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Properties

Label 1-800-800.187-r0-0-0
Degree $1$
Conductor $800$
Sign $0.389 + 0.920i$
Analytic cond. $3.71518$
Root an. cond. $3.71518$
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  + (0.987 + 0.156i)3-s + 7-s + (0.951 + 0.309i)9-s + (−0.453 + 0.891i)11-s + (−0.891 + 0.453i)13-s + (−0.587 + 0.809i)17-s + (−0.987 + 0.156i)19-s + (0.987 + 0.156i)21-s + (0.309 + 0.951i)23-s + (0.891 + 0.453i)27-s + (−0.156 + 0.987i)29-s + (0.809 + 0.587i)31-s + (−0.587 + 0.809i)33-s + (0.453 + 0.891i)37-s + (−0.951 + 0.309i)39-s + ⋯
L(s)  = 1  + (0.987 + 0.156i)3-s + 7-s + (0.951 + 0.309i)9-s + (−0.453 + 0.891i)11-s + (−0.891 + 0.453i)13-s + (−0.587 + 0.809i)17-s + (−0.987 + 0.156i)19-s + (0.987 + 0.156i)21-s + (0.309 + 0.951i)23-s + (0.891 + 0.453i)27-s + (−0.156 + 0.987i)29-s + (0.809 + 0.587i)31-s + (−0.587 + 0.809i)33-s + (0.453 + 0.891i)37-s + (−0.951 + 0.309i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.389 + 0.920i$
Analytic conductor: \(3.71518\)
Root analytic conductor: \(3.71518\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (0:\ ),\ 0.389 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.726075801 + 1.143548052i\)
\(L(\frac12)\) \(\approx\) \(1.726075801 + 1.143548052i\)
\(L(1)\) \(\approx\) \(1.453615010 + 0.3737225048i\)
\(L(1)\) \(\approx\) \(1.453615010 + 0.3737225048i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.987 + 0.156i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.453 + 0.891i)T \)
13 \( 1 + (-0.891 + 0.453i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (-0.987 + 0.156i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (-0.156 + 0.987i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.453 + 0.891i)T \)
41 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (0.156 - 0.987i)T \)
59 \( 1 + (0.891 - 0.453i)T \)
61 \( 1 + (-0.891 - 0.453i)T \)
67 \( 1 + (-0.156 - 0.987i)T \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.156 - 0.987i)T \)
89 \( 1 + (0.951 - 0.309i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−21.90016030046702137511106428929, −21.10627937474908943431453731364, −20.6434769399795242116590434559, −19.69792998445435869832151836098, −18.99317690512703652879489593180, −18.21539112650498552510784916, −17.42471291064806202076813183028, −16.42353108336385645590708576593, −15.351519065718779249397341469845, −14.83800340709082862909762591905, −13.99584209814555860541155437702, −13.30681373009942706232028005937, −12.44081863808081299456008167214, −11.368903477285905029111477188544, −10.55926148553007108812826724725, −9.55857412570912667713055146245, −8.60431089266163687029008535140, −8.02648622260554285780162277033, −7.24897646323885490257736349453, −6.10897198837728224309429927437, −4.84535540049956175414450190802, −4.182787995419184380963357356218, −2.729174276440612748101871229354, −2.32170500053484747637916023185, −0.824209712927794866432819899994, 1.728057078829216268503804564451, 2.16294662934622006400482468353, 3.46948894853398049444949420958, 4.552321893887670631638559156700, 5.038142423696457927317142217881, 6.67808390131948533867819525102, 7.47246708281300224242050745472, 8.28139708042815260234660428285, 8.990214346590885042358252459315, 10.02732593449150005388044430255, 10.6687982827017702352341300254, 11.83723126761555573815582088284, 12.73012712561517667792009311029, 13.528700109116520977314611525304, 14.506416554093526872502909910975, 14.99764112397806843682026474106, 15.61281162009337822120475924351, 16.917015182299566307497265278015, 17.58570837252707983588304790586, 18.485182115704082488679331265137, 19.38787280932291702616376420574, 20.00466392794614681910312176162, 20.85475519810575843561653963685, 21.45123460027172338065630615577, 22.12763887908340710126793391109

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