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Properties

Label 1-1800-1800.923-r1-0-0
Degree $1$
Conductor $1800$
Sign $-0.0906 + 0.995i$
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 + (−0.669 − 0.743i)11-s + (−0.743 − 0.669i)13-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.207 − 0.978i)23-s + (−0.913 + 0.406i)29-s + (−0.913 − 0.406i)31-s + (−0.951 + 0.309i)37-s + (−0.669 + 0.743i)41-s + (−0.866 − 0.5i)43-s + (−0.406 − 0.913i)47-s + (0.5 + 0.866i)49-s + (0.587 − 0.809i)53-s + (0.669 − 0.743i)59-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (−0.669 − 0.743i)11-s + (−0.743 − 0.669i)13-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.207 − 0.978i)23-s + (−0.913 + 0.406i)29-s + (−0.913 − 0.406i)31-s + (−0.951 + 0.309i)37-s + (−0.669 + 0.743i)41-s + (−0.866 − 0.5i)43-s + (−0.406 − 0.913i)47-s + (0.5 + 0.866i)49-s + (0.587 − 0.809i)53-s + (0.669 − 0.743i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0906 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0906 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.0906 + 0.995i$
Analytic conductor: \(193.436\)
Root analytic conductor: \(193.436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (923, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (1:\ ),\ -0.0906 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2701515025 - 0.2958536459i\)
\(L(\frac12)\) \(\approx\) \(-0.2701515025 - 0.2958536459i\)
\(L(1)\) \(\approx\) \(0.6612964836 - 0.2668662541i\)
\(L(1)\) \(\approx\) \(0.6612964836 - 0.2668662541i\)

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 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.743 - 0.669i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (-0.207 - 0.978i)T \)
29 \( 1 + (-0.913 + 0.406i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.406 - 0.913i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (-0.669 - 0.743i)T \)
67 \( 1 + (0.406 - 0.913i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (-0.994 + 0.104i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.406 - 0.913i)T \)
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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

−20.50159204501211661413335722578, −19.71149727535284008214641919046, −19.16556814310426895032817753464, −18.35830809465868753018006219242, −17.680760419955145283634615157069, −16.81887161493733073324130805568, −16.10805444691479716108478149833, −15.36553613479841259235763255891, −14.81513526098729495503652619140, −13.784820186066341989437243107079, −13.062784668477185596674883901041, −12.35320939852617360996897539669, −11.7701746958583277006938769321, −10.71647971427943432945071119859, −9.86535441824764927728308537157, −9.39758177281357473729349252105, −8.49525296407400031054581720815, −7.42984743936856820600830649421, −6.94007336822949579980064397669, −5.83119014310644273107570180753, −5.2588022625642385490453512575, −4.1505000456041771438450919776, −3.34494481554870937384497875531, −2.31488595493021492980690017533, −1.59963590788173566580261856042, 0.12887188766405523465735337794, 0.48547471822741802941220210004, 2.0554101017630721928997874732, 3.03825560004981839454408487338, 3.56537329060930855660974529297, 4.88589159961841474491337985223, 5.40660214186184900234864878529, 6.549844684187774737725675805796, 7.1530069400169377118178259209, 7.98147508957136291267463030568, 8.91609779696064331005288894599, 9.721828392509480636224666680715, 10.387606578944995187310566638989, 11.15079942091634315975152813182, 12.00644484342122646505282414628, 12.99479610868810997743439999740, 13.35830578733958061947572833971, 14.18497440158671643136301820441, 15.14648904123138765859335328112, 15.85026482353452592436658303619, 16.5199720052981260811142150288, 17.111130959162665152361027957293, 18.26362528097644001719372563295, 18.531155559875837028948248949499, 19.67389641151947662363566313179

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