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Properties

Label 1-6003-6003.3587-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.997 - 0.0705i$
Analytic cond. $27.8778$
Root an. cond. $27.8778$
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  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (−0.988 − 0.149i)5-s + (0.733 + 0.680i)7-s + (0.900 + 0.433i)8-s + (0.222 + 0.974i)10-s + (0.0747 − 0.997i)11-s + (0.826 + 0.563i)13-s + (0.365 − 0.930i)14-s + (0.0747 − 0.997i)16-s + 17-s + (0.222 + 0.974i)19-s + (0.826 − 0.563i)20-s + (−0.955 + 0.294i)22-s + (0.955 + 0.294i)25-s + (0.222 − 0.974i)26-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (−0.988 − 0.149i)5-s + (0.733 + 0.680i)7-s + (0.900 + 0.433i)8-s + (0.222 + 0.974i)10-s + (0.0747 − 0.997i)11-s + (0.826 + 0.563i)13-s + (0.365 − 0.930i)14-s + (0.0747 − 0.997i)16-s + 17-s + (0.222 + 0.974i)19-s + (0.826 − 0.563i)20-s + (−0.955 + 0.294i)22-s + (0.955 + 0.294i)25-s + (0.222 − 0.974i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.997 - 0.0705i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (3587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.997 - 0.0705i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.415624537 - 0.05000117000i\)
\(L(\frac12)\) \(\approx\) \(1.415624537 - 0.05000117000i\)
\(L(1)\) \(\approx\) \(0.8574923150 - 0.2197313782i\)
\(L(1)\) \(\approx\) \(0.8574923150 - 0.2197313782i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.365 - 0.930i)T \)
5 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (0.733 + 0.680i)T \)
11 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (0.826 + 0.563i)T \)
17 \( 1 + T \)
19 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + (-0.988 - 0.149i)T \)
37 \( 1 + (0.900 + 0.433i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.988 - 0.149i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.733 + 0.680i)T \)
67 \( 1 + (-0.0747 - 0.997i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (-0.826 + 0.563i)T \)
83 \( 1 + (0.955 + 0.294i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + (-0.955 - 0.294i)T \)
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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

−17.684886871914594015931621691099, −17.10386362765351153672200591620, −16.28527426819368170314259109257, −15.8369993690660552667853674435, −15.099894920298381776894898080650, −14.652587886315206479160191115865, −14.0758749693531511600464106988, −13.20295548457377687177605837474, −12.59529702562531316434211120367, −11.6864356283496091358701271280, −10.89757791723953793173279958799, −10.5182074625604025607119678127, −9.620413309417600960223521877344, −8.87944692867324927552989075568, −8.1531622280924743604254022078, −7.52134713421698514409555762102, −7.26794675019035250742961323198, −6.44342788305455458132429086567, −5.40311214651794889186987561259, −4.93036134265724387432941094465, −4.00752944941001926048030703674, −3.6860655591527280498662561956, −2.33489011426147441775807722220, −1.1852070796438278874972769486, −0.60240326118901851224555240848, 0.90596756713775802884518353572, 1.3827602039434934212026922269, 2.429529561174979417088658534902, 3.27684623865783387425719325946, 3.821546109297137713845495930132, 4.472368988384625727587411609010, 5.412867283614385755442674405686, 6.011859892884969125179168533356, 7.27201258171418702195031926125, 8.04459968671548202238766047827, 8.29271745967949301661547961134, 9.02133226263663802863469236786, 9.67006308013599122456820096692, 10.68846379978187580069744158870, 11.27111546299060312996567900743, 11.56912373263496049811698005793, 12.29698559845726970636100104135, 12.81368424294969486739130046763, 13.75777150839500930207082316536, 14.38334641063310859978857957530, 14.93949187189258021584640002910, 16.08222514771912024553303027210, 16.375242167217903151684104898727, 16.99589950907498989967713507056, 18.15582979620702908895789269549

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