Properties

Label 2-55440-1.1-c1-0-26
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 − 6·13-s − 2·17-s + 6·23-s + 25-s − 6·29-s + 2·31-s + 35-s + 10·37-s + 8·41-s + 8·43-s + 4·47-s + 49-s − 6·53-s + 55-s − 6·59-s − 8·61-s − 6·65-s − 14·67-s − 8·71-s − 2·73-s + 77-s + 16·79-s + 12·83-s − 2·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.301·11-s − 1.66·13-s − 0.485·17-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s + 0.169·35-s + 1.64·37-s + 1.24·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.824·53-s + 0.134·55-s − 0.781·59-s − 1.02·61-s − 0.744·65-s − 1.71·67-s − 0.949·71-s − 0.234·73-s + 0.113·77-s + 1.80·79-s + 1.31·83-s − 0.216·85-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

\(L(1)\) \(\approx\) \(2.411713456\)
\(L(\frac12)\) \(\approx\) \(2.411713456\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−14.62325529132014, −13.89399629846242, −13.35132822514109, −12.94690323141380, −12.26861786667656, −12.03438643940053, −11.19666490448583, −10.88226514523395, −10.37388728784594, −9.568241608050571, −9.189064593076081, −9.050007693885288, −7.934983633648613, −7.558309093163323, −7.211721015500132, −6.345245700525977, −5.971505782549208, −5.238744015957115, −4.613294673478875, −4.378714334060888, −3.361304389963972, −2.615115194584999, −2.249990861709461, −1.366141314328262, −0.5467092855353353, 0.5467092855353353, 1.366141314328262, 2.249990861709461, 2.615115194584999, 3.361304389963972, 4.378714334060888, 4.613294673478875, 5.238744015957115, 5.971505782549208, 6.345245700525977, 7.211721015500132, 7.558309093163323, 7.934983633648613, 9.050007693885288, 9.189064593076081, 9.568241608050571, 10.37388728784594, 10.88226514523395, 11.19666490448583, 12.03438643940053, 12.26861786667656, 12.94690323141380, 13.35132822514109, 13.89399629846242, 14.62325529132014

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