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Properties

Label 2-147-7.2-c3-0-12
Degree $2$
Conductor $147$
Sign $0.386 + 0.922i$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 3.46i)2-s + (−1.5 − 2.59i)3-s + (−3.99 − 6.92i)4-s + (−9 + 15.5i)5-s + 12·6-s + (−4.5 + 7.79i)9-s + (−36 − 62.3i)10-s + (25 + 43.3i)11-s + (−12.0 + 20.7i)12-s − 36·13-s + 54·15-s + (31.9 − 55.4i)16-s + (−63 − 109. i)17-s + (−18 − 31.1i)18-s + (36 − 62.3i)19-s + 144·20-s + ⋯
L(s)  = 1  + (−0.707 + 1.22i)2-s + (−0.288 − 0.499i)3-s + (−0.499 − 0.866i)4-s + (−0.804 + 1.39i)5-s + 0.816·6-s + (−0.166 + 0.288i)9-s + (−1.13 − 1.97i)10-s + (0.685 + 1.18i)11-s + (−0.288 + 0.499i)12-s − 0.768·13-s + 0.929·15-s + (0.499 − 0.866i)16-s + (−0.898 − 1.55i)17-s + (−0.235 − 0.408i)18-s + (0.434 − 0.752i)19-s + 1.60·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 2.59i)T \)
7 \( 1 \)
good2 \( 1 + (2 - 3.46i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (9 - 15.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-25 - 43.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 36T + 2.19e3T^{2} \)
17 \( 1 + (63 + 109. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-36 + 62.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (7 - 12.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 158T + 2.43e4T^{2} \)
31 \( 1 + (-18 - 31.1i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-81 + 140. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 270T + 6.89e4T^{2} \)
43 \( 1 + 324T + 7.95e4T^{2} \)
47 \( 1 + (-36 + 62.3i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-11 - 19.0i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (234 + 405. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (396 - 685. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (116 + 200. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 734T + 3.57e5T^{2} \)
73 \( 1 + (90 + 155. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (118 - 204. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 36T + 5.71e5T^{2} \)
89 \( 1 + (117 - 202. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 468T + 9.12e5T^{2} \)
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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

−12.07653512680595501408641903162, −11.53639251519702535670550255020, −10.14595375311847695350886023968, −9.095293354547065102967212782462, −7.65761356988255321816198631900, −7.03951605894775838375946492292, −6.61204974425279908054834065742, −4.81945259426370886347912121054, −2.77723989039261325963493674381, 0, 1.33704890239877054658380872613, 3.47096534539461245038430210247, 4.54117402065392770454252524608, 6.07962032875324163357809601084, 8.280910372127769070634519980474, 8.713437068392087788187797026833, 9.794404910288988365472095613191, 10.80167456299345189206957689926, 11.79686476357514307127734961111

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