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Properties

Label 1-6004-6004.271-r0-0-0
Degree $1$
Conductor $6004$
Sign $-0.667 - 0.744i$
Analytic cond. $27.8824$
Root an. cond. $27.8824$
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  + (−0.711 + 0.702i)3-s + (0.830 + 0.556i)5-s + (0.428 − 0.903i)7-s + (0.0134 − 0.999i)9-s + (0.970 − 0.239i)11-s + (−0.545 + 0.837i)13-s + (−0.982 + 0.186i)15-s + (−0.730 − 0.682i)17-s + (0.329 + 0.944i)21-s + (−0.766 − 0.642i)23-s + (0.379 + 0.925i)25-s + (0.692 + 0.721i)27-s + (−0.653 − 0.757i)29-s + (0.200 − 0.979i)31-s + (−0.523 + 0.852i)33-s + ⋯
L(s)  = 1  + (−0.711 + 0.702i)3-s + (0.830 + 0.556i)5-s + (0.428 − 0.903i)7-s + (0.0134 − 0.999i)9-s + (0.970 − 0.239i)11-s + (−0.545 + 0.837i)13-s + (−0.982 + 0.186i)15-s + (−0.730 − 0.682i)17-s + (0.329 + 0.944i)21-s + (−0.766 − 0.642i)23-s + (0.379 + 0.925i)25-s + (0.692 + 0.721i)27-s + (−0.653 − 0.757i)29-s + (0.200 − 0.979i)31-s + (−0.523 + 0.852i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
Sign: $-0.667 - 0.744i$
Analytic conductor: \(27.8824\)
Root analytic conductor: \(27.8824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6004} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6004,\ (0:\ ),\ -0.667 - 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1547845143 - 0.3464122651i\)
\(L(\frac12)\) \(\approx\) \(0.1547845143 - 0.3464122651i\)
\(L(1)\) \(\approx\) \(0.8559206247 + 0.09441565979i\)
\(L(1)\) \(\approx\) \(0.8559206247 + 0.09441565979i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
79 \( 1 \)
good3 \( 1 + (-0.711 + 0.702i)T \)
5 \( 1 + (0.830 + 0.556i)T \)
7 \( 1 + (0.428 - 0.903i)T \)
11 \( 1 + (0.970 - 0.239i)T \)
13 \( 1 + (-0.545 + 0.837i)T \)
17 \( 1 + (-0.730 - 0.682i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.653 - 0.757i)T \)
31 \( 1 + (0.200 - 0.979i)T \)
37 \( 1 + (0.845 + 0.534i)T \)
41 \( 1 + (-0.830 - 0.556i)T \)
43 \( 1 + (-0.897 + 0.440i)T \)
47 \( 1 + (-0.991 - 0.133i)T \)
53 \( 1 + (-0.964 + 0.265i)T \)
59 \( 1 + (-0.452 + 0.891i)T \)
61 \( 1 + (0.303 + 0.952i)T \)
67 \( 1 + (0.999 - 0.0268i)T \)
71 \( 1 + (0.909 + 0.416i)T \)
73 \( 1 + (-0.545 - 0.837i)T \)
83 \( 1 + (-0.799 + 0.600i)T \)
89 \( 1 + (-0.815 - 0.579i)T \)
97 \( 1 + (0.379 - 0.925i)T \)
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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

−17.79380523225154173661304715755, −17.45973280309893305105487653805, −16.88584073909636709394375451088, −16.12530364946518088090890544207, −15.36366527391241226029043460372, −14.56291845599033428214691152720, −14.0269644203959798038053744015, −13.10248772291285175222703501262, −12.66437878579120644322286099796, −12.1798132074276397745012476772, −11.43879068600819990832741163236, −10.841614064701117696766849693058, −9.879344399189285601368783111296, −9.40417090829282983719903039833, −8.3968700318310510663699666938, −8.123555497935710476208449757230, −6.985493421664487780591681612983, −6.39316009259646950480676641803, −5.78059954620700846213222058870, −5.12394021099874398560579894372, −4.67812931762812736204217903698, −3.4671177753809978635831381511, −2.36652882882958405875874102082, −1.74861731296604935533706454176, −1.291463722752540737187352473187, 0.09909500334512388097790792928, 1.263537809583318094636557692810, 2.049454117170370141195398164439, 3.0085379083199960004060983468, 4.03576195607217356614536150443, 4.372821398449988193108310819147, 5.16892112178983752760093948436, 6.06566105561350193105747595805, 6.623117597768955928410756775680, 7.052559701430652290782490715310, 8.10819224822489532668266557257, 9.10918239205711784251143597767, 9.68711087992129211344040346603, 10.068313486713448145191805452520, 10.918372942788748826642994731594, 11.51006195716997082794465565490, 11.799367076665271102961518985228, 12.98734813220839605922628845349, 13.69919908098446888001860106658, 14.25597934012882487702783073144, 14.78607747056139398541331965545, 15.45484823749087311408913008376, 16.586631514665631813975841794964, 16.74856690870638341497806432949, 17.31780537739764367392860973930

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