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Properties

Label 1-1800-1800.1283-r1-0-0
Degree $1$
Conductor $1800$
Sign $0.959 - 0.282i$
Analytic cond. $193.436$
Root an. cond. $193.436$
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  + (−0.866 − 0.5i)7-s + (0.104 + 0.994i)11-s + (0.994 + 0.104i)13-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.406 + 0.913i)23-s + (−0.669 − 0.743i)29-s + (−0.669 + 0.743i)31-s + (0.587 − 0.809i)37-s + (0.104 − 0.994i)41-s + (−0.866 − 0.5i)43-s + (0.743 − 0.669i)47-s + (0.5 + 0.866i)49-s + (0.951 + 0.309i)53-s + (−0.104 + 0.994i)59-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (0.104 + 0.994i)11-s + (0.994 + 0.104i)13-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.406 + 0.913i)23-s + (−0.669 − 0.743i)29-s + (−0.669 + 0.743i)31-s + (0.587 − 0.809i)37-s + (0.104 − 0.994i)41-s + (−0.866 − 0.5i)43-s + (0.743 − 0.669i)47-s + (0.5 + 0.866i)49-s + (0.951 + 0.309i)53-s + (−0.104 + 0.994i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.959 - 0.282i$
Analytic conductor: \(193.436\)
Root analytic conductor: \(193.436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (1:\ ),\ 0.959 - 0.282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.396105814 - 0.2011819212i\)
\(L(\frac12)\) \(\approx\) \(1.396105814 - 0.2011819212i\)
\(L(1)\) \(\approx\) \(0.9111748450 + 0.01014740076i\)
\(L(1)\) \(\approx\) \(0.9111748450 + 0.01014740076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (0.994 + 0.104i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.406 + 0.913i)T \)
29 \( 1 + (-0.669 - 0.743i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.104 - 0.994i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.743 - 0.669i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (-0.743 - 0.669i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (-0.207 + 0.978i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.743 - 0.669i)T \)
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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

−20.16523607234844509230946038178, −19.158859573195598503159336698412, −18.548099844110186738335063804006, −18.15440169815541847372442658952, −16.84067658900011121318671377771, −16.36431953600429216593431282223, −15.7663982232458220295037872146, −14.8938240999023269046233853543, −14.10296412997299509688851937903, −13.14174853578864782075605636161, −12.863128177186761954744919061617, −11.71090991800941630103592138037, −11.120583828328580405284834239587, −10.27609455943559649408338188462, −9.394379946537614391129970437909, −8.65763969903573166813192529936, −8.08895836948620124398847087969, −6.825340191805774215375162494779, −6.15572891601115992141568314967, −5.643829651053278219096830807578, −4.37311116397945287594604766769, −3.53839460485051529680977951058, −2.80445943961299580538026838845, −1.7293457560660890822548185424, −0.532949921877731040445815922009, 0.43941312518706280762603594972, 1.66646419632620416678911382046, 2.53738211422828211398152382866, 3.7879656627469126824972114839, 4.13534588878000719281724538649, 5.34579589972143048962607244416, 6.25284581901652851448253954955, 6.97906521536552337779978416219, 7.58638834405464795684624397226, 8.87552183704290704655387077958, 9.253684356152409818267491541125, 10.28878234083016532757974991382, 10.849716279424144990273292667920, 11.76952537755217725656620095250, 12.63343692674608178546696782871, 13.40568474075843608160792748541, 13.73416990714926969604434381100, 15.08036355289514423909635813423, 15.447104320551767607363937740033, 16.29227568900657340925738371845, 17.04619797259349972839046561163, 17.81055662668628648924748071032, 18.38550730054748083631149916020, 19.61287743242857376382297843726, 19.734252645364948367680385066653

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