Properties

Label 2-147-147.101-c3-0-10
Degree $2$
Conductor $147$
Sign $0.926 - 0.376i$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.96 − 0.770i)2-s + (1.25 − 5.04i)3-s + (−2.60 − 2.41i)4-s + (10.6 + 7.23i)5-s + (−6.34 + 8.93i)6-s + (3.24 + 18.2i)7-s + (10.5 + 21.9i)8-s + (−23.8 − 12.6i)9-s + (−15.2 − 22.3i)10-s + (7.14 + 47.4i)11-s + (−15.4 + 10.1i)12-s + (−39.7 + 31.7i)13-s + (7.68 − 38.2i)14-s + (49.7 − 44.4i)15-s + (−1.71 − 22.8i)16-s + (74.8 + 23.0i)17-s + ⋯
L(s)  = 1  + (−0.693 − 0.272i)2-s + (0.241 − 0.970i)3-s + (−0.325 − 0.302i)4-s + (0.948 + 0.646i)5-s + (−0.431 + 0.607i)6-s + (0.175 + 0.984i)7-s + (0.467 + 0.970i)8-s + (−0.883 − 0.468i)9-s + (−0.482 − 0.707i)10-s + (0.195 + 1.29i)11-s + (−0.371 + 0.243i)12-s + (−0.848 + 0.676i)13-s + (0.146 − 0.730i)14-s + (0.856 − 0.764i)15-s + (−0.0267 − 0.357i)16-s + (1.06 + 0.329i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.926 - 0.376i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ 0.926 - 0.376i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.12320 + 0.219648i\)
\(L(\frac12)\) \(\approx\) \(1.12320 + 0.219648i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.25 + 5.04i)T \)
7 \( 1 + (-3.24 - 18.2i)T \)
good2 \( 1 + (1.96 + 0.770i)T + (5.86 + 5.44i)T^{2} \)
5 \( 1 + (-10.6 - 7.23i)T + (45.6 + 116. i)T^{2} \)
11 \( 1 + (-7.14 - 47.4i)T + (-1.27e3 + 392. i)T^{2} \)
13 \( 1 + (39.7 - 31.7i)T + (488. - 2.14e3i)T^{2} \)
17 \( 1 + (-74.8 - 23.0i)T + (4.05e3 + 2.76e3i)T^{2} \)
19 \( 1 + (-3.72 + 2.15i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (3.58 + 11.6i)T + (-1.00e4 + 6.85e3i)T^{2} \)
29 \( 1 + (59.0 - 13.4i)T + (2.19e4 - 1.05e4i)T^{2} \)
31 \( 1 + (-243. - 140. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (165. - 153. i)T + (3.78e3 - 5.05e4i)T^{2} \)
41 \( 1 + (-306. + 147. i)T + (4.29e4 - 5.38e4i)T^{2} \)
43 \( 1 + (171. + 82.6i)T + (4.95e4 + 6.21e4i)T^{2} \)
47 \( 1 + (-128. + 327. i)T + (-7.61e4 - 7.06e4i)T^{2} \)
53 \( 1 + (101. - 109. i)T + (-1.11e4 - 1.48e5i)T^{2} \)
59 \( 1 + (327. - 223. i)T + (7.50e4 - 1.91e5i)T^{2} \)
61 \( 1 + (-114. - 123. i)T + (-1.69e4 + 2.26e5i)T^{2} \)
67 \( 1 + (28.2 - 48.8i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-493. - 112. i)T + (3.22e5 + 1.55e5i)T^{2} \)
73 \( 1 + (-337. + 132. i)T + (2.85e5 - 2.64e5i)T^{2} \)
79 \( 1 + (638. + 1.10e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (395. - 495. i)T + (-1.27e5 - 5.57e5i)T^{2} \)
89 \( 1 + (-113. - 17.0i)T + (6.73e5 + 2.07e5i)T^{2} \)
97 \( 1 - 1.36e3iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.46013780235457573382705014367, −11.79606397037185567018112085369, −10.28235131883666960589286673708, −9.583610412582545050578555780439, −8.656817957328339005971461128534, −7.42883966830135833487516269810, −6.26681479430180617238910208374, −5.10487904972002850949380428068, −2.47490738056125358611643766444, −1.65289365528713037918219957249, 0.72302141448750632711568896738, 3.30056759063262406869239133696, 4.63181875913636223636552453554, 5.80302759621912896817493535497, 7.64915485893897832529707111902, 8.460548446075400786158497481000, 9.610455022803763295232009024336, 9.961537290635377762971645780816, 11.15479526608600041167148384740, 12.72675667757618538828509647572

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