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Properties

Label 1-147-147.11-r1-0-0
Degree $1$
Conductor $147$
Sign $-0.462 - 0.886i$
Analytic cond. $15.7973$
Root an. cond. $15.7973$
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  + (−0.0747 + 0.997i)2-s + (−0.988 − 0.149i)4-s + (0.733 + 0.680i)5-s + (0.222 − 0.974i)8-s + (−0.733 + 0.680i)10-s + (−0.826 − 0.563i)11-s + (−0.900 + 0.433i)13-s + (0.955 + 0.294i)16-s + (−0.365 + 0.930i)17-s + (−0.5 + 0.866i)19-s + (−0.623 − 0.781i)20-s + (0.623 − 0.781i)22-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.365 − 0.930i)26-s + ⋯
L(s)  = 1  + (−0.0747 + 0.997i)2-s + (−0.988 − 0.149i)4-s + (0.733 + 0.680i)5-s + (0.222 − 0.974i)8-s + (−0.733 + 0.680i)10-s + (−0.826 − 0.563i)11-s + (−0.900 + 0.433i)13-s + (0.955 + 0.294i)16-s + (−0.365 + 0.930i)17-s + (−0.5 + 0.866i)19-s + (−0.623 − 0.781i)20-s + (0.623 − 0.781i)22-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.365 − 0.930i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.462 - 0.886i$
Analytic conductor: \(15.7973\)
Root analytic conductor: \(15.7973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (1:\ ),\ -0.462 - 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1643137304 + 0.2710526849i\)
\(L(\frac12)\) \(\approx\) \(-0.1643137304 + 0.2710526849i\)
\(L(1)\) \(\approx\) \(0.6068039247 + 0.4430444893i\)
\(L(1)\) \(\approx\) \(0.6068039247 + 0.4430444893i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.0747 + 0.997i)T \)
5 \( 1 + (0.733 + 0.680i)T \)
11 \( 1 + (-0.826 - 0.563i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (-0.365 + 0.930i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.365 - 0.930i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.988 + 0.149i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (0.988 + 0.149i)T \)
59 \( 1 + (0.733 - 0.680i)T \)
61 \( 1 + (-0.988 + 0.149i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (0.0747 + 0.997i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
97 \( 1 + T \)
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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

−27.61284599227213653821803989651, −26.41162852288026438637503305234, −25.46027591841176124867487126597, −24.24495651297950947763922705279, −23.166211211857080607966286219576, −22.01423816303001961011558780535, −21.279265159510193639084823892216, −20.28326627625354249438060331326, −19.62295205271742177278884565481, −18.06022056277613772119369138329, −17.66131233303724574432778533819, −16.390959347397110438979268322643, −14.89875059210527499457466819669, −13.57030954548728418697833407007, −12.907413007966607373124511503892, −11.91713341793497662686946739314, −10.57120338817236836578943444378, −9.6715040454263509523516642830, −8.78051831209663550339359799307, −7.36926902248786952887986563385, −5.36994237158208266651574541012, −4.68633168642923456822767937652, −2.88188704191740949481711125784, −1.77589761709716038689037095112, −0.11662559917347098874539480307, 2.19029817327759210482988907906, 3.95331182656333224826774926372, 5.46382620977225317274767974, 6.31183742725239979979479398670, 7.45940656508831650013242039990, 8.60308969788404365948163894995, 9.87447814901301528068362608500, 10.704924418756836521801886353488, 12.56063553980061487057449939821, 13.62158589278141853270237516663, 14.51048244029233651112807545730, 15.35058952680080459812333806295, 16.67582378497440442815551611435, 17.38066532326463018882517804150, 18.54705880257312675778567396194, 19.12617395700853083990550918626, 20.95937931499895623425567362754, 21.94729979344030118107611072348, 22.68478660105913656622860418427, 23.94585896613118477823334937128, 24.606308804097184801723003553835, 25.87977226776734560068002044032, 26.312911185268235140474275542564, 27.27365395195596133529121638260, 28.57200254951345092668846642954

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