Properties

Label 2-2880-1.1-c1-0-30
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 $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·7-s + 4·11-s + 2·13-s − 2·17-s − 4·19-s − 4·23-s + 25-s − 2·29-s − 8·31-s − 4·35-s − 6·37-s + 6·41-s + 8·43-s − 4·47-s + 9·49-s + 6·53-s + 4·55-s − 4·59-s + 2·61-s + 2·65-s − 8·67-s − 6·73-s − 16·77-s − 16·83-s − 2·85-s + 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s + 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-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 + 1.21·43-s − 0.583·47-s + 9/7·49-s + 0.824·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s − 0.702·73-s − 1.82·77-s − 1.75·83-s − 0.216·85-s + 0.635·89-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: \(1\)
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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−8.751357480986822885848373141185, −7.48336835165842113395842122254, −6.68092210870308353326240648663, −6.21553430939489124152139730965, −5.57665228598311567295093326166, −4.15385036455655232210799544286, −3.71775993896911012431428454191, −2.62554243893362178335475900376, −1.54848190261935741643202258178, 0, 1.54848190261935741643202258178, 2.62554243893362178335475900376, 3.71775993896911012431428454191, 4.15385036455655232210799544286, 5.57665228598311567295093326166, 6.21553430939489124152139730965, 6.68092210870308353326240648663, 7.48336835165842113395842122254, 8.751357480986822885848373141185

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