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Properties

Label 1-6004-6004.47-r0-0-0
Degree $1$
Conductor $6004$
Sign $0.0121 + 0.999i$
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  + (0.977 + 0.213i)3-s + (0.998 + 0.0536i)5-s + (−0.0402 + 0.999i)7-s + (0.909 + 0.416i)9-s + (−0.120 + 0.992i)11-s + (0.930 − 0.367i)13-s + (0.964 + 0.265i)15-s + (−0.476 + 0.879i)17-s + (−0.252 + 0.967i)21-s + (0.939 + 0.342i)23-s + (0.994 + 0.107i)25-s + (0.799 + 0.600i)27-s + (0.711 − 0.702i)29-s + (−0.987 − 0.160i)31-s + (−0.329 + 0.944i)33-s + ⋯
L(s)  = 1  + (0.977 + 0.213i)3-s + (0.998 + 0.0536i)5-s + (−0.0402 + 0.999i)7-s + (0.909 + 0.416i)9-s + (−0.120 + 0.992i)11-s + (0.930 − 0.367i)13-s + (0.964 + 0.265i)15-s + (−0.476 + 0.879i)17-s + (−0.252 + 0.967i)21-s + (0.939 + 0.342i)23-s + (0.994 + 0.107i)25-s + (0.799 + 0.600i)27-s + (0.711 − 0.702i)29-s + (−0.987 − 0.160i)31-s + (−0.329 + 0.944i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0121 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0121 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6004\)    =    \(2^{2} \cdot 19 \cdot 79\)
Sign: $0.0121 + 0.999i$
Analytic conductor: \(27.8824\)
Root analytic conductor: \(27.8824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6004} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6004,\ (0:\ ),\ 0.0121 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.558167894 + 2.589331611i\)
\(L(\frac12)\) \(\approx\) \(2.558167894 + 2.589331611i\)
\(L(1)\) \(\approx\) \(1.757904206 + 0.6452774105i\)
\(L(1)\) \(\approx\) \(1.757904206 + 0.6452774105i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
79 \( 1 \)
good3 \( 1 + (0.977 + 0.213i)T \)
5 \( 1 + (0.998 + 0.0536i)T \)
7 \( 1 + (-0.0402 + 0.999i)T \)
11 \( 1 + (-0.120 + 0.992i)T \)
13 \( 1 + (0.930 - 0.367i)T \)
17 \( 1 + (-0.476 + 0.879i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.711 - 0.702i)T \)
31 \( 1 + (-0.987 - 0.160i)T \)
37 \( 1 + (-0.692 + 0.721i)T \)
41 \( 1 + (-0.998 - 0.0536i)T \)
43 \( 1 + (-0.452 - 0.891i)T \)
47 \( 1 + (-0.404 - 0.914i)T \)
53 \( 1 + (0.673 + 0.739i)T \)
59 \( 1 + (-0.783 + 0.621i)T \)
61 \( 1 + (0.897 - 0.440i)T \)
67 \( 1 + (-0.653 + 0.757i)T \)
71 \( 1 + (0.379 + 0.925i)T \)
73 \( 1 + (0.930 + 0.367i)T \)
83 \( 1 + (0.200 + 0.979i)T \)
89 \( 1 + (0.611 + 0.791i)T \)
97 \( 1 + (0.994 - 0.107i)T \)
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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

−17.75004110374752704005195537107, −16.74497239246592332355775580394, −16.303402063664782401227755100470, −15.6833848201436049897767131284, −14.55396890278030663188439400061, −14.23649955200183034411277621464, −13.523179144469130131965928117094, −13.30853944385094038479729913733, −12.62542493530874983355377127183, −11.47317338001245705563607928474, −10.74351599854078919258246578672, −10.31227016944007875768618472181, −9.299918242498911788983466800717, −8.9964907450695491868770087936, −8.30690586500470386498896840301, −7.45969395726887119072104245181, −6.66897694859709725641108015684, −6.377721386121108268931691287010, −5.1850010358123694132071705317, −4.58768902423597845171813648186, −3.442219595753808056475530748405, −3.230123092883572710464116366281, −2.18150368881514614549156729883, −1.422949414287348529694272809527, −0.7457984158972245089179178595, 1.37423337241999320289474450340, 1.92322910935870337100531058805, 2.5438644056782306300443494583, 3.29776360236139270949019099829, 4.09176104596792710856041140018, 5.07087929903206119718339398618, 5.53197250676524565352454038931, 6.52166216157477244502592269855, 7.00602104678680233270392378877, 8.09366441232058809968431495339, 8.68417675330254046491569104722, 9.08759256647451972599071807899, 9.92279358639093198895121179745, 10.31814033221444191682295833917, 11.13114842869818223461167400923, 12.16452470866629135594799504454, 12.82788256993950000737676957768, 13.33577535265133491578916904472, 13.85295973017870864303879889072, 14.79535537311525784169588622239, 15.257459546923190156642775730246, 15.532961492414319186473363982721, 16.57749226800473728306270772127, 17.31587619373992168836445002239, 17.9995379454271110176046800933

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