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Properties

Label 2-2880-1.1-c1-0-6
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 − 4·7-s + 2·13-s + 6·17-s + 4·23-s + 25-s − 2·29-s − 8·31-s − 4·35-s − 6·37-s + 6·41-s − 12·43-s + 12·47-s + 9·49-s − 10·53-s + 8·59-s + 10·61-s + 2·65-s + 12·67-s − 8·71-s + 10·73-s + 16·79-s + 12·83-s + 6·85-s + 6·89-s − 8·91-s + 18·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s + 0.554·13-s + 1.45·17-s + 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.937·41-s − 1.82·43-s + 1.75·47-s + 9/7·49-s − 1.37·53-s + 1.04·59-s + 1.28·61-s + 0.248·65-s + 1.46·67-s − 0.949·71-s + 1.17·73-s + 1.80·79-s + 1.31·83-s + 0.650·85-s + 0.635·89-s − 0.838·91-s + 1.82·97-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.665322264\)
\(L(\frac12)\) \(\approx\) \(1.665322264\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−9.030286901197179914237932277925, −8.001601195013891794739315195655, −7.12027585547906577096357002702, −6.50980878973832103181127313269, −5.72523264417157553873051016989, −5.12484102972813105731809337260, −3.60890987853244258057621629934, −3.37218459266436962553706504644, −2.13130095374382376963731883215, −0.792978888625782745854172461096, 0.792978888625782745854172461096, 2.13130095374382376963731883215, 3.37218459266436962553706504644, 3.60890987853244258057621629934, 5.12484102972813105731809337260, 5.72523264417157553873051016989, 6.50980878973832103181127313269, 7.12027585547906577096357002702, 8.001601195013891794739315195655, 9.030286901197179914237932277925

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