Properties

Label 2-132-1.1-c1-0-1
Degree $2$
Conductor $132$
Sign $1$
Analytic cond. $1.05402$
Root an. cond. $1.02665$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s − 2·7-s + 9-s + 11-s − 2·13-s + 2·15-s + 4·17-s − 6·19-s − 2·21-s − 25-s + 27-s − 8·29-s − 8·31-s + 33-s − 4·35-s + 10·37-s − 2·39-s + 8·41-s − 2·43-s + 2·45-s − 8·47-s − 3·49-s + 4·51-s − 2·53-s + 2·55-s − 6·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.516·15-s + 0.970·17-s − 1.37·19-s − 0.436·21-s − 1/5·25-s + 0.192·27-s − 1.48·29-s − 1.43·31-s + 0.174·33-s − 0.676·35-s + 1.64·37-s − 0.320·39-s + 1.24·41-s − 0.304·43-s + 0.298·45-s − 1.16·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s + 0.269·55-s − 0.794·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $1$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.328164578\)
\(L(\frac12)\) \(\approx\) \(1.328164578\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
11 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.10795468637363568116956190325, −12.70566797794869228083627204121, −11.13297096888404845423920902820, −9.796917351475791676316160488949, −9.413820553443583654003786378045, −7.996057650348538062805293058353, −6.70453643193197308588262041684, −5.57273521417705379772496968279, −3.79294235641559942977546839028, −2.21252199435586523776406167130, 2.21252199435586523776406167130, 3.79294235641559942977546839028, 5.57273521417705379772496968279, 6.70453643193197308588262041684, 7.996057650348538062805293058353, 9.413820553443583654003786378045, 9.796917351475791676316160488949, 11.13297096888404845423920902820, 12.70566797794869228083627204121, 13.10795468637363568116956190325

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