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Properties

Label 1-6003-6003.1123-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.0412 - 0.999i$
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  + (−0.649 − 0.760i)2-s + (−0.155 + 0.987i)4-s + (−0.694 + 0.719i)5-s + (0.833 − 0.552i)7-s + (0.852 − 0.523i)8-s + (0.998 + 0.0611i)10-s + (0.993 − 0.115i)11-s + (0.352 − 0.935i)13-s + (−0.961 − 0.275i)14-s + (−0.951 − 0.307i)16-s + (−0.989 − 0.142i)17-s + (−0.359 − 0.933i)19-s + (−0.601 − 0.798i)20-s + (−0.733 − 0.680i)22-s + (−0.0339 − 0.999i)25-s + (−0.940 + 0.339i)26-s + ⋯
L(s)  = 1  + (−0.649 − 0.760i)2-s + (−0.155 + 0.987i)4-s + (−0.694 + 0.719i)5-s + (0.833 − 0.552i)7-s + (0.852 − 0.523i)8-s + (0.998 + 0.0611i)10-s + (0.993 − 0.115i)11-s + (0.352 − 0.935i)13-s + (−0.961 − 0.275i)14-s + (−0.951 − 0.307i)16-s + (−0.989 − 0.142i)17-s + (−0.359 − 0.933i)19-s + (−0.601 − 0.798i)20-s + (−0.733 − 0.680i)22-s + (−0.0339 − 0.999i)25-s + (−0.940 + 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0412 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0412 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.0412 - 0.999i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.0412 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8657630866 - 0.9022851875i\)
\(L(\frac12)\) \(\approx\) \(0.8657630866 - 0.9022851875i\)
\(L(1)\) \(\approx\) \(0.7602910971 - 0.2968014158i\)
\(L(1)\) \(\approx\) \(0.7602910971 - 0.2968014158i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.649 - 0.760i)T \)
5 \( 1 + (-0.694 + 0.719i)T \)
7 \( 1 + (0.833 - 0.552i)T \)
11 \( 1 + (0.993 - 0.115i)T \)
13 \( 1 + (0.352 - 0.935i)T \)
17 \( 1 + (-0.989 - 0.142i)T \)
19 \( 1 + (-0.359 - 0.933i)T \)
31 \( 1 + (0.980 + 0.195i)T \)
37 \( 1 + (0.639 + 0.768i)T \)
41 \( 1 + (0.0950 - 0.995i)T \)
43 \( 1 + (0.980 - 0.195i)T \)
47 \( 1 + (-0.563 + 0.826i)T \)
53 \( 1 + (0.377 - 0.925i)T \)
59 \( 1 + (0.723 - 0.690i)T \)
61 \( 1 + (-0.268 + 0.963i)T \)
67 \( 1 + (-0.115 + 0.993i)T \)
71 \( 1 + (-0.262 + 0.965i)T \)
73 \( 1 + (0.728 + 0.685i)T \)
79 \( 1 + (0.977 + 0.209i)T \)
83 \( 1 + (-0.314 - 0.949i)T \)
89 \( 1 + (0.994 + 0.101i)T \)
97 \( 1 + (-0.384 + 0.923i)T \)
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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

−17.90637510602514152211075978987, −17.00898685851405958265589836242, −16.6977409515541963652208535868, −16.0060051751549971269617359798, −15.277708024285064750342437252005, −14.850817566739378081050034932642, −14.14751616077744768828198481005, −13.50438671667417731116567484464, −12.495846972314986101796818046431, −11.81007282965152530529958835585, −11.29209981630344185214562998543, −10.66133234004448349079560953469, −9.46909423461574217344281284202, −9.143598798232264833548520344633, −8.45404517591978145537255038454, −8.01770982473987339435682144748, −7.24585153943939321242275863852, −6.366248485526555530443119858698, −5.936479612886896061214580626658, −4.84004324469313935504931191285, −4.444049241418312689621532616, −3.7873446153430402670194872341, −2.22875018417746209954000160021, −1.61389222683394073561189082829, −0.8688948062841493828137070814, 0.544355041809653796156589435337, 1.1971780495368257663285408374, 2.30019340760442942247346103774, 2.8738734624973620390105402197, 3.83894699313817033724644609008, 4.20965434982349415200049919390, 5.01265606615308560861006646588, 6.367652841534345973410776514433, 6.9428047091674356521654715025, 7.58718452907957069866947175162, 8.34257986159999171381138168432, 8.724995521890279800459558709401, 9.67477802983252423232261103409, 10.48593938132058664662547550863, 10.937066398547622741824727915572, 11.47735518054114648293282634643, 11.918472903933721378415172562725, 12.89290399923793767569782205865, 13.51963496377446846575485035570, 14.22637825220035057007143189507, 14.964207502630577015072498123760, 15.64826411131530183787753007117, 16.28205801107912233073503073219, 17.33506925719058704952323265832, 17.574307987447309593186397328750

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