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 + 2·13-s − 6·17-s − 4·19-s + 25-s − 6·29-s + 4·31-s − 2·37-s + 6·41-s − 8·43-s − 12·47-s + 6·53-s + 12·59-s + 2·61-s + 2·65-s − 8·67-s − 14·73-s − 16·79-s − 12·83-s − 6·85-s + 6·89-s − 4·95-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 0.824·53-s + 1.56·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s − 1.63·73-s − 1.80·79-s − 1.31·83-s − 0.650·85-s + 0.635·89-s − 0.410·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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

\(L(1)\) \(\approx\) \(1.228516895\)
\(L(\frac12)\) \(\approx\) \(1.228516895\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.20839686796045, −13.11114373138380, −12.68739422728992, −11.69080321397635, −11.59678087026484, −11.04677050982566, −10.43434104109302, −10.15476944717242, −9.513980936367765, −8.985619398309626, −8.537320419186253, −8.263879718789318, −7.434239083120781, −6.903323164714281, −6.513054029381040, −5.985663224563098, −5.497819591593251, −4.827522184244023, −4.278036760384572, −3.875281678773427, −3.060695176347681, −2.507421750448578, −1.856607093240321, −1.384061817206284, −0.3162910625696713, 0.3162910625696713, 1.384061817206284, 1.856607093240321, 2.507421750448578, 3.060695176347681, 3.875281678773427, 4.278036760384572, 4.827522184244023, 5.497819591593251, 5.985663224563098, 6.513054029381040, 6.903323164714281, 7.434239083120781, 8.263879718789318, 8.537320419186253, 8.985619398309626, 9.513980936367765, 10.15476944717242, 10.43434104109302, 11.04677050982566, 11.59678087026484, 11.69080321397635, 12.68739422728992, 13.11114373138380, 13.20839686796045

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