Properties

Label 2-2880-1.1-c1-0-13
Degree $2$
Conductor $2880$
Sign $1$
Analytic cond. $22.9969$
Root an. cond. $4.79550$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 2·29-s + 10·37-s − 10·41-s + 4·43-s + 8·47-s − 7·49-s − 10·53-s + 4·55-s + 4·59-s + 2·61-s + 2·65-s + 12·67-s − 8·71-s + 10·73-s − 12·83-s − 2·85-s + 6·89-s + 4·95-s + 2·97-s + 6·101-s + 16·103-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 − 0.371·29-s + 1.64·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s − 49-s − 1.37·53-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 + 1.17·73-s − 1.31·83-s − 0.216·85-s + 0.635·89-s + 0.410·95-s + 0.203·97-s + 0.597·101-s + 1.57·103-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: $1$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2880} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.287175190\)
\(L(\frac12)\) \(\approx\) \(2.287175190\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−8.881388845996159681294188149673, −8.084180434635331166491599942631, −7.16842258862004714094187855827, −6.44491636719189125157029592476, −5.83817914073203780391805721828, −4.88120268416850114670329243377, −3.99423297820309292802284752251, −3.16799571044027062920603606790, −1.98364241717056996622491365555, −0.991628865642481429121344680829, 0.991628865642481429121344680829, 1.98364241717056996622491365555, 3.16799571044027062920603606790, 3.99423297820309292802284752251, 4.88120268416850114670329243377, 5.83817914073203780391805721828, 6.44491636719189125157029592476, 7.16842258862004714094187855827, 8.084180434635331166491599942631, 8.881388845996159681294188149673

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