Properties

Label 2-147-1.1-c3-0-19
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  + 4·2-s − 3·3-s + 8·4-s − 18·5-s − 12·6-s + 9·9-s − 72·10-s − 50·11-s − 24·12-s + 36·13-s + 54·15-s − 64·16-s − 126·17-s + 36·18-s + 72·19-s − 144·20-s − 200·22-s + 14·23-s + 199·25-s + 144·26-s − 27·27-s + 158·29-s + 216·30-s + 36·31-s − 256·32-s + 150·33-s − 504·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 1.60·5-s − 0.816·6-s + 1/3·9-s − 2.27·10-s − 1.37·11-s − 0.577·12-s + 0.768·13-s + 0.929·15-s − 16-s − 1.79·17-s + 0.471·18-s + 0.869·19-s − 1.60·20-s − 1.93·22-s + 0.126·23-s + 1.59·25-s + 1.08·26-s − 0.192·27-s + 1.01·29-s + 1.31·30-s + 0.208·31-s − 1.41·32-s + 0.791·33-s − 2.54·34-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 \( 1 - p^{2} T + p^{3} T^{2} \)
5 \( 1 + 18 T + p^{3} T^{2} \)
11 \( 1 + 50 T + p^{3} T^{2} \)
13 \( 1 - 36 T + p^{3} T^{2} \)
17 \( 1 + 126 T + p^{3} T^{2} \)
19 \( 1 - 72 T + p^{3} T^{2} \)
23 \( 1 - 14 T + p^{3} T^{2} \)
29 \( 1 - 158 T + p^{3} T^{2} \)
31 \( 1 - 36 T + p^{3} T^{2} \)
37 \( 1 + 162 T + p^{3} T^{2} \)
41 \( 1 - 270 T + p^{3} T^{2} \)
43 \( 1 + 324 T + p^{3} T^{2} \)
47 \( 1 - 72 T + p^{3} T^{2} \)
53 \( 1 + 22 T + p^{3} T^{2} \)
59 \( 1 + 468 T + p^{3} T^{2} \)
61 \( 1 + 792 T + p^{3} T^{2} \)
67 \( 1 - 232 T + p^{3} T^{2} \)
71 \( 1 + 734 T + p^{3} T^{2} \)
73 \( 1 + 180 T + p^{3} T^{2} \)
79 \( 1 - 236 T + p^{3} T^{2} \)
83 \( 1 + 36 T + p^{3} T^{2} \)
89 \( 1 + 234 T + p^{3} T^{2} \)
97 \( 1 + 468 T + p^{3} T^{2} \)
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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

−12.16645973406122493451025689508, −11.38309769590287406411963194947, −10.69689359716016265096272270130, −8.720431635188126265043856803254, −7.52407927167921603983619499483, −6.39552663713415537798935085043, −5.03430412213801933359197332846, −4.25302697626860858694008665577, −3.01745726904532705415108523385, 0, 3.01745726904532705415108523385, 4.25302697626860858694008665577, 5.03430412213801933359197332846, 6.39552663713415537798935085043, 7.52407927167921603983619499483, 8.720431635188126265043856803254, 10.69689359716016265096272270130, 11.38309769590287406411963194947, 12.16645973406122493451025689508

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