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Properties

Label 1-6003-6003.202-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.845 + 0.534i$
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  + (0.888 − 0.458i)2-s + (0.580 − 0.814i)4-s + (0.235 + 0.971i)5-s + (−0.327 − 0.945i)7-s + (0.142 − 0.989i)8-s + (0.654 + 0.755i)10-s + (−0.0475 − 0.998i)11-s + (−0.327 + 0.945i)13-s + (−0.723 − 0.690i)14-s + (−0.327 − 0.945i)16-s + (−0.415 + 0.909i)17-s + (−0.415 − 0.909i)19-s + (0.928 + 0.371i)20-s + (−0.5 − 0.866i)22-s + (−0.888 + 0.458i)25-s + (0.142 + 0.989i)26-s + ⋯
L(s)  = 1  + (0.888 − 0.458i)2-s + (0.580 − 0.814i)4-s + (0.235 + 0.971i)5-s + (−0.327 − 0.945i)7-s + (0.142 − 0.989i)8-s + (0.654 + 0.755i)10-s + (−0.0475 − 0.998i)11-s + (−0.327 + 0.945i)13-s + (−0.723 − 0.690i)14-s + (−0.327 − 0.945i)16-s + (−0.415 + 0.909i)17-s + (−0.415 − 0.909i)19-s + (0.928 + 0.371i)20-s + (−0.5 − 0.866i)22-s + (−0.888 + 0.458i)25-s + (0.142 + 0.989i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.845 + 0.534i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.845 + 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1582025797 - 0.5460571527i\)
\(L(\frac12)\) \(\approx\) \(-0.1582025797 - 0.5460571527i\)
\(L(1)\) \(\approx\) \(1.289746396 - 0.4836825911i\)
\(L(1)\) \(\approx\) \(1.289746396 - 0.4836825911i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.888 - 0.458i)T \)
5 \( 1 + (0.235 + 0.971i)T \)
7 \( 1 + (-0.327 - 0.945i)T \)
11 \( 1 + (-0.0475 - 0.998i)T \)
13 \( 1 + (-0.327 + 0.945i)T \)
17 \( 1 + (-0.415 + 0.909i)T \)
19 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (-0.928 + 0.371i)T \)
37 \( 1 + (0.959 - 0.281i)T \)
41 \( 1 + (-0.235 - 0.971i)T \)
43 \( 1 + (-0.928 - 0.371i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.654 + 0.755i)T \)
59 \( 1 + (-0.327 + 0.945i)T \)
61 \( 1 + (0.786 + 0.618i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (-0.415 - 0.909i)T \)
79 \( 1 + (-0.981 + 0.189i)T \)
83 \( 1 + (0.235 - 0.971i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (-0.723 + 0.690i)T \)
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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

−17.91809851364390093185640798986, −17.33128141677755115961082839494, −16.61006597388674934170889250808, −16.0401958270228144634628341180, −15.46367854505777899271110586842, −14.86649335356231373595217797575, −14.3018814556505236277096058533, −13.28351272499207636767145546695, −12.77185427700151561919997310125, −12.53595714602545841602547201454, −11.76524531055698434119560654419, −11.1298559300649210851944489258, −9.83937517092491797911315761553, −9.58216879134191163895216372959, −8.52508190258899796203039847036, −8.07026947203300767835756145542, −7.31008655587293815678759119959, −6.426847547008225416181123532305, −5.80360931134252996817931519425, −5.13175542531616638804754079569, −4.71164949614055409100525655113, −3.86122229379208264642199039676, −2.90326268969482671014806531450, −2.27634461352009226803987466302, −1.50585779563907958276531526720, 0.08837830656369355305559357339, 1.3100276273945156076297616243, 2.16892331107808760680238570493, 2.82060203057184484964116816748, 3.73044515034664226839008230694, 3.98983067805300364771025593714, 4.97599853124311605303421110800, 5.861354654412506274386411275371, 6.50959829726355000165890806056, 6.92864373625433804324264478589, 7.62822486490647608707633538960, 8.82950000309685157443301091660, 9.51285806881151283909296259313, 10.44300200768755017707149388115, 10.72165675266910586651943223147, 11.2916133552165979424996190290, 11.98945078256156161460693884960, 12.96204872406483286173308250398, 13.47266072889468858424089374100, 13.93876535448381826739802579467, 14.566178611519208423513923701255, 15.14231282568387054558019936097, 15.894058234526766580286985592360, 16.6566454502300372477427548829, 17.1792610066734675268616231

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