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Properties

Label 2-55440-1.1-c1-0-40
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 − 4·19-s − 2·23-s + 25-s + 2·29-s + 10·31-s − 35-s + 2·37-s + 12·41-s + 49-s + 6·53-s + 55-s + 6·59-s − 2·65-s − 2·67-s − 8·71-s − 10·73-s − 77-s + 4·83-s − 6·85-s + 6·89-s + 2·91-s + 4·95-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.417·23-s + 1/5·25-s + 0.371·29-s + 1.79·31-s − 0.169·35-s + 0.328·37-s + 1.87·41-s + 1/7·49-s + 0.824·53-s + 0.134·55-s + 0.781·59-s − 0.248·65-s − 0.244·67-s − 0.949·71-s − 1.17·73-s − 0.113·77-s + 0.439·83-s − 0.650·85-s + 0.635·89-s + 0.209·91-s + 0.410·95-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

\(L(1)\) \(\approx\) \(2.683031137\)
\(L(\frac12)\) \(\approx\) \(2.683031137\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−14.40192267134807, −14.02988346748680, −13.28703701991240, −12.96501578900396, −12.14740247270242, −12.02440868034873, −11.34559120092983, −10.84528956782780, −10.21342165035706, −9.992888198006496, −9.146986929224561, −8.546392622553755, −8.171033576979948, −7.652331204015201, −7.186485467304250, −6.315025064472349, −5.963552373413981, −5.347360153679294, −4.510299106456686, −4.243673952306199, −3.448686689249835, −2.821213495404971, −2.165003859818216, −1.193929843205792, −0.6410352068450334, 0.6410352068450334, 1.193929843205792, 2.165003859818216, 2.821213495404971, 3.448686689249835, 4.243673952306199, 4.510299106456686, 5.347360153679294, 5.963552373413981, 6.315025064472349, 7.186485467304250, 7.652331204015201, 8.171033576979948, 8.546392622553755, 9.146986929224561, 9.992888198006496, 10.21342165035706, 10.84528956782780, 11.34559120092983, 12.02440868034873, 12.14740247270242, 12.96501578900396, 13.28703701991240, 14.02988346748680, 14.40192267134807

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