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Properties

Label 1-4003-4003.129-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.0183 - 0.999i$
Analytic cond. $18.5898$
Root an. cond. $18.5898$
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  + (0.803 + 0.595i)2-s + (0.291 + 0.956i)3-s + (0.291 + 0.956i)4-s + (0.113 + 0.993i)5-s + (−0.334 + 0.942i)6-s + (−0.998 + 0.0455i)7-s + (−0.334 + 0.942i)8-s + (−0.829 + 0.557i)9-s + (−0.5 + 0.866i)10-s + (0.962 + 0.269i)11-s + (−0.829 + 0.557i)12-s + (−0.0682 + 0.997i)13-s + (−0.829 − 0.557i)14-s + (−0.917 + 0.398i)15-s + (−0.829 + 0.557i)16-s + (−0.334 − 0.942i)17-s + ⋯
L(s)  = 1  + (0.803 + 0.595i)2-s + (0.291 + 0.956i)3-s + (0.291 + 0.956i)4-s + (0.113 + 0.993i)5-s + (−0.334 + 0.942i)6-s + (−0.998 + 0.0455i)7-s + (−0.334 + 0.942i)8-s + (−0.829 + 0.557i)9-s + (−0.5 + 0.866i)10-s + (0.962 + 0.269i)11-s + (−0.829 + 0.557i)12-s + (−0.0682 + 0.997i)13-s + (−0.829 − 0.557i)14-s + (−0.917 + 0.398i)15-s + (−0.829 + 0.557i)16-s + (−0.334 − 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0183 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0183 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.0183 - 0.999i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.0183 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.553749140 + 1.525498777i\)
\(L(\frac12)\) \(\approx\) \(-1.553749140 + 1.525498777i\)
\(L(1)\) \(\approx\) \(0.6121060323 + 1.399486483i\)
\(L(1)\) \(\approx\) \(0.6121060323 + 1.399486483i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (0.803 + 0.595i)T \)
3 \( 1 + (0.291 + 0.956i)T \)
5 \( 1 + (0.113 + 0.993i)T \)
7 \( 1 + (-0.998 + 0.0455i)T \)
11 \( 1 + (0.962 + 0.269i)T \)
13 \( 1 + (-0.0682 + 0.997i)T \)
17 \( 1 + (-0.334 - 0.942i)T \)
19 \( 1 + (0.113 + 0.993i)T \)
23 \( 1 + (0.377 + 0.926i)T \)
29 \( 1 + (-0.998 - 0.0455i)T \)
31 \( 1 + (0.995 + 0.0909i)T \)
37 \( 1 + (-0.419 - 0.907i)T \)
41 \( 1 + (-0.576 + 0.816i)T \)
43 \( 1 + (0.995 - 0.0909i)T \)
47 \( 1 + (-0.715 + 0.699i)T \)
53 \( 1 + (0.983 + 0.181i)T \)
59 \( 1 + (-0.998 - 0.0455i)T \)
61 \( 1 + (0.538 - 0.842i)T \)
67 \( 1 + (0.934 + 0.356i)T \)
71 \( 1 + (0.377 - 0.926i)T \)
73 \( 1 + (0.682 + 0.730i)T \)
79 \( 1 + (-0.974 + 0.225i)T \)
83 \( 1 + (-0.715 - 0.699i)T \)
89 \( 1 + (-0.974 + 0.225i)T \)
97 \( 1 + (0.995 + 0.0909i)T \)
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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

−18.14469231564282751506801400145, −17.11856507473803061732576544791, −16.856982848763726857024355936551, −15.62450204253528246670057442823, −15.2388318478982445596419123167, −14.299205126855820297190516726532, −13.57271175128819744256035919572, −13.04767213416984709426507606072, −12.7033114874172816824803784343, −12.05102128520083253280395594615, −11.38950030367757164454971659568, −10.43202055032578564357210881147, −9.65288615868974900240132305358, −8.88261137099320232917364039108, −8.42397263531378859951812081504, −7.192064977471420610120547044817, −6.5414867021424428262911886976, −5.952553369196380210766090663254, −5.26104436996502343870816766294, −4.24315108224509556712374122436, −3.52633487501474461663656998571, −2.78481367351331995613456000986, −1.97634129328841710871828850711, −1.04888703567864930443194125652, −0.44693191296520444188396065639, 1.92611339364029736424799470786, 2.72755405169823182185741163733, 3.49521702591382344863582855625, 3.87111058429863759183940886897, 4.68935032376861042676099697894, 5.675176100638361405652614536916, 6.265464039268612820065318386225, 6.97699288737867530411224480991, 7.52867343135178911164197755593, 8.63538601448422835236173705375, 9.53972232097066290112481765844, 9.66705758955825736315664870535, 10.8788305044631550967926956116, 11.50877815748830365528980902618, 12.04388101188266486086231487868, 13.09059529773479135854197573358, 13.95142739762337830629206598461, 14.18568262844414952855542257607, 14.87303446690205573134049920670, 15.63268670346600444307916540976, 15.99204850956806651492373155758, 16.859127354325151755145406633, 17.2086505992609303572324466424, 18.285521761761566459946120734097, 19.11960774154016748494184158729

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