Properties

Label 2-55440-1.1-c1-0-61
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 11-s − 2·13-s − 2·17-s − 4·19-s + 25-s − 6·29-s + 35-s + 6·37-s + 6·41-s + 4·43-s + 49-s + 2·53-s + 55-s + 4·59-s + 6·61-s + 2·65-s − 12·67-s + 10·73-s + 77-s − 8·79-s − 4·83-s + 2·85-s − 10·89-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 − 0.485·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.169·35-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.274·53-s + 0.134·55-s + 0.520·59-s + 0.768·61-s + 0.248·65-s − 1.46·67-s + 1.17·73-s + 0.113·77-s − 0.900·79-s − 0.439·83-s + 0.216·85-s − 1.05·89-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: $\chi_{55440} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 55440,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−14.57189119328099, −14.37496449358589, −13.50344633375626, −12.97381621511852, −12.80573785506899, −12.13382244075851, −11.57888621192827, −11.05659948907847, −10.65636938709552, −10.00201398672524, −9.508536410978810, −8.947319519023762, −8.464984091844646, −7.758012479562649, −7.393965836262679, −6.797303350767550, −6.171555501451658, −5.651152822050950, −4.972425951228071, −4.230927672243893, −3.977788627191748, −3.049913189907070, −2.493501098823466, −1.863060676971774, −0.7631993363583716, 0, 0.7631993363583716, 1.863060676971774, 2.493501098823466, 3.049913189907070, 3.977788627191748, 4.230927672243893, 4.972425951228071, 5.651152822050950, 6.171555501451658, 6.797303350767550, 7.393965836262679, 7.758012479562649, 8.464984091844646, 8.947319519023762, 9.508536410978810, 10.00201398672524, 10.65636938709552, 11.05659948907847, 11.57888621192827, 12.13382244075851, 12.80573785506899, 12.97381621511852, 13.50344633375626, 14.37496449358589, 14.57189119328099

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