Properties

Degree $2$
Conductor $141120$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

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: \(0\)
Selberg data: \((2,\ 141120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8749057864\)
\(L(\frac12)\) \(\approx\) \(0.8749057864\)
\(L(\frac{3}{2})\) 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

−13.39470026203441, −13.06518500132425, −12.43150096421898, −11.99992130320511, −11.34383411716719, −11.07525523693288, −10.60284455918603, −10.15212141316022, −9.474748060925962, −8.885745929982214, −8.629552060834559, −8.129782485787572, −7.355615543631413, −7.162862563658393, −6.492828967373403, −6.047619568815975, −5.257480953175800, −4.893120976902089, −4.406999734761228, −3.633709947520644, −3.188280067341639, −2.658938926328513, −1.814051735912688, −1.268365508491681, −0.2821056194218139, 0.2821056194218139, 1.268365508491681, 1.814051735912688, 2.658938926328513, 3.188280067341639, 3.633709947520644, 4.406999734761228, 4.893120976902089, 5.257480953175800, 6.047619568815975, 6.492828967373403, 7.162862563658393, 7.355615543631413, 8.129782485787572, 8.629552060834559, 8.885745929982214, 9.474748060925962, 10.15212141316022, 10.60284455918603, 11.07525523693288, 11.34383411716719, 11.99992130320511, 12.43150096421898, 13.06518500132425, 13.39470026203441

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