Properties

Degree $2$
Conductor $720$
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 − 2·7-s − 2·11-s + 4·13-s − 2·17-s − 4·19-s − 8·23-s + 25-s − 10·29-s − 4·31-s + 2·35-s + 8·43-s − 8·47-s − 3·49-s + 6·53-s + 2·55-s + 14·59-s − 14·61-s − 4·65-s + 4·67-s − 12·71-s + 6·73-s + 4·77-s + 12·79-s − 4·83-s + 2·85-s − 12·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 0.603·11-s + 1.10·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 1.85·29-s − 0.718·31-s + 0.338·35-s + 1.21·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s + 0.269·55-s + 1.82·59-s − 1.79·61-s − 0.496·65-s + 0.488·67-s − 1.42·71-s + 0.702·73-s + 0.455·77-s + 1.35·79-s − 0.439·83-s + 0.216·85-s − 1.27·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{720} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−10.02523263988524277353338918623, −9.081759070126685363465935730485, −8.273767264975110551561533420364, −7.41534942734669416840578685752, −6.36049438647887892819733581821, −5.65023019086191961009411991273, −4.21999895332902477463481579150, −3.48524593863450834073159469636, −2.06566091715443158914144420460, 0, 2.06566091715443158914144420460, 3.48524593863450834073159469636, 4.21999895332902477463481579150, 5.65023019086191961009411991273, 6.36049438647887892819733581821, 7.41534942734669416840578685752, 8.273767264975110551561533420364, 9.081759070126685363465935730485, 10.02523263988524277353338918623

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