Properties

Label 2-54720-1.1-c1-0-56
Degree $2$
Conductor $54720$
Sign $-1$
Analytic cond. $436.941$
Root an. cond. $20.9031$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·11-s − 2·13-s − 2·17-s − 19-s + 4·23-s + 25-s + 6·29-s − 4·31-s + 6·37-s − 10·41-s − 4·43-s − 12·47-s − 7·49-s + 6·53-s + 4·55-s + 12·59-s + 2·61-s + 2·65-s + 4·67-s + 8·71-s − 6·73-s + 4·79-s + 12·83-s + 2·85-s − 10·89-s + 95-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.229·19-s + 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s + 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.75·47-s − 49-s + 0.824·53-s + 0.539·55-s + 1.56·59-s + 0.256·61-s + 0.248·65-s + 0.488·67-s + 0.949·71-s − 0.702·73-s + 0.450·79-s + 1.31·83-s + 0.216·85-s − 1.05·89-s + 0.102·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54720\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(436.941\)
Root analytic conductor: \(20.9031\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 54720,\ (\ :1/2),\ -1)\)

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

−14.70256732075949, −14.30910047057656, −13.42931427499342, −13.16427482824578, −12.75119460101451, −12.13687968974046, −11.51544033856593, −11.18056321347570, −10.58444741635190, −9.955150041980534, −9.743496419889676, −8.764710053330327, −8.437830101448229, −7.933562616240356, −7.323427181905562, −6.792484911037260, −6.326714861914369, −5.420994344500249, −4.925532731299856, −4.636323608129330, −3.670508019885303, −3.150666955757874, −2.476939306821640, −1.864226832490663, −0.7783375238680073, 0, 0.7783375238680073, 1.864226832490663, 2.476939306821640, 3.150666955757874, 3.670508019885303, 4.636323608129330, 4.925532731299856, 5.420994344500249, 6.326714861914369, 6.792484911037260, 7.323427181905562, 7.933562616240356, 8.437830101448229, 8.764710053330327, 9.743496419889676, 9.955150041980534, 10.58444741635190, 11.18056321347570, 11.51544033856593, 12.13687968974046, 12.75119460101451, 13.16427482824578, 13.42931427499342, 14.30910047057656, 14.70256732075949

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