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Properties

Label 1-1800-1800.1021-r0-0-0
Degree $1$
Conductor $1800$
Sign $-0.400 - 0.916i$
Analytic cond. $8.35916$
Root an. cond. $8.35916$
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  + (−0.5 + 0.866i)7-s + (−0.913 + 0.406i)11-s + (−0.913 − 0.406i)13-s + (0.309 − 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.104 + 0.994i)23-s + (0.978 + 0.207i)29-s + (−0.978 + 0.207i)31-s + (0.809 − 0.587i)37-s + (0.913 + 0.406i)41-s + (0.5 − 0.866i)43-s + (−0.978 − 0.207i)47-s + (−0.5 − 0.866i)49-s + (−0.309 − 0.951i)53-s + (−0.913 − 0.406i)59-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (−0.913 + 0.406i)11-s + (−0.913 − 0.406i)13-s + (0.309 − 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.104 + 0.994i)23-s + (0.978 + 0.207i)29-s + (−0.978 + 0.207i)31-s + (0.809 − 0.587i)37-s + (0.913 + 0.406i)41-s + (0.5 − 0.866i)43-s + (−0.978 − 0.207i)47-s + (−0.5 − 0.866i)49-s + (−0.309 − 0.951i)53-s + (−0.913 − 0.406i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.400 - 0.916i$
Analytic conductor: \(8.35916\)
Root analytic conductor: \(8.35916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (0:\ ),\ -0.400 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1996030181 - 0.3050253838i\)
\(L(\frac12)\) \(\approx\) \(0.1996030181 - 0.3050253838i\)
\(L(1)\) \(\approx\) \(0.7629034789 + 0.04730855568i\)
\(L(1)\) \(\approx\) \(0.7629034789 + 0.04730855568i\)

Euler product

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

−20.283488313243207121463425251710, −19.68211523777665085502948756536, −19.085006802942297160515955747686, −18.25098037110785573773630746604, −17.363956467795407491876352640, −16.75359822493892609808448193057, −16.125235323777538553086004043623, −15.27352424730885009395030066220, −14.44148454005162840967894114945, −13.77520774767211682610558448494, −12.844847172375811610459080285968, −12.53398608420896316271935190585, −11.25376981410122035857509922041, −10.6613233909308600391743599869, −9.971071735340560710735714639, −9.17876147920165232241215185375, −8.15200957199051457383036394195, −7.51955854111256700787298961114, −6.63269329813213884848340361196, −5.9416343113384610380483577264, −4.76456727373161894442221900020, −4.22172175647755017019394473144, −3.06284185575478326677160889841, −2.39346838313217888247782639510, −1.04538988746704214851966616480, 0.13798281064263391231960135062, 1.75208999333379420817277533400, 2.65190237416191349059472883817, 3.28021929224850461048314992118, 4.5478198560107238983738146229, 5.38231060332351784343035920176, 5.910970826409079080927695703334, 7.10403436458250397125578999575, 7.70284771919395020963715359491, 8.572678551701085837876450187116, 9.62312108715020415227299936508, 9.9191828142929948469195496131, 10.96384811632637881934277964483, 11.89531598007165404014313958411, 12.55838486297744815991943812140, 13.03601236534866024814147854029, 14.16966743984670433796094203188, 14.78541881316916708939566028288, 15.67347836746482050684995920604, 16.0904674833471124892021589091, 17.0074223157739702718484481350, 18.01058793238524474176720030934, 18.33445248694013046513231124749, 19.27772092727912822647121187629, 19.86776098545257720567096942316

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