Properties

Degree $2$
Conductor $13680$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s − 4·11-s − 2·13-s + 2·17-s + 19-s − 8·23-s + 25-s − 6·29-s − 4·31-s + 4·35-s − 10·37-s + 2·41-s − 12·43-s + 9·49-s − 6·53-s + 4·55-s − 10·61-s + 2·65-s + 4·67-s − 8·71-s + 2·73-s + 16·77-s + 12·79-s − 8·83-s − 2·85-s − 6·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.229·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.676·35-s − 1.64·37-s + 0.312·41-s − 1.82·43-s + 9/7·49-s − 0.824·53-s + 0.539·55-s − 1.28·61-s + 0.248·65-s + 0.488·67-s − 0.949·71-s + 0.234·73-s + 1.82·77-s + 1.35·79-s − 0.878·83-s − 0.216·85-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13680\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{13680} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 13680,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
19 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−16.55713211441445, −16.17267708266861, −15.68341254846806, −15.25488444581209, −14.54298021279244, −13.82843585375863, −13.35131683006762, −12.70441761959627, −12.36741547947524, −11.79058992102475, −11.01624018053720, −10.22218855354513, −10.04246268216197, −9.364011634391960, −8.680113275279937, −7.838077997115796, −7.514720864501513, −6.774431200154739, −6.098283514470747, −5.450962035925957, −4.835370725334566, −3.741782461821015, −3.406050080543009, −2.611529424630720, −1.730904613668053, 0, 0, 1.730904613668053, 2.611529424630720, 3.406050080543009, 3.741782461821015, 4.835370725334566, 5.450962035925957, 6.098283514470747, 6.774431200154739, 7.514720864501513, 7.838077997115796, 8.680113275279937, 9.364011634391960, 10.04246268216197, 10.22218855354513, 11.01624018053720, 11.79058992102475, 12.36741547947524, 12.70441761959627, 13.35131683006762, 13.82843585375863, 14.54298021279244, 15.25488444581209, 15.68341254846806, 16.17267708266861, 16.55713211441445

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