0.669 + 0.743i)13-s + (0.913 − 0.406i)14-s + (0.339 + 0.940i)15-s + (0.594 − 0.804i)16-s + (0.0884 + 0.996i)17-s + (0.853 − 0.520i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0625 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0625 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.0625 + 0.998i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.0625 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.723210548 + 1.834652625i\)
\(L(\frac12)\) \(\approx\) \(1.723210548 + 1.834652625i\)
\(L(1)\) \(\approx\) \(1.369352371 + 0.2954315574i\)
\(L(1)\) \(\approx\) \(1.369352371 + 0.2954315574i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.231 - 0.972i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.827 + 0.561i)T \)
7 \( 1 + (0.184 + 0.982i)T \)
13 \( 1 + (-0.669 + 0.743i)T \)
17 \( 1 + (0.0884 + 0.996i)T \)
19 \( 1 + (0.953 + 0.301i)T \)
23 \( 1 + (0.0402 - 0.999i)T \)
29 \( 1 + (0.989 - 0.144i)T \)
31 \( 1 + (0.861 + 0.506i)T \)
37 \( 1 + (-0.619 - 0.784i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.632 + 0.774i)T \)
47 \( 1 + (0.997 + 0.0643i)T \)
53 \( 1 + (-0.471 + 0.881i)T \)
59 \( 1 + (0.953 - 0.301i)T \)
61 \( 1 + (-0.384 + 0.923i)T \)
67 \( 1 + (-0.632 + 0.774i)T \)
71 \( 1 + (-0.870 - 0.493i)T \)
73 \( 1 + (0.789 + 0.613i)T \)
79 \( 1 + (0.779 - 0.626i)T \)
83 \( 1 + (0.899 - 0.435i)T \)
89 \( 1 + (-0.568 - 0.822i)T \)
97 \( 1 + (-0.324 + 0.945i)T \)
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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

−17.50310088984397920065022780719, −17.1205355868546567305111951234, −16.20197180351836032995067060811, −15.58420941393258378440131322925, −14.90956630859705057393866002463, −14.02862386807157105680371013887, −13.64578670128893257957095844046, −13.50315145911286413454846364354, −12.49639844882515423324399950625, −11.86557707800195953158545822434, −10.55842077853423876687779110694, −9.82060734288438386997393013422, −9.57579565306531950772104757657, −8.71916710226796698213517517941, −8.06580548790703066902621277583, −7.44616092179414116672259499185, −6.95530356589518593830719021024, −6.24368913183451148983636616363, −5.17242418657427428103351158648, −4.9535949243107972489162304811, −3.84414846049887559765522522407, −3.0555562244607256296291570827, −2.062308965554045230559965414959, −1.06965976639337643334632851063, −0.661009885304705703514206979172, 1.40989010570724621945402616008, 1.94405550916561872920464478088, 2.851079608342564867733403137673, 2.91349816119881592005672718091, 4.09933691646764222223600956100, 4.746038128172710429429208054870, 5.44992552541050125163276495992, 6.33111796828573701747489960844, 7.337133912625828322665891489779, 8.2098495684039625075121754069, 8.73807049364490522376996916838, 9.35973006328821946603199085275, 9.950698407312959654808510973127, 10.40528428575125564434268424747, 11.119104214831976207615302068597, 11.99815890404117883881180559479, 12.482451659416469691719442773577, 13.383436097579880481682161989430, 13.96611812853563579710850652289, 14.55441805037934082903417254722, 14.92412760264942723073324146615, 15.95000129537089278176906114445, 16.64181465956183913252473603543, 17.42533832474153113011522416560, 18.06149133061026072470748129976

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