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Properties

Label 1-147-147.8-r1-0-0
Degree $1$
Conductor $147$
Sign $0.718 - 0.695i$
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  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.623 + 0.781i)5-s + (−0.623 + 0.781i)8-s + (0.623 + 0.781i)10-s + (0.222 − 0.974i)11-s + (−0.222 + 0.974i)13-s + (0.623 + 0.781i)16-s + (0.900 − 0.433i)17-s + 19-s + (0.900 − 0.433i)20-s + (−0.900 − 0.433i)22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)26-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.623 + 0.781i)5-s + (−0.623 + 0.781i)8-s + (0.623 + 0.781i)10-s + (0.222 − 0.974i)11-s + (−0.222 + 0.974i)13-s + (0.623 + 0.781i)16-s + (0.900 − 0.433i)17-s + 19-s + (0.900 − 0.433i)20-s + (−0.900 − 0.433i)22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.718 - 0.695i$
Analytic conductor: \(15.7973\)
Root analytic conductor: \(15.7973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (1:\ ),\ 0.718 - 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.478938088 - 0.5987556728i\)
\(L(\frac12)\) \(\approx\) \(1.478938088 - 0.5987556728i\)
\(L(1)\) \(\approx\) \(1.003327853 - 0.3800528324i\)
\(L(1)\) \(\approx\) \(1.003327853 - 0.3800528324i\)

Euler product

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

−27.70577462095759108548826507712, −27.1742596387447925920985819592, −25.88175497578801182520526029095, −24.9764257004706649695958565614, −24.266236786729656767721372557998, −23.111435257746167759249202763510, −22.625903316306262042638887269336, −21.15677853448402173995170434860, −20.16567899313917280908791945518, −18.98001519820438345308284594974, −17.667743541604200344085932381882, −16.944859382781929936782642876284, −15.80389203092996956253944793781, −15.13334229282943936957088726740, −13.95990174646749091961925303292, −12.631428457555186039753910938046, −12.14429637867857594693190327223, −10.18790418252490913039514447957, −8.98477978625934364592443469354, −7.94139954370942916281976527514, −7.07882453602700507997186696347, −5.49707169654080879575190002091, −4.64629362837759255068539074892, −3.34515164728306834772091759657, −0.83620572290159702921690193524, 0.964643417684805153191457609405, 2.790106746523592839056708943223, 3.64983625301675173593742337904, 5.02801318467265742287022522755, 6.51995898179697726570854083606, 7.979250921854328853454207852521, 9.28183694570767220136253817630, 10.3922802921330239958794522488, 11.537868056096666380321733298958, 11.978661958448186145797792097650, 13.67918295326423554920781688149, 14.24999061308258221368653099743, 15.4990184177812365696438115848, 16.83327122407688631027664725346, 18.26235564353760807347158410726, 19.02146278113624128634391195166, 19.66083230707708641794017141475, 21.01948915907002659690192631541, 21.78387401944925430754985488727, 22.80393343439968959539113903266, 23.52730628540947174129602741521, 24.66919278320577757525482532655, 26.35907149037805326354780241431, 26.91016550693761459884901266342, 27.813439594355242470853958797773

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