Properties

Degree $2$
Conductor $141120$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·11-s − 2·13-s + 4·17-s + 8·23-s + 25-s + 2·31-s − 8·37-s + 2·41-s − 2·43-s + 10·47-s − 2·53-s − 2·55-s − 4·59-s + 10·61-s − 2·65-s + 2·67-s − 12·71-s + 10·73-s − 16·79-s − 16·83-s + 4·85-s − 14·89-s + 6·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.603·11-s − 0.554·13-s + 0.970·17-s + 1.66·23-s + 1/5·25-s + 0.359·31-s − 1.31·37-s + 0.312·41-s − 0.304·43-s + 1.45·47-s − 0.274·53-s − 0.269·55-s − 0.520·59-s + 1.28·61-s − 0.248·65-s + 0.244·67-s − 1.42·71-s + 1.17·73-s − 1.80·79-s − 1.75·83-s + 0.433·85-s − 1.48·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(141120\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{141120} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 141120,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.65934474994259, −13.09602711012918, −12.65773539066116, −12.39212331189707, −11.68201697383263, −11.28212368914052, −10.66117439624873, −10.25055958975948, −9.877737192252514, −9.334164022239856, −8.738698685948677, −8.426749960682950, −7.684544130147959, −7.192107968912818, −6.934022909472190, −6.153855565554566, −5.513236785630575, −5.302275509279976, −4.683567386884593, −4.079480075898516, −3.263603776048168, −2.874036950494800, −2.313735942905510, −1.494449553106535, −0.9258359340040602, 0, 0.9258359340040602, 1.494449553106535, 2.313735942905510, 2.874036950494800, 3.263603776048168, 4.079480075898516, 4.683567386884593, 5.302275509279976, 5.513236785630575, 6.153855565554566, 6.934022909472190, 7.192107968912818, 7.684544130147959, 8.426749960682950, 8.738698685948677, 9.334164022239856, 9.877737192252514, 10.25055958975948, 10.66117439624873, 11.28212368914052, 11.68201697383263, 12.39212331189707, 12.65773539066116, 13.09602711012918, 13.65934474994259

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