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Properties

Label 1-147-147.107-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: $\chi_{147} (107, \cdot )$
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

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

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