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Properties

Label 1-4003-4003.1038-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.837 + 0.546i$
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  + (−0.939 − 0.343i)2-s + (0.423 + 0.905i)3-s + (0.764 + 0.644i)4-s + (−0.978 + 0.204i)5-s + (−0.0870 − 0.996i)6-s + (0.847 − 0.530i)7-s + (−0.497 − 0.867i)8-s + (−0.641 + 0.767i)9-s + (0.989 + 0.143i)10-s + (0.946 − 0.323i)11-s + (−0.259 + 0.965i)12-s + (0.982 + 0.187i)13-s + (−0.978 + 0.207i)14-s + (−0.599 − 0.800i)15-s + (0.169 + 0.985i)16-s + (0.997 − 0.0752i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.343i)2-s + (0.423 + 0.905i)3-s + (0.764 + 0.644i)4-s + (−0.978 + 0.204i)5-s + (−0.0870 − 0.996i)6-s + (0.847 − 0.530i)7-s + (−0.497 − 0.867i)8-s + (−0.641 + 0.767i)9-s + (0.989 + 0.143i)10-s + (0.946 − 0.323i)11-s + (−0.259 + 0.965i)12-s + (0.982 + 0.187i)13-s + (−0.978 + 0.207i)14-s + (−0.599 − 0.800i)15-s + (0.169 + 0.985i)16-s + (0.997 − 0.0752i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.837 + 0.546i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (1038, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.837 + 0.546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.340482703 + 0.3984063373i\)
\(L(\frac12)\) \(\approx\) \(1.340482703 + 0.3984063373i\)
\(L(1)\) \(\approx\) \(0.8700661523 + 0.1350571035i\)
\(L(1)\) \(\approx\) \(0.8700661523 + 0.1350571035i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (-0.939 - 0.343i)T \)
3 \( 1 + (0.423 + 0.905i)T \)
5 \( 1 + (-0.978 + 0.204i)T \)
7 \( 1 + (0.847 - 0.530i)T \)
11 \( 1 + (0.946 - 0.323i)T \)
13 \( 1 + (0.982 + 0.187i)T \)
17 \( 1 + (0.997 - 0.0752i)T \)
19 \( 1 + (0.862 - 0.506i)T \)
23 \( 1 + (-0.548 - 0.836i)T \)
29 \( 1 + (-0.785 - 0.618i)T \)
31 \( 1 + (-0.997 + 0.0737i)T \)
37 \( 1 + (-0.0525 + 0.998i)T \)
41 \( 1 + (-0.124 + 0.992i)T \)
43 \( 1 + (-0.936 + 0.351i)T \)
47 \( 1 + (0.237 + 0.971i)T \)
53 \( 1 + (0.835 - 0.548i)T \)
59 \( 1 + (-0.0713 + 0.997i)T \)
61 \( 1 + (0.986 - 0.162i)T \)
67 \( 1 + (0.132 + 0.991i)T \)
71 \( 1 + (0.973 - 0.230i)T \)
73 \( 1 + (0.716 - 0.697i)T \)
79 \( 1 + (-0.867 + 0.496i)T \)
83 \( 1 + (-0.997 - 0.0768i)T \)
89 \( 1 + (-0.140 - 0.990i)T \)
97 \( 1 + (-0.401 + 0.915i)T \)
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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

−18.290008083359368523902745744856, −18.11322082949560018582486411596, −17.0684436384905023135054137225, −16.570251326858235305606944606028, −15.61598068335090012920110565490, −15.149856080904819126568430851811, −14.34666322907684435007063335575, −14.02626141994316332896179368539, −12.67502046819424757293058813155, −12.10521467814006934572993967549, −11.51455064068661882621853066336, −11.08975890724040831956280659275, −9.9026996294253406521435820980, −9.01535398943080057463297294315, −8.631841810183882065313673882596, −7.91332681709532615339903301773, −7.42446418805466793932892884178, −6.853982893993193303892194866324, −5.671427749554196836285196382732, −5.424534192742017498907310390889, −3.78980472496672856001001765138, −3.38114638093214662539921395773, −1.98741993480647610527572584218, −1.52511898808396974514346825162, −0.74342776671649426598241187801, 0.802394121042994044281055105104, 1.58620866050392839238900030213, 2.765738450272133006723870458882, 3.597478637308380386365177783582, 3.89985200970326181856762314100, 4.778943747239124782524234651440, 5.91319535667592741960417547693, 6.918493555815216726702963425664, 7.65713311238468944072953960344, 8.28078494286051262161430243444, 8.69900925624345987624033310938, 9.57681013320204693848299780673, 10.22957738973451088333733979515, 11.0087495711460850578980076344, 11.51164820032247778195992854195, 11.791570185485097016893024795319, 13.04502595181820376645246068287, 14.008860036923381300103814442650, 14.61360468822481542051388533872, 15.23303008753426587633324440449, 16.02983270752899790367271675339, 16.58969101223485252811111365616, 16.91779484993020777842124214848, 18.036512141417784978503153079823, 18.5801502811040332529954370280

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