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Properties

Label 1-1200-1200.1139-r0-0-0
Degree $1$
Conductor $1200$
Sign $0.868 + 0.495i$
Analytic cond. $5.57277$
Root an. cond. $5.57277$
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  − 7-s + (−0.951 − 0.309i)11-s + (0.951 − 0.309i)13-s + (−0.809 + 0.587i)17-s + (0.587 + 0.809i)19-s + (0.309 − 0.951i)23-s + (−0.587 + 0.809i)29-s + (0.809 − 0.587i)31-s + (−0.951 + 0.309i)37-s + (0.309 + 0.951i)41-s − i·43-s + (0.809 + 0.587i)47-s + 49-s + (−0.587 + 0.809i)53-s + (0.951 − 0.309i)59-s + ⋯
L(s)  = 1  − 7-s + (−0.951 − 0.309i)11-s + (0.951 − 0.309i)13-s + (−0.809 + 0.587i)17-s + (0.587 + 0.809i)19-s + (0.309 − 0.951i)23-s + (−0.587 + 0.809i)29-s + (0.809 − 0.587i)31-s + (−0.951 + 0.309i)37-s + (0.309 + 0.951i)41-s − i·43-s + (0.809 + 0.587i)47-s + 49-s + (−0.587 + 0.809i)53-s + (0.951 − 0.309i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.868 + 0.495i$
Analytic conductor: \(5.57277\)
Root analytic conductor: \(5.57277\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1200,\ (0:\ ),\ 0.868 + 0.495i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096462459 + 0.2907217533i\)
\(L(\frac12)\) \(\approx\) \(1.096462459 + 0.2907217533i\)
\(L(1)\) \(\approx\) \(0.9234181412 + 0.04473211213i\)
\(L(1)\) \(\approx\) \(0.9234181412 + 0.04473211213i\)

Euler product

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

−20.999155891294061165101477155875, −20.45386198022039352047566096384, −19.46952031413221266019325177934, −18.94523030139099485210570602637, −17.97926576213325720022398524217, −17.45978235117933700987542115372, −16.21528691925920257006380966026, −15.79065164246648488674435590052, −15.23834326947116559288453717394, −13.79931935782687208242926640642, −13.43858374912940628840772675626, −12.69797256300225770111336946410, −11.65517827991813444251957423689, −10.95114219746725515255265719026, −10.00552287608800175243954966524, −9.29494374243933241824744655840, −8.52380175672179103002954477082, −7.367278751899527458997473947213, −6.775943931426594666853488410020, −5.78358047439168398346545376162, −4.95294409061828559794947755073, −3.84732116946784083364734845271, −2.9956446106475732094334808052, −2.08345140077348233475469914698, −0.60782282288936992067490256355, 0.86033314894241046079758588141, 2.24994978808044752614762516818, 3.19961876134217547972335628045, 3.93001245982197973453662463642, 5.14708099570187216726056076930, 6.0280192501792059901253537482, 6.65852673333408457979934890801, 7.77456992241031408683705198253, 8.54449460743905745216692688571, 9.35777909407605889654339530068, 10.43198236688918472312782054363, 10.76821850546079695686892574166, 11.962752742016715324122201831771, 12.86999727687419145442354971995, 13.295971418715612051216587682262, 14.16872925621005948503915227421, 15.30124444289165674891978212779, 15.84053493017606727324352960698, 16.50311770310120516335886089231, 17.37501071688267211157826180548, 18.49241297859898433267571444669, 18.71251946219135413640461899496, 19.743391162822259396056740018822, 20.55871134184415361023689987812, 21.07556261122832513148621679665

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