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Properties

Label 2-55440-1.1-c1-0-6
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 + 6·17-s − 8·19-s − 6·23-s + 25-s − 6·29-s − 2·31-s − 35-s + 2·37-s − 8·43-s − 12·47-s + 49-s − 6·53-s − 55-s + 6·59-s + 8·61-s + 2·65-s − 2·67-s − 10·73-s + 77-s − 8·79-s + 12·83-s + 6·85-s − 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.301·11-s + 0.554·13-s + 1.45·17-s − 1.83·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s − 0.169·35-s + 0.328·37-s − 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s − 0.134·55-s + 0.781·59-s + 1.02·61-s + 0.248·65-s − 0.244·67-s − 1.17·73-s + 0.113·77-s − 0.900·79-s + 1.31·83-s + 0.650·85-s − 0.635·89-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

\(L(1)\) \(\approx\) \(1.339773229\)
\(L(\frac12)\) \(\approx\) \(1.339773229\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.48439723550094, −13.91974607651316, −13.14265670240400, −13.05424532469851, −12.50338107849826, −11.84694727570040, −11.35350970815807, −10.74169391225169, −10.19086538357776, −9.881103268952037, −9.328790593331576, −8.606839254106936, −8.153855386592378, −7.745796472705043, −6.873862271193050, −6.463141669711482, −5.820205553640217, −5.513388292909876, −4.701466171768155, −3.997948351062312, −3.493187363492930, −2.830969318681183, −1.947346509218819, −1.566199002557290, −0.3826578840254059, 0.3826578840254059, 1.566199002557290, 1.947346509218819, 2.830969318681183, 3.493187363492930, 3.997948351062312, 4.701466171768155, 5.513388292909876, 5.820205553640217, 6.463141669711482, 6.873862271193050, 7.745796472705043, 8.153855386592378, 8.606839254106936, 9.328790593331576, 9.881103268952037, 10.19086538357776, 10.74169391225169, 11.35350970815807, 11.84694727570040, 12.50338107849826, 13.05424532469851, 13.14265670240400, 13.91974607651316, 14.48439723550094

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