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Properties

Label 1-1800-1800.371-r0-0-0
Degree $1$
Conductor $1800$
Sign $0.959 + 0.282i$
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 + (0.104 − 0.994i)11-s + (0.104 + 0.994i)13-s + (−0.309 + 0.951i)17-s + (0.309 − 0.951i)19-s + (0.913 − 0.406i)23-s + (0.669 − 0.743i)29-s + (−0.669 − 0.743i)31-s + (0.809 − 0.587i)37-s + (0.104 + 0.994i)41-s + (−0.5 − 0.866i)43-s + (0.669 − 0.743i)47-s + (−0.5 + 0.866i)49-s + (0.309 + 0.951i)53-s + (0.104 + 0.994i)59-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s + (0.104 − 0.994i)11-s + (0.104 + 0.994i)13-s + (−0.309 + 0.951i)17-s + (0.309 − 0.951i)19-s + (0.913 − 0.406i)23-s + (0.669 − 0.743i)29-s + (−0.669 − 0.743i)31-s + (0.809 − 0.587i)37-s + (0.104 + 0.994i)41-s + (−0.5 − 0.866i)43-s + (0.669 − 0.743i)47-s + (−0.5 + 0.866i)49-s + (0.309 + 0.951i)53-s + (0.104 + 0.994i)59-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.959 + 0.282i$
Analytic conductor: \(8.35916\)
Root analytic conductor: \(8.35916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (0:\ ),\ 0.959 + 0.282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.759554714 + 0.2535557077i\)
\(L(\frac12)\) \(\approx\) \(1.759554714 + 0.2535557077i\)
\(L(1)\) \(\approx\) \(1.188166106 + 0.08678488697i\)
\(L(1)\) \(\approx\) \(1.188166106 + 0.08678488697i\)

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

−20.31769681038171396966522477742, −19.63228149013610155346792794234, −18.54511012409298378239593900083, −17.82762939582480868896357115450, −17.41355790218552314228706570650, −16.47012098017569267276192564726, −15.78984370284282948874010780349, −14.84737239131098550973864812272, −14.346621673733952080000049888344, −13.45469682459760165317762953357, −12.763207316503517087572049160907, −11.9635045194877071584248823706, −11.086966304344001737427410157881, −10.3916020210057211072421999333, −9.7254605226036977272766916812, −8.80118947914087871158539087709, −7.78651805528252078913332254640, −7.32124056867254639677471705063, −6.49215028709643826843001239178, −5.24786259301647408530786192560, −4.78530589596228940959797807513, −3.742684991896335812762490428922, −2.92025873747630888290018918719, −1.72527547100092491031411791742, −0.8606745948746860729536106508, 0.900046445456895273829818705892, 2.05505737891670183408208720571, 2.78085501189164093919793187807, 3.912925034109424307491870966688, 4.70803208451950802127820670806, 5.67157625639123768528517084702, 6.30727630142096983330828362950, 7.21574379180619511055484701467, 8.29786085813350075828122747178, 8.81917902840999379145999500344, 9.45420418849279789757972699321, 10.65675314224221897812069779264, 11.34613395321964245360001906527, 11.8104293117783363618917352095, 12.85711063425353387818241718900, 13.53173451855245171645321450297, 14.36496206599711199841074416063, 15.074419216899465376146227454201, 15.724550296944190343258134175762, 16.63704298118605262496239921615, 17.20513241536625156009614855492, 18.15868971930268820774893091612, 18.78565049047088805673571463087, 19.34249614256485559089181632816, 20.20489716999850082197217458024

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