↑

Properties

Label 2-2880-20.19-c0-0-1
Degree $2$
Conductor $2880$
Sign $1$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 25-s + 2·29-s + 2·41-s − 49-s + 2·61-s − 2·89-s + 2·101-s + 2·109-s + ⋯
L(s)  = 1  − 5-s + 25-s + 2·29-s + 2·41-s − 49-s + 2·61-s − 2·89-s + 2·101-s + 2·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2880} (1279, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.021270862\)
\(L(\frac12)\) \(\approx\) \(1.021270862\)
\(L(1)\) 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 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( ( 1 - T )^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )^{2} \)
67 \( 1 + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
89 \( ( 1 + T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.747109020645146241067807094402, −8.244972095725996922240332460033, −7.48749522557563405723087764915, −6.78696642123288121277914585054, −5.97228156561733104924169586111, −4.89819154702732173523668456399, −4.27801326148291986573619654185, −3.36821152264418884516930075847, −2.47915149575835169695364294516, −0.938223458969756128047204406514, 0.938223458969756128047204406514, 2.47915149575835169695364294516, 3.36821152264418884516930075847, 4.27801326148291986573619654185, 4.89819154702732173523668456399, 5.97228156561733104924169586111, 6.78696642123288121277914585054, 7.48749522557563405723087764915, 8.244972095725996922240332460033, 8.747109020645146241067807094402

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