Properties

Label 2-147-3.2-c2-0-12
Degree $2$
Conductor $147$
Sign $1$
Analytic cond. $4.00545$
Root an. cond. $2.00136$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 4·4-s + 9·9-s + 12·12-s − 23·13-s + 16·16-s − 11·19-s + 25·25-s + 27·27-s + 13·31-s + 36·36-s − 73·37-s − 69·39-s − 61·43-s + 48·48-s − 92·52-s − 33·57-s − 74·61-s + 64·64-s − 13·67-s + 97·73-s + 75·75-s − 44·76-s + 11·79-s + 81·81-s + 39·93-s − 2·97-s + ⋯
L(s)  = 1  + 3-s + 4-s + 9-s + 12-s − 1.76·13-s + 16-s − 0.578·19-s + 25-s + 27-s + 0.419·31-s + 36-s − 1.97·37-s − 1.76·39-s − 1.41·43-s + 48-s − 1.76·52-s − 0.578·57-s − 1.21·61-s + 64-s − 0.194·67-s + 1.32·73-s + 75-s − 0.578·76-s + 0.139·79-s + 81-s + 0.419·93-s − 0.0206·97-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4.00545\)
Root analytic conductor: \(2.00136\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{147} (50, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.281132653\)
\(L(\frac12)\) \(\approx\) \(2.281132653\)
\(L(2)\) 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 \( ( 1 - p T )( 1 + p T ) \)
5 \( ( 1 - p T )( 1 + p T ) \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( 1 + 23 T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 + 11 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 - 13 T + p^{2} T^{2} \)
37 \( 1 + 73 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 + 61 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( ( 1 - p T )( 1 + p T ) \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 + 74 T + p^{2} T^{2} \)
67 \( 1 + 13 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 - 97 T + p^{2} T^{2} \)
79 \( 1 - 11 T + p^{2} T^{2} \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 + 2 T + p^{2} T^{2} \)
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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

−12.68335225490082405301353772104, −12.02495324137181980318558862894, −10.61475239491124312656255385715, −9.824413471321259790039875562075, −8.560827121366394777761342863881, −7.45783461989240393020086773671, −6.69878656362012781762069178695, −4.89314783644220342530353574901, −3.17415659105302014468945646645, −2.03103217404730035474201708510, 2.03103217404730035474201708510, 3.17415659105302014468945646645, 4.89314783644220342530353574901, 6.69878656362012781762069178695, 7.45783461989240393020086773671, 8.560827121366394777761342863881, 9.824413471321259790039875562075, 10.61475239491124312656255385715, 12.02495324137181980318558862894, 12.68335225490082405301353772104

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