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Properties

Label 1-4003-4003.1165-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.899 - 0.437i$
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  + (0.989 + 0.143i)2-s + (−0.922 − 0.386i)3-s + (0.958 + 0.284i)4-s + (0.403 − 0.915i)5-s + (−0.856 − 0.515i)6-s + (0.935 + 0.353i)7-s + (0.907 + 0.419i)8-s + (0.700 + 0.713i)9-s + (0.530 − 0.847i)10-s + (0.0541 − 0.998i)11-s + (−0.773 − 0.633i)12-s + (0.647 + 0.762i)13-s + (0.874 + 0.484i)14-s + (−0.725 + 0.687i)15-s + (0.837 + 0.546i)16-s + (−0.561 + 0.827i)17-s + ⋯
L(s)  = 1  + (0.989 + 0.143i)2-s + (−0.922 − 0.386i)3-s + (0.958 + 0.284i)4-s + (0.403 − 0.915i)5-s + (−0.856 − 0.515i)6-s + (0.935 + 0.353i)7-s + (0.907 + 0.419i)8-s + (0.700 + 0.713i)9-s + (0.530 − 0.847i)10-s + (0.0541 − 0.998i)11-s + (−0.773 − 0.633i)12-s + (0.647 + 0.762i)13-s + (0.874 + 0.484i)14-s + (−0.725 + 0.687i)15-s + (0.837 + 0.546i)16-s + (−0.561 + 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.899 - 0.437i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (1165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.899 - 0.437i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.421664127 - 0.7884851892i\)
\(L(\frac12)\) \(\approx\) \(3.421664127 - 0.7884851892i\)
\(L(1)\) \(\approx\) \(1.900160806 - 0.2312812235i\)
\(L(1)\) \(\approx\) \(1.900160806 - 0.2312812235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (0.989 + 0.143i)T \)
3 \( 1 + (-0.922 - 0.386i)T \)
5 \( 1 + (0.403 - 0.915i)T \)
7 \( 1 + (0.935 + 0.353i)T \)
11 \( 1 + (0.0541 - 0.998i)T \)
13 \( 1 + (0.647 + 0.762i)T \)
17 \( 1 + (-0.561 + 0.827i)T \)
19 \( 1 + (-0.436 - 0.899i)T \)
23 \( 1 + (0.837 + 0.546i)T \)
29 \( 1 + (-0.436 + 0.899i)T \)
31 \( 1 + (-0.232 - 0.972i)T \)
37 \( 1 + (-0.436 + 0.899i)T \)
41 \( 1 + (0.0541 + 0.998i)T \)
43 \( 1 + (0.126 - 0.992i)T \)
47 \( 1 + (0.837 - 0.546i)T \)
53 \( 1 + (0.837 - 0.546i)T \)
59 \( 1 + (-0.0901 - 0.995i)T \)
61 \( 1 + (-0.0180 - 0.999i)T \)
67 \( 1 + (0.874 - 0.484i)T \)
71 \( 1 + (-0.922 - 0.386i)T \)
73 \( 1 + (0.976 + 0.214i)T \)
79 \( 1 + (-0.0180 + 0.999i)T \)
83 \( 1 + (0.989 + 0.143i)T \)
89 \( 1 + (0.874 + 0.484i)T \)
97 \( 1 + (0.874 - 0.484i)T \)
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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

−18.34767525675330372157928770593, −17.75338300473439767603606379536, −17.269375452331088015901824427823, −16.38497176848364958240845229263, −15.61151663081311708769199977733, −15.00903967190778616352213107805, −14.580787071867355273059741639061, −13.78319867136884377860997476143, −13.05083121708813661602659242655, −12.28933821524451471798771945737, −11.656307233643349387854389433713, −10.8298853846088396467130645280, −10.644113535079505235337893259416, −9.99653385198213364786922840521, −8.94844821799388141917506435556, −7.52909394241562213581288991065, −7.20838180655816014101132057637, −6.372922392428157011216389187741, −5.69452285974251243266183240504, −5.08234382811081028299899716797, −4.29035497870912019791159093652, −3.75201257288985718766875007516, −2.67520301332092075933490032400, −1.89093504516691299391448374638, −0.99866072538946212852413011698, 0.92529366585492409406251212704, 1.709868157846573362725574574502, 2.25692386999586159352882269599, 3.660611666850835755110637357060, 4.3965664679156027301004234024, 5.14641262224107732908932654532, 5.49738485853466309683607104092, 6.34356062453767482050837088605, 6.82098656178895026930358317997, 7.93964620991491366925032119203, 8.516066784236039059383046547079, 9.27752534328444968799002292016, 10.68721232722007828344781691044, 11.07469254760256730320849126046, 11.63153645695032624804852636875, 12.27955903806090906703603761527, 13.20084218162383159901929641073, 13.35244226675329854240968421316, 14.128998457829270261631024431417, 15.12738804110417102865722199422, 15.6894001486628239910952652515, 16.48830477811983969858166574459, 17.05099175042841017351076172625, 17.37796904332903706806875453136, 18.40312632304743649126922224554

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