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Properties

Label 1-6003-6003.101-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.369 - 0.929i$
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  + (−0.968 − 0.248i)2-s + (0.876 + 0.482i)4-s + (−0.999 − 0.0135i)5-s + (0.704 − 0.709i)7-s + (−0.728 − 0.685i)8-s + (0.965 + 0.262i)10-s + (0.659 − 0.751i)11-s + (0.942 + 0.333i)13-s + (−0.859 + 0.511i)14-s + (0.534 + 0.844i)16-s + (−0.909 + 0.415i)17-s + (0.999 − 0.0203i)19-s + (−0.869 − 0.494i)20-s + (−0.826 + 0.563i)22-s + (0.999 + 0.0271i)25-s + (−0.830 − 0.557i)26-s + ⋯
L(s)  = 1  + (−0.968 − 0.248i)2-s + (0.876 + 0.482i)4-s + (−0.999 − 0.0135i)5-s + (0.704 − 0.709i)7-s + (−0.728 − 0.685i)8-s + (0.965 + 0.262i)10-s + (0.659 − 0.751i)11-s + (0.942 + 0.333i)13-s + (−0.859 + 0.511i)14-s + (0.534 + 0.844i)16-s + (−0.909 + 0.415i)17-s + (0.999 − 0.0203i)19-s + (−0.869 − 0.494i)20-s + (−0.826 + 0.563i)22-s + (0.999 + 0.0271i)25-s + (−0.830 − 0.557i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.369 - 0.929i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.369 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9289006560 - 0.6305808771i\)
\(L(\frac12)\) \(\approx\) \(0.9289006560 - 0.6305808771i\)
\(L(1)\) \(\approx\) \(0.7107146185 - 0.1788330005i\)
\(L(1)\) \(\approx\) \(0.7107146185 - 0.1788330005i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.968 - 0.248i)T \)
5 \( 1 + (-0.999 - 0.0135i)T \)
7 \( 1 + (0.704 - 0.709i)T \)
11 \( 1 + (0.659 - 0.751i)T \)
13 \( 1 + (0.942 + 0.333i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (0.999 - 0.0203i)T \)
31 \( 1 + (-0.987 - 0.155i)T \)
37 \( 1 + (0.925 - 0.377i)T \)
41 \( 1 + (-0.971 - 0.235i)T \)
43 \( 1 + (0.987 - 0.155i)T \)
47 \( 1 + (-0.149 + 0.988i)T \)
53 \( 1 + (0.591 - 0.806i)T \)
59 \( 1 + (0.327 + 0.945i)T \)
61 \( 1 + (-0.215 - 0.976i)T \)
67 \( 1 + (0.751 - 0.659i)T \)
71 \( 1 + (0.101 - 0.994i)T \)
73 \( 1 + (-0.607 + 0.794i)T \)
79 \( 1 + (0.885 - 0.464i)T \)
83 \( 1 + (0.634 + 0.773i)T \)
89 \( 1 + (0.0815 + 0.996i)T \)
97 \( 1 + (0.307 - 0.951i)T \)
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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

−17.93941440917948217868621105172, −17.35197663978640802939619660075, −16.390211180425834212780031450842, −15.964279791417072420481872075045, −15.26695231864454927245425523625, −14.92204115285771566182958384118, −14.202012765061257669540064220879, −13.18763409155411915103112608943, −12.218946890244336637547915358006, −11.71774026764497461915050704180, −11.255284471013885137612625816151, −10.656218133549611556014105595114, −9.68257011943753638943045677075, −8.99836053945961977689875395330, −8.552632102409820174026409699261, −7.83994686000066143398187876373, −7.23953659561615954709976055191, −6.61655039329667179619146498934, −5.72165450095637254162369186485, −5.004291103045137561592952254328, −4.1647520844122141234033756675, −3.26419819813127366977666511851, −2.407050226983999743833884987846, −1.559272760938569608563862977962, −0.80781139335477776639008763813, 0.575870511243458048311116999424, 1.20292102911675254593664778322, 2.01530074981598174826427449643, 3.17004310867133781270184184583, 3.80713749802664987997946116417, 4.24605302971304430153545465902, 5.426076845064838431020696017523, 6.429390998460074209790258052249, 6.95205073366943169505619317226, 7.770545109344841705258412407260, 8.16765322834431991929803982782, 8.93305141594565216205793138531, 9.37539914332383907898063616777, 10.588695037589722175025897373348, 10.97679865755085786882432708473, 11.44093605512328970683361942248, 11.9572781331567200999325174123, 12.89358073906964532015920825155, 13.63856602947256965259881423589, 14.421881974391358345821071817593, 15.13302783906872307972725692390, 15.88668553830507493930080211712, 16.36755465120341830677327707001, 16.89379795183752687536339673223, 17.68726300777272683205704713376

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