Properties

Label 1-147-147.5-r0-0-0
Degree $1$
Conductor $147$
Sign $0.969 - 0.243i$
Analytic cond. $0.682665$
Root an. cond. $0.682665$
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  + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.988 − 0.149i)5-s + (−0.623 + 0.781i)8-s + (0.988 − 0.149i)10-s + (0.733 + 0.680i)11-s + (0.222 − 0.974i)13-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.900 + 0.433i)20-s + (−0.900 − 0.433i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (0.0747 + 0.997i)26-s + ⋯
L(s)  = 1  + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.988 − 0.149i)5-s + (−0.623 + 0.781i)8-s + (0.988 − 0.149i)10-s + (0.733 + 0.680i)11-s + (0.222 − 0.974i)13-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.900 + 0.433i)20-s + (−0.900 − 0.433i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (0.0747 + 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.969 - 0.243i$
Analytic conductor: \(0.682665\)
Root analytic conductor: \(0.682665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (0:\ ),\ 0.969 - 0.243i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6250444034 - 0.07719782382i\)
\(L(\frac12)\) \(\approx\) \(0.6250444034 - 0.07719782382i\)
\(L(1)\) \(\approx\) \(0.6552868890 + 0.0009362437754i\)
\(L(1)\) \(\approx\) \(0.6552868890 + 0.0009362437754i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.955 + 0.294i)T \)
5 \( 1 + (-0.988 - 0.149i)T \)
11 \( 1 + (0.733 + 0.680i)T \)
13 \( 1 + (0.222 - 0.974i)T \)
17 \( 1 + (0.0747 - 0.997i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.0747 - 0.997i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.826 + 0.563i)T \)
41 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (-0.826 + 0.563i)T \)
59 \( 1 + (-0.988 + 0.149i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
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

−28.13926534446631907622176821806, −27.172515589079867601211803057907, −26.545772763894717950015486767861, −25.54676585311772250689339401022, −24.30485167810207031878722876658, −23.54999644457151434499181838205, −22.0071939792379622262353650607, −21.24116044956404921607997645443, −19.6548888680024260122316428270, −19.53565775949467285037782111689, −18.36977482769106405468311511456, −17.2120860939200907188946549894, −16.22668023672981476184052017852, −15.42958897451449410154894477896, −14.06291916643086568097971037765, −12.44325857761025363957453602883, −11.517320842769418112902378193625, −10.82225489165289237703008909990, −9.31767899475512979250887062048, −8.46450658374618659091809185254, −7.33517944920092291418922859258, −6.29907634036503508575571752740, −4.16248026633437961815498503993, −3.06219033330936147764369548291, −1.24921177535168384842669961887, 0.933619498113316701513380037826, 2.83591344808341412822941986809, 4.49470973843552403020916699381, 6.07223297071615687757020675000, 7.36622195083247284699307676917, 8.115523554539010566436685334733, 9.31099777023602157169333632618, 10.41017620853175416293938825011, 11.60648395510431179029541264144, 12.405114931784344989100526658, 14.267837122534313743324318634798, 15.29354695262048661489943615459, 16.08018140171176998600694535543, 17.086282952151063001550605098832, 18.1896594646744135069519049078, 19.06304361471190519887724071558, 20.190735897709280242219010795724, 20.58458397124500475068754824820, 22.56838729755449426632251807824, 23.244628767147287698009182332964, 24.59673573124088339830962417284, 25.09430818580623156011943389818, 26.381751404750168399255618148057, 27.310913395141092741979223169965, 27.80214809187897961224510007464

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