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Properties

Label 1-10e3-1000.477-r1-0-0
Degree $1$
Conductor $1000$
Sign $0.459 + 0.888i$
Analytic cond. $107.464$
Root an. cond. $107.464$
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

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

Invariants

Degree: \(1\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $0.459 + 0.888i$
Analytic conductor: \(107.464\)
Root analytic conductor: \(107.464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1000,\ (1:\ ),\ 0.459 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.040469401 + 1.850088350i\)
\(L(\frac12)\) \(\approx\) \(3.040469401 + 1.850088350i\)
\(L(1)\) \(\approx\) \(1.533880434 + 0.4913515131i\)
\(L(1)\) \(\approx\) \(1.533880434 + 0.4913515131i\)

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

−21.21638238492289227962303298402, −20.42198112392720490085920855338, −19.807292882043299376081190078033, −18.78233747771723413760058073977, −18.344248972238912100813645586517, −17.68228909114198214094458031348, −16.67226819468181458652484112383, −15.72405578329547163013454192802, −14.76168074959080031113569144134, −14.259763096921072972295620052, −13.39652226836990458830903200668, −12.59510425462467846058914048414, −11.67964267522795669054580232051, −11.29906714868991145933121902700, −9.85943704197170875520345506179, −8.83430571962557705359160223973, −8.48641513852494046403193147457, −7.46292137050841697258907155761, −6.602517134992685218659560552802, −5.75421865667193027612538077654, −4.82739772740006525161684978158, −3.3345797215476667124811505044, −2.81977198125453705164173682758, −1.46813096513125022920548752569, −0.873342911059895358553563259311, 1.0428400233178503362537608647, 1.961880465936680238470111145368, 3.48555135417507001052172615445, 3.90868796988918309094881992315, 4.81425021099639486391142445341, 5.8241493402885794751534160254, 6.94785081232158858351008565963, 7.94873629642930278037482561333, 8.581369933474302009595660158992, 9.719696257663002810039285885778, 10.06317884281120197716966880505, 11.35730278867170525395699748795, 11.56355885509048807310163205793, 13.11746989655723136880518341643, 13.844117932551289391787534298247, 14.534099590825937934338951088485, 15.14185624723424609815068343032, 16.14487968556856419241627728891, 16.88871048539200781727066036290, 17.38228178128482077724237079056, 18.599869039582734082227708935761, 19.44653230747045785620345607809, 20.14891867055755772783126448394, 20.92949568809324088049201831434, 21.30467988460909476532135649985

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