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Properties

Label 1-1800-1800.853-r1-0-0
Degree $1$
Conductor $1800$
Sign $-0.282 + 0.959i$
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

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

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.282 + 0.959i$
Analytic conductor: \(193.436\)
Root analytic conductor: \(193.436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (1:\ ),\ -0.282 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2732115592 + 0.3652094290i\)
\(L(\frac12)\) \(\approx\) \(0.2732115592 + 0.3652094290i\)
\(L(1)\) \(\approx\) \(0.7999272201 - 0.04293758480i\)
\(L(1)\) \(\approx\) \(0.7999272201 - 0.04293758480i\)

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

−19.92033228462816747035106914878, −19.16242927426470063431163850797, −18.32057526542962905426785559712, −17.51786563741875147277655920505, −16.83237572421828175462451494127, −16.24738972435321664234031885638, −15.21578131488296002878462470212, −14.78111540016723847214743310665, −13.7812711244234571054657620088, −13.047968976949877396407275145271, −12.36630688237187558271215077452, −11.75003889566294032068296708110, −10.452271877759568817062746298666, −10.12775357290988434751583531490, −9.33318548694216168941701472088, −8.40603375229916285635296308118, −7.43322968533372943530407705174, −6.79428391763835919243051214185, −6.11857819257914916147337447735, −4.850648674674115623667237948010, −4.339771909377720641570199186476, −3.244138557958172661138320735186, −2.4288710747081091027247844708, −1.35663366440624915987334002755, −0.11230598987243357002591631219, 0.7030789188460441315912878329, 2.16155752107494937644200038692, 2.90125004234951858455607805548, 3.682779583465494444956996144563, 4.876956043132721841423335505845, 5.50825841987727933753743810035, 6.59921201090841677787500179764, 6.99458089492031573515508030544, 8.19409143355363406646825140605, 8.98270228261915412325824930416, 9.55237566995030720746787902456, 10.40656981863811164167845521957, 11.405127381880067275988523285371, 11.91129144791690302193326996814, 12.8681906101781043387887728165, 13.53116356621321562054638586149, 14.168973556455973583324578170760, 15.32987061505866629358721200556, 15.69446260803760088769757040502, 16.48722125821161284789445843239, 17.34098720254952371575769898062, 17.955165124787301213683965789146, 18.96554668950252509345788246857, 19.521331427715392993772331318273, 19.902360862856520118068217793818

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