Properties

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

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 − 8·23-s + 25-s − 6·29-s − 8·31-s − 2·37-s + 2·41-s + 12·43-s + 8·47-s − 6·53-s + 4·55-s − 4·59-s + 2·61-s + 2·65-s − 12·67-s + 8·71-s + 14·73-s − 12·83-s + 2·85-s + 2·89-s − 4·95-s − 10·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.312·41-s + 1.82·43-s + 1.16·47-s − 0.824·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.248·65-s − 1.46·67-s + 0.949·71-s + 1.63·73-s − 1.31·83-s + 0.216·85-s + 0.211·89-s − 0.410·95-s − 1.01·97-s + 0.0995·101-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: \(1\)
Selberg data: \((2,\ 35280,\ (\ :1/2),\ -1)\)

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 \( 1 \)
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 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−15.16795006148422, −14.49722938803314, −14.18256531469436, −13.81528179061438, −13.03704333189100, −12.54184828900961, −12.16522546628426, −11.49307992936691, −10.83460493726009, −10.62451425894387, −9.693859274235424, −9.364293371310721, −8.880600999786153, −8.235924559134461, −7.584420305374129, −7.050631837309314, −6.263031867194771, −5.931433732237611, −5.443717927033889, −4.416452717023179, −3.925363061550828, −3.500111789336166, −2.397397754713740, −1.837860052810143, −1.142436955087004, 0, 1.142436955087004, 1.837860052810143, 2.397397754713740, 3.500111789336166, 3.925363061550828, 4.416452717023179, 5.443717927033889, 5.931433732237611, 6.263031867194771, 7.050631837309314, 7.584420305374129, 8.235924559134461, 8.880600999786153, 9.364293371310721, 9.693859274235424, 10.62451425894387, 10.83460493726009, 11.49307992936691, 12.16522546628426, 12.54184828900961, 13.03704333189100, 13.81528179061438, 14.18256531469436, 14.49722938803314, 15.16795006148422

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