Properties

Degree $2$
Conductor $2880$
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 + 4·7-s − 2·13-s − 6·17-s − 4·19-s + 25-s − 6·29-s − 8·31-s − 4·35-s − 2·37-s + 6·41-s − 4·43-s + 9·49-s − 6·53-s + 10·61-s + 2·65-s − 4·67-s + 2·73-s − 8·79-s − 12·83-s + 6·85-s − 18·89-s − 8·91-s + 4·95-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.676·35-s − 0.328·37-s + 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.824·53-s + 1.28·61-s + 0.248·65-s − 0.488·67-s + 0.234·73-s − 0.900·79-s − 1.31·83-s + 0.650·85-s − 1.90·89-s − 0.838·91-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{2880} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -1)\)

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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.04871238703468, −18.18510169189027, −17.89905654046853, −17.08676660207245, −16.71610691786525, −15.63116317685125, −15.26038697954290, −14.46842963768342, −14.25232318960650, −13.05157407176890, −12.74759148778105, −11.69575814395274, −11.19212710336686, −10.85647287034924, −9.859053367151805, −8.839725981578544, −8.525273921993025, −7.564482475719698, −7.141365571297185, −6.068760038786950, −5.116229128798402, −4.507800107153761, −3.788921009046561, −2.385456951272076, −1.670765276044539, 0, 1.670765276044539, 2.385456951272076, 3.788921009046561, 4.507800107153761, 5.116229128798402, 6.068760038786950, 7.141365571297185, 7.564482475719698, 8.525273921993025, 8.839725981578544, 9.859053367151805, 10.85647287034924, 11.19212710336686, 11.69575814395274, 12.74759148778105, 13.05157407176890, 14.25232318960650, 14.46842963768342, 15.26038697954290, 15.63116317685125, 16.71610691786525, 17.08676660207245, 17.89905654046853, 18.18510169189027, 19.04871238703468

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