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 + 2·13-s − 2·19-s + 25-s − 6·29-s + 8·31-s + 4·37-s − 6·41-s + 2·43-s − 6·47-s − 6·53-s − 12·59-s + 8·61-s − 2·65-s + 2·67-s − 6·71-s − 2·73-s + 16·79-s − 6·89-s + 2·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.554·13-s − 0.458·19-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.657·37-s − 0.937·41-s + 0.304·43-s − 0.875·47-s − 0.824·53-s − 1.56·59-s + 1.02·61-s − 0.248·65-s + 0.244·67-s − 0.712·71-s − 0.234·73-s + 1.80·79-s − 0.635·89-s + 0.205·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.49439382623064, −13.27644849730775, −12.63055304623518, −12.24063619894471, −11.70670598182109, −11.10006330528940, −11.03397861944859, −10.21716200943764, −9.840785766305583, −9.308673003653411, −8.612577414469318, −8.429188737732475, −7.687723183054449, −7.452145362743334, −6.622977015250715, −6.275856973819153, −5.816103414401025, −4.947123552369656, −4.695005922722396, −3.963466165762091, −3.489166486344579, −2.938332309220514, −2.197482602302538, −1.554233081712647, −0.8054575590821579, 0, 0.8054575590821579, 1.554233081712647, 2.197482602302538, 2.938332309220514, 3.489166486344579, 3.963466165762091, 4.695005922722396, 4.947123552369656, 5.816103414401025, 6.275856973819153, 6.622977015250715, 7.452145362743334, 7.687723183054449, 8.429188737732475, 8.612577414469318, 9.308673003653411, 9.840785766305583, 10.21716200943764, 11.03397861944859, 11.10006330528940, 11.70670598182109, 12.24063619894471, 12.63055304623518, 13.27644849730775, 13.49439382623064

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