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Properties

Label 1-10e3-1000.421-r0-0-0
Degree $1$
Conductor $1000$
Sign $0.979 - 0.199i$
Analytic cond. $4.64398$
Root an. cond. $4.64398$
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  + (−0.0627 − 0.998i)3-s + (0.309 + 0.951i)7-s + (−0.992 + 0.125i)9-s + (0.425 − 0.904i)11-s + (0.992 − 0.125i)13-s + (−0.637 + 0.770i)17-s + (−0.0627 + 0.998i)19-s + (0.929 − 0.368i)21-s + (0.876 − 0.481i)23-s + (0.187 + 0.982i)27-s + (−0.535 + 0.844i)29-s + (−0.637 + 0.770i)31-s + (−0.929 − 0.368i)33-s + (0.187 − 0.982i)37-s + (−0.187 − 0.982i)39-s + ⋯
L(s)  = 1  + (−0.0627 − 0.998i)3-s + (0.309 + 0.951i)7-s + (−0.992 + 0.125i)9-s + (0.425 − 0.904i)11-s + (0.992 − 0.125i)13-s + (−0.637 + 0.770i)17-s + (−0.0627 + 0.998i)19-s + (0.929 − 0.368i)21-s + (0.876 − 0.481i)23-s + (0.187 + 0.982i)27-s + (−0.535 + 0.844i)29-s + (−0.637 + 0.770i)31-s + (−0.929 − 0.368i)33-s + (0.187 − 0.982i)37-s + (−0.187 − 0.982i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $0.979 - 0.199i$
Analytic conductor: \(4.64398\)
Root analytic conductor: \(4.64398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1000,\ (0:\ ),\ 0.979 - 0.199i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.503251645 - 0.1516345131i\)
\(L(\frac12)\) \(\approx\) \(1.503251645 - 0.1516345131i\)
\(L(1)\) \(\approx\) \(1.104411895 - 0.1798051497i\)
\(L(1)\) \(\approx\) \(1.104411895 - 0.1798051497i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.0627 - 0.998i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.425 - 0.904i)T \)
13 \( 1 + (0.992 - 0.125i)T \)
17 \( 1 + (-0.637 + 0.770i)T \)
19 \( 1 + (-0.0627 + 0.998i)T \)
23 \( 1 + (0.876 - 0.481i)T \)
29 \( 1 + (-0.535 + 0.844i)T \)
31 \( 1 + (-0.637 + 0.770i)T \)
37 \( 1 + (0.187 - 0.982i)T \)
41 \( 1 + (0.876 + 0.481i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (0.968 + 0.248i)T \)
53 \( 1 + (0.929 - 0.368i)T \)
59 \( 1 + (-0.728 + 0.684i)T \)
61 \( 1 + (-0.876 + 0.481i)T \)
67 \( 1 + (-0.535 - 0.844i)T \)
71 \( 1 + (0.968 + 0.248i)T \)
73 \( 1 + (0.728 + 0.684i)T \)
79 \( 1 + (0.0627 + 0.998i)T \)
83 \( 1 + (-0.0627 + 0.998i)T \)
89 \( 1 + (0.728 + 0.684i)T \)
97 \( 1 + (0.535 - 0.844i)T \)
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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

−21.614394870351292086387911629255, −20.76382833842224253125867267007, −20.32340331554001451222962041672, −19.654500246755476156229514531966, −18.46115840217749935253273074909, −17.41977892568940924683969666664, −17.1256427258918975504785283737, −16.1039872257760260733319140195, −15.39641109755326483014223893555, −14.73266894617055944367603466873, −13.73968128915577603046079016526, −13.20152977437498826696951433785, −11.78963364205247598727085210589, −11.119342379810370661740139518955, −10.5769543948829537961928035811, −9.39370008775738713000645922574, −9.1177023801441578722765350260, −7.80643942249953683912224025228, −6.98155948730770010626282466044, −5.98180431149492110328466841809, −4.795574926483138598696083377689, −4.3027036491990487597179668977, −3.4359567007829088005958895664, −2.23025842216712208229988261968, −0.80100977000179222419266828009, 1.0680766999140418791644950129, 1.930066345789099760690026140181, 2.972378977085853689744222640762, 3.96799634744354561877133876026, 5.5137313956737041281185528799, 5.927042537278345409806447184651, 6.79439136332164705181736447635, 7.87921769140240046369736188531, 8.72099466282487508913248168340, 9.02461991048670544346046378894, 10.92539560725374248305888400008, 11.03395857599603097827278443296, 12.32465490085210373762050467047, 12.670937646364230172220792657863, 13.7043472985402844478250343946, 14.40611117276722598258128414067, 15.19735864501217654744603829515, 16.28060178173598266536906922113, 16.952664951084586267272021063439, 18.04013898973429169011964741705, 18.41248611717051747023678985918, 19.177741689038976804349974446017, 19.86157669026610672773431584554, 20.92419658334346888372872055144, 21.60463836007167375302319595808

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