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Properties

Label 1-10e3-1000.363-r0-0-0
Degree $1$
Conductor $1000$
Sign $0.893 - 0.448i$
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  + (0.684 + 0.728i)3-s + (−0.587 − 0.809i)7-s + (−0.0627 + 0.998i)9-s + (0.535 − 0.844i)11-s + (−0.998 − 0.0627i)13-s + (0.904 + 0.425i)17-s + (−0.728 − 0.684i)19-s + (0.187 − 0.982i)21-s + (0.248 − 0.968i)23-s + (−0.770 + 0.637i)27-s + (0.876 + 0.481i)29-s + (0.425 − 0.904i)31-s + (0.982 − 0.187i)33-s + (−0.770 − 0.637i)37-s + (−0.637 − 0.770i)39-s + ⋯
L(s)  = 1  + (0.684 + 0.728i)3-s + (−0.587 − 0.809i)7-s + (−0.0627 + 0.998i)9-s + (0.535 − 0.844i)11-s + (−0.998 − 0.0627i)13-s + (0.904 + 0.425i)17-s + (−0.728 − 0.684i)19-s + (0.187 − 0.982i)21-s + (0.248 − 0.968i)23-s + (−0.770 + 0.637i)27-s + (0.876 + 0.481i)29-s + (0.425 − 0.904i)31-s + (0.982 − 0.187i)33-s + (−0.770 − 0.637i)37-s + (−0.637 − 0.770i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $0.893 - 0.448i$
Analytic conductor: \(4.64398\)
Root analytic conductor: \(4.64398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1000,\ (0:\ ),\ 0.893 - 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.599320208 - 0.3786527898i\)
\(L(\frac12)\) \(\approx\) \(1.599320208 - 0.3786527898i\)
\(L(1)\) \(\approx\) \(1.234331500 + 0.02054258418i\)
\(L(1)\) \(\approx\) \(1.234331500 + 0.02054258418i\)

Euler product

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

−21.54576037815293928614258387846, −20.95841814966634257108601371418, −19.923634765503534279775366810110, −19.26696827419990789118192711633, −18.90183911345850317806115144766, −17.77956761568607288437044703882, −17.2720675450129207067651579247, −16.09917239756177144542512348066, −15.21779884335935479204485406573, −14.57767844538529278926925292765, −13.892172038980388395188129717973, −12.77144866436055905967840061005, −12.26302497224391463277197790156, −11.76157333036217220999902114820, −10.122251919938876158644495269113, −9.55465144000629273315048668490, −8.77548575560172137502440039720, −7.81678358460854199221616846892, −7.04967651743969383242508826180, −6.27644151709969195598742440455, −5.26455237334344923995593435607, −4.04899873860645998036270147836, −2.98380752942310336275927181353, −2.28393652142198675627615848364, −1.22436753014478162536017268806, 0.69434921700493065917943145435, 2.30733497246393798840496754193, 3.1779877808958334266292384384, 4.005193008157990068352802680889, 4.76690865531024723643268494815, 5.94112562875993381221115033308, 6.95295749260838692738191899734, 7.83762720710041726463848508651, 8.75302399392261766535653899356, 9.470453992094269845971968162, 10.39140720216314633169775528879, 10.80196100640012260329325244808, 12.08810027896793360570618575841, 12.98290365419872836324639272917, 13.86547873621348582778362622656, 14.455716159375556062432416219203, 15.1889983230595026570042744297, 16.231989015582937746448083067881, 16.73172797644935048670362179330, 17.38697069820713097116489953900, 18.81865576208673855586392990525, 19.56304514554615129413540732146, 19.775367550767287221437077061207, 20.959962013213120181710193266707, 21.44526312420099273334707290867

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