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 − 4·11-s + 2·13-s + 2·17-s − 4·19-s + 25-s + 10·29-s + 6·37-s − 6·41-s + 4·43-s + 8·47-s − 6·53-s − 4·55-s + 4·59-s + 10·61-s + 2·65-s − 4·67-s − 16·71-s + 14·73-s − 8·79-s + 4·83-s + 2·85-s + 10·89-s − 4·95-s − 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 1/5·25-s + 1.85·29-s + 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s − 0.824·53-s − 0.539·55-s + 0.520·59-s + 1.28·61-s + 0.248·65-s − 0.488·67-s − 1.89·71-s + 1.63·73-s − 0.900·79-s + 0.439·83-s + 0.216·85-s + 1.05·89-s − 0.410·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-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

\(L(1)\) \(\approx\) \(2.290414788\)
\(L(\frac12)\) \(\approx\) \(2.290414788\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
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

−14.95645058054208, −14.38986133742240, −13.78424154766915, −13.44923575126419, −12.79939557681492, −12.48316015439512, −11.79647464403792, −11.14811566094233, −10.60054143749968, −10.19653068235252, −9.781628493364061, −8.896451967387262, −8.525777851826983, −7.949488567324191, −7.404244709463908, −6.611323457435462, −6.163854782966423, −5.560176619335859, −4.945022929911820, −4.369639642345230, −3.597462918791808, −2.751496774660772, −2.390997463964714, −1.409323173761413, −0.5803230790800490, 0.5803230790800490, 1.409323173761413, 2.390997463964714, 2.751496774660772, 3.597462918791808, 4.369639642345230, 4.945022929911820, 5.560176619335859, 6.163854782966423, 6.611323457435462, 7.404244709463908, 7.949488567324191, 8.525777851826983, 8.896451967387262, 9.781628493364061, 10.19653068235252, 10.60054143749968, 11.14811566094233, 11.79647464403792, 12.48316015439512, 12.79939557681492, 13.44923575126419, 13.78424154766915, 14.38986133742240, 14.95645058054208

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