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Properties

Label 2-55440-1.1-c1-0-36
Degree $2$
Conductor $55440$
Sign $1$
Analytic cond. $442.690$
Root an. cond. $21.0402$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s − 11-s − 4·13-s + 4·19-s + 25-s + 6·29-s + 10·31-s − 35-s + 2·37-s + 12·41-s + 4·43-s + 6·47-s + 49-s + 6·53-s − 55-s − 6·59-s − 4·61-s − 4·65-s + 4·67-s + 12·71-s − 4·73-s + 77-s − 8·79-s + 12·83-s − 18·89-s + 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.301·11-s − 1.10·13-s + 0.917·19-s + 1/5·25-s + 1.11·29-s + 1.79·31-s − 0.169·35-s + 0.328·37-s + 1.87·41-s + 0.609·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 0.134·55-s − 0.781·59-s − 0.512·61-s − 0.496·65-s + 0.488·67-s + 1.42·71-s − 0.468·73-s + 0.113·77-s − 0.900·79-s + 1.31·83-s − 1.90·89-s + 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55440\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(442.690\)
Root analytic conductor: \(21.0402\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 55440,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.607076081\)
\(L(\frac12)\) \(\approx\) \(2.607076081\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−14.20007244522986, −13.95145954841800, −13.52222843597957, −12.77253303428386, −12.37742168682848, −12.01466588881524, −11.35130579868061, −10.73792186667283, −10.19431960272593, −9.767027722094628, −9.376256352414275, −8.776503264088690, −8.022265343804276, −7.654498952062453, −6.976711117785587, −6.524740421559234, −5.785146032979559, −5.438697708498672, −4.581080976820813, −4.325340348544197, −3.291818177296951, −2.604595803544789, −2.428006813214349, −1.208277632426073, −0.6120223541366488, 0.6120223541366488, 1.208277632426073, 2.428006813214349, 2.604595803544789, 3.291818177296951, 4.325340348544197, 4.581080976820813, 5.438697708498672, 5.785146032979559, 6.524740421559234, 6.976711117785587, 7.654498952062453, 8.022265343804276, 8.776503264088690, 9.376256352414275, 9.767027722094628, 10.19431960272593, 10.73792186667283, 11.35130579868061, 12.01466588881524, 12.37742168682848, 12.77253303428386, 13.52222843597957, 13.95145954841800, 14.20007244522986

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