Properties

Label 2-147-1.1-c3-0-9
Degree $2$
Conductor $147$
Sign $-1$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3·3-s + 4-s + 3·5-s + 9·6-s + 21·8-s + 9·9-s − 9·10-s − 15·11-s − 3·12-s + 64·13-s − 9·15-s − 71·16-s − 84·17-s − 27·18-s + 16·19-s + 3·20-s + 45·22-s − 84·23-s − 63·24-s − 116·25-s − 192·26-s − 27·27-s − 297·29-s + 27·30-s + 253·31-s + 45·32-s + ⋯
L(s)  = 1  − 1.06·2-s − 0.577·3-s + 1/8·4-s + 0.268·5-s + 0.612·6-s + 0.928·8-s + 1/3·9-s − 0.284·10-s − 0.411·11-s − 0.0721·12-s + 1.36·13-s − 0.154·15-s − 1.10·16-s − 1.19·17-s − 0.353·18-s + 0.193·19-s + 0.0335·20-s + 0.436·22-s − 0.761·23-s − 0.535·24-s − 0.927·25-s − 1.44·26-s − 0.192·27-s − 1.90·29-s + 0.164·30-s + 1.46·31-s + 0.248·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{147} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ -1)\)

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 \( 1 + p T \)
7 \( 1 \)
good2 \( 1 + 3 T + p^{3} T^{2} \)
5 \( 1 - 3 T + p^{3} T^{2} \)
11 \( 1 + 15 T + p^{3} T^{2} \)
13 \( 1 - 64 T + p^{3} T^{2} \)
17 \( 1 + 84 T + p^{3} T^{2} \)
19 \( 1 - 16 T + p^{3} T^{2} \)
23 \( 1 + 84 T + p^{3} T^{2} \)
29 \( 1 + 297 T + p^{3} T^{2} \)
31 \( 1 - 253 T + p^{3} T^{2} \)
37 \( 1 + 316 T + p^{3} T^{2} \)
41 \( 1 + 360 T + p^{3} T^{2} \)
43 \( 1 - 26 T + p^{3} T^{2} \)
47 \( 1 - 30 T + p^{3} T^{2} \)
53 \( 1 - 363 T + p^{3} T^{2} \)
59 \( 1 - 15 T + p^{3} T^{2} \)
61 \( 1 - 118 T + p^{3} T^{2} \)
67 \( 1 + 370 T + p^{3} T^{2} \)
71 \( 1 + 342 T + p^{3} T^{2} \)
73 \( 1 + 362 T + p^{3} T^{2} \)
79 \( 1 - 467 T + p^{3} T^{2} \)
83 \( 1 + 477 T + p^{3} T^{2} \)
89 \( 1 + 906 T + p^{3} T^{2} \)
97 \( 1 + 503 T + p^{3} T^{2} \)
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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

−11.74663933791515619400125166220, −10.83144801609188006745059735686, −10.04964402843276502625090531371, −8.975138931525456771633189572416, −8.079074557276253008223699212346, −6.80856713564553332987415802987, −5.56753673269928998684754335972, −4.08710906384776057019120087316, −1.70657780781485673545787325476, 0, 1.70657780781485673545787325476, 4.08710906384776057019120087316, 5.56753673269928998684754335972, 6.80856713564553332987415802987, 8.079074557276253008223699212346, 8.975138931525456771633189572416, 10.04964402843276502625090531371, 10.83144801609188006745059735686, 11.74663933791515619400125166220

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