Properties

Degree $2$
Conductor $35280$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 2·11-s + 2·13-s + 4·17-s + 8·23-s + 25-s + 2·31-s + 8·37-s + 2·41-s + 2·43-s + 10·47-s + 2·53-s − 2·55-s + 4·59-s − 10·61-s − 2·65-s − 2·67-s − 12·71-s + 10·73-s − 16·79-s + 16·83-s − 4·85-s − 14·89-s + 6·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.603·11-s + 0.554·13-s + 0.970·17-s + 1.66·23-s + 1/5·25-s + 0.359·31-s + 1.31·37-s + 0.312·41-s + 0.304·43-s + 1.45·47-s + 0.274·53-s − 0.269·55-s + 0.520·59-s − 1.28·61-s − 0.248·65-s − 0.244·67-s − 1.42·71-s + 1.17·73-s − 1.80·79-s + 1.75·83-s − 0.433·85-s − 1.48·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35280\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{35280} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 35280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.945719159\)
\(L(\frac12)\) \(\approx\) \(2.945719159\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.00164138655172, −14.37359850890480, −14.04097299981379, −13.24198207367703, −12.91258146053771, −12.24467657275233, −11.79563125311983, −11.23955169826699, −10.80148118024097, −10.18767518435292, −9.548567758717300, −8.949633834986394, −8.617849862467296, −7.807598066575872, −7.365250028278257, −6.851861030943796, −5.991442146287297, −5.742134413630214, −4.725994357334290, −4.385076949046866, −3.505417024711309, −3.115859826292636, −2.263137835081931, −1.184773792281958, −0.7633668472663102, 0.7633668472663102, 1.184773792281958, 2.263137835081931, 3.115859826292636, 3.505417024711309, 4.385076949046866, 4.725994357334290, 5.742134413630214, 5.991442146287297, 6.851861030943796, 7.365250028278257, 7.807598066575872, 8.617849862467296, 8.949633834986394, 9.548567758717300, 10.18767518435292, 10.80148118024097, 11.23955169826699, 11.79563125311983, 12.24467657275233, 12.91258146053771, 13.24198207367703, 14.04097299981379, 14.37359850890480, 15.00164138655172

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