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Properties

Label 1-6004-6004.511-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: $\chi_{6004} (511, \cdot )$
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

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

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