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Properties

Label 1-6017-6017.3210-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.835 + 0.548i$
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  + (−0.443 − 0.896i)2-s + (0.309 − 0.951i)3-s + (−0.607 + 0.794i)4-s + (0.485 + 0.873i)5-s + (−0.989 + 0.144i)6-s + (0.715 + 0.698i)7-s + (0.981 + 0.192i)8-s + (−0.809 − 0.587i)9-s + (0.568 − 0.822i)10-s + (0.568 + 0.822i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (0.981 − 0.192i)15-s + (−0.262 − 0.964i)16-s + (0.995 + 0.0965i)17-s + (−0.168 + 0.985i)18-s + ⋯
L(s)  = 1  + (−0.443 − 0.896i)2-s + (0.309 − 0.951i)3-s + (−0.607 + 0.794i)4-s + (0.485 + 0.873i)5-s + (−0.989 + 0.144i)6-s + (0.715 + 0.698i)7-s + (0.981 + 0.192i)8-s + (−0.809 − 0.587i)9-s + (0.568 − 0.822i)10-s + (0.568 + 0.822i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (0.981 − 0.192i)15-s + (−0.262 − 0.964i)16-s + (0.995 + 0.0965i)17-s + (−0.168 + 0.985i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.835 + 0.548i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (3210, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.835 + 0.548i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8875093374 + 0.2652851434i\)
\(L(\frac12)\) \(\approx\) \(0.8875093374 + 0.2652851434i\)
\(L(1)\) \(\approx\) \(0.8107050170 - 0.3364152791i\)
\(L(1)\) \(\approx\) \(0.8107050170 - 0.3364152791i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.443 - 0.896i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.485 + 0.873i)T \)
7 \( 1 + (0.715 + 0.698i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.995 + 0.0965i)T \)
19 \( 1 + (-0.998 + 0.0483i)T \)
23 \( 1 + (-0.970 - 0.239i)T \)
29 \( 1 + (-0.527 - 0.849i)T \)
31 \( 1 + (0.836 - 0.548i)T \)
37 \( 1 + (-0.527 - 0.849i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.568 + 0.822i)T \)
47 \( 1 + (-0.0724 + 0.997i)T \)
53 \( 1 + (-0.681 + 0.732i)T \)
59 \( 1 + (-0.998 - 0.0483i)T \)
61 \( 1 + (-0.989 + 0.144i)T \)
67 \( 1 + (0.568 + 0.822i)T \)
71 \( 1 + (0.836 + 0.548i)T \)
73 \( 1 + (-0.906 + 0.421i)T \)
79 \( 1 + (0.0241 - 0.999i)T \)
83 \( 1 + (0.485 + 0.873i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (0.215 + 0.976i)T \)
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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

−17.291209890509909645069521082372, −16.873141164403068093218526412508, −16.554034327424730034317840142830, −15.80098068352436290611220190128, −15.062191073147316123619777490631, −14.401926273807359622959073279820, −14.015483081313594757476204360929, −13.4448082965300376937840397280, −12.42349282820225920588760699152, −11.63243760891932928311008841826, −10.58279838921503495040786276113, −10.23854512090962340534449839376, −9.526666247048262530086558441886, −9.03330494195915326455290810835, −8.16296790938864825412477158257, −7.941699840076257820153660968571, −6.930232731057898605526029382229, −6.0752955873212955055695073681, −5.26996820692822034154465876698, −4.76674548769773775642075796787, −4.33034806897662364807110081023, −3.4298218260332481944052783730, −2.04716640740485281678658369649, −1.492804431844828601660059233988, −0.270678357970581945766405707913, 0.97766398034208781186103000665, 1.95532225219456335885293754424, 2.32002979521702051593992802116, 2.88801718367093029843635446366, 3.75286948212292565831139514489, 4.72669722343362087081210916474, 5.808594063406655533280788176441, 6.15978679472991728643159043098, 7.48919139874555301058912370460, 7.658625507069826713358594536772, 8.37284118906844584829107316473, 9.18870051695763917049783081684, 9.81211669193775642608051045320, 10.55242223337182515107157170097, 11.187561859878199972449456076694, 11.914927367835612013302585131624, 12.42811838660648290780292802855, 12.93979970224897239058247390225, 13.908022163565795733896186331255, 14.30270425566056642110141453757, 14.85819039634102185839340653803, 15.7045810524190994244687377713, 17.089647729686950602920018122017, 17.31502673112729874461917380683, 17.92877446567596269373109665919

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