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Properties

Label 1-10e3-1000.163-r0-0-0
Degree $1$
Conductor $1000$
Sign $-0.459 + 0.888i$
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  + (−0.481 + 0.876i)3-s + (−0.587 − 0.809i)7-s + (−0.535 − 0.844i)9-s + (−0.929 + 0.368i)11-s + (0.844 − 0.535i)13-s + (0.684 − 0.728i)17-s + (−0.876 + 0.481i)19-s + (0.992 − 0.125i)21-s + (−0.770 + 0.637i)23-s + (0.998 − 0.0627i)27-s + (−0.187 + 0.982i)29-s + (−0.728 − 0.684i)31-s + (0.125 − 0.992i)33-s + (0.998 + 0.0627i)37-s + (0.0627 + 0.998i)39-s + ⋯
L(s)  = 1  + (−0.481 + 0.876i)3-s + (−0.587 − 0.809i)7-s + (−0.535 − 0.844i)9-s + (−0.929 + 0.368i)11-s + (0.844 − 0.535i)13-s + (0.684 − 0.728i)17-s + (−0.876 + 0.481i)19-s + (0.992 − 0.125i)21-s + (−0.770 + 0.637i)23-s + (0.998 − 0.0627i)27-s + (−0.187 + 0.982i)29-s + (−0.728 − 0.684i)31-s + (0.125 − 0.992i)33-s + (0.998 + 0.0627i)37-s + (0.0627 + 0.998i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $-0.459 + 0.888i$
Analytic conductor: \(4.64398\)
Root analytic conductor: \(4.64398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1000,\ (0:\ ),\ -0.459 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3357707943 + 0.5518119315i\)
\(L(\frac12)\) \(\approx\) \(0.3357707943 + 0.5518119315i\)
\(L(1)\) \(\approx\) \(0.7040478022 + 0.1926309287i\)
\(L(1)\) \(\approx\) \(0.7040478022 + 0.1926309287i\)

Euler product

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

−21.53211796005016008230417379948, −20.695715580860141059044753745911, −19.51050791057022009380488881044, −18.91112651788756120849834115561, −18.449542066888847253082594687659, −17.62797649362313374034131682284, −16.63879909690000794096580317047, −16.05652080257896010002379606514, −15.17096935930759446838219261754, −14.09224041816455744710197203703, −13.26960384662175770325548117418, −12.64706528070662702693621411562, −11.98611634289515540169660599207, −10.99865836316278005934448025348, −10.37006084658144797349637821222, −9.06990302545754463818343782730, −8.33597128692076000202573974608, −7.55437923804340414313054858669, −6.27605655196851099103077345930, −6.07105845142471950708402180432, −5.0452548031454137414891707399, −3.739085137562846013582198481982, −2.56741710910950424668516889486, −1.83214684828369149806207355827, −0.335861174472381957564956279769, 1.04762454707555317807411795993, 2.7263018385018494968547668052, 3.65273146738317247103981038989, 4.348377031167646791383679552343, 5.47640650221419967364938900653, 6.06057261093571611219840256378, 7.2258072820811217723224762854, 8.06442307707207829605819840847, 9.21957262624426993783076222130, 10.01596433040380698991644672809, 10.565367252352929352404128230869, 11.27935663866590552282437709635, 12.39419286699863700767014677292, 13.11504711250494499018477708340, 14.030315885405730145342592544481, 14.98788039526587841628140798652, 15.75550854780836986211317938252, 16.40317409924597007086836652465, 16.97791008424373277935097131242, 18.026604232079322710694816922084, 18.56434743543885982284056803850, 19.88134045498244955083634789822, 20.448937674869315194480789298851, 21.05201926011919744504352041677, 21.947361510217924287480788526647

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