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Properties

Label 2-55440-1.1-c1-0-42
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 + 2·13-s + 7·17-s − 3·19-s + 3·23-s + 25-s + 5·29-s − 35-s + 2·37-s − 43-s + 8·47-s + 49-s + 9·53-s − 55-s − 9·59-s + 5·61-s + 2·65-s + 2·67-s − 12·71-s + 77-s + 14·79-s + 9·83-s + 7·85-s − 5·89-s − 2·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.301·11-s + 0.554·13-s + 1.69·17-s − 0.688·19-s + 0.625·23-s + 1/5·25-s + 0.928·29-s − 0.169·35-s + 0.328·37-s − 0.152·43-s + 1.16·47-s + 1/7·49-s + 1.23·53-s − 0.134·55-s − 1.17·59-s + 0.640·61-s + 0.248·65-s + 0.244·67-s − 1.42·71-s + 0.113·77-s + 1.57·79-s + 0.987·83-s + 0.759·85-s − 0.529·89-s − 0.209·91-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

\(L(1)\) \(\approx\) \(3.024230891\)
\(L(\frac12)\) \(\approx\) \(3.024230891\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−14.39832570204096, −13.79210921508461, −13.47873483464558, −12.87274922328686, −12.36462463277361, −12.01914041374151, −11.31131365471185, −10.62512895317111, −10.36940935699411, −9.822898512204035, −9.247422341589675, −8.706426020303561, −8.196811171422387, −7.565682766018923, −7.057554315158171, −6.338355951295148, −5.937784508286156, −5.375769917515607, −4.782271933724237, −4.034159046858743, −3.387240526648696, −2.837814904180858, −2.157840552087934, −1.246847720287983, −0.6612161057584924, 0.6612161057584924, 1.246847720287983, 2.157840552087934, 2.837814904180858, 3.387240526648696, 4.034159046858743, 4.782271933724237, 5.375769917515607, 5.937784508286156, 6.338355951295148, 7.057554315158171, 7.565682766018923, 8.196811171422387, 8.706426020303561, 9.247422341589675, 9.822898512204035, 10.36940935699411, 10.62512895317111, 11.31131365471185, 12.01914041374151, 12.36462463277361, 12.87274922328686, 13.47873483464558, 13.79210921508461, 14.39832570204096

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