Properties

Label 2-58800-1.1-c1-0-107
Degree $2$
Conductor $58800$
Sign $-1$
Analytic cond. $469.520$
Root an. cond. $21.6684$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 6·11-s − 4·13-s + 6·17-s + 2·19-s − 27-s + 6·29-s − 10·31-s + 6·33-s − 2·37-s + 4·39-s + 6·41-s − 4·43-s − 6·51-s + 12·53-s − 2·57-s − 14·61-s − 4·67-s − 6·71-s − 4·73-s + 16·79-s + 81-s + 12·83-s − 6·87-s − 6·89-s + 10·93-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s − 1.10·13-s + 1.45·17-s + 0.458·19-s − 0.192·27-s + 1.11·29-s − 1.79·31-s + 1.04·33-s − 0.328·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s − 0.840·51-s + 1.64·53-s − 0.264·57-s − 1.79·61-s − 0.488·67-s − 0.712·71-s − 0.468·73-s + 1.80·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s − 0.635·89-s + 1.03·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58800\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(469.520\)
Root analytic conductor: \(21.6684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{58800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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

−14.60977090381359, −14.09351230703984, −13.49746085339343, −13.01972507153627, −12.42067309576720, −12.16857047040196, −11.67937289937000, −10.78246761124332, −10.60015961533211, −10.07888428196846, −9.603625175354443, −9.010627629641824, −8.182258534582508, −7.701285563428535, −7.385143293739456, −6.846172207470253, −5.846655365213737, −5.606093223701030, −5.018521570691428, −4.655739211138632, −3.702670504275367, −3.034623695113309, −2.511139338748717, −1.716484593829331, −0.7639584344386294, 0, 0.7639584344386294, 1.716484593829331, 2.511139338748717, 3.034623695113309, 3.702670504275367, 4.655739211138632, 5.018521570691428, 5.606093223701030, 5.846655365213737, 6.846172207470253, 7.385143293739456, 7.701285563428535, 8.182258534582508, 9.010627629641824, 9.603625175354443, 10.07888428196846, 10.60015961533211, 10.78246761124332, 11.67937289937000, 12.16857047040196, 12.42067309576720, 13.01972507153627, 13.49746085339343, 14.09351230703984, 14.60977090381359

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