Properties

Label 2-24e2-1.1-c3-0-24
Degree $2$
Conductor $576$
Sign $-1$
Analytic cond. $33.9851$
Root an. cond. $5.82967$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 18·13-s − 104·17-s − 109·25-s + 284·29-s − 214·37-s − 472·41-s − 343·49-s + 572·53-s − 830·61-s − 72·65-s − 1.09e3·73-s − 416·85-s + 176·89-s − 594·97-s − 1.94e3·101-s − 1.74e3·109-s + 1.32e3·113-s + ⋯
L(s)  = 1  + 0.357·5-s − 0.384·13-s − 1.48·17-s − 0.871·25-s + 1.81·29-s − 0.950·37-s − 1.79·41-s − 49-s + 1.48·53-s − 1.74·61-s − 0.137·65-s − 1.76·73-s − 0.530·85-s + 0.209·89-s − 0.621·97-s − 1.91·101-s − 1.53·109-s + 1.10·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{576} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5 \( 1 - 4 T + p^{3} T^{2} \)
7 \( 1 + p^{3} T^{2} \)
11 \( 1 + p^{3} T^{2} \)
13 \( 1 + 18 T + p^{3} T^{2} \)
17 \( 1 + 104 T + p^{3} T^{2} \)
19 \( 1 + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 - 284 T + p^{3} T^{2} \)
31 \( 1 + p^{3} T^{2} \)
37 \( 1 + 214 T + p^{3} T^{2} \)
41 \( 1 + 472 T + p^{3} T^{2} \)
43 \( 1 + p^{3} T^{2} \)
47 \( 1 + p^{3} T^{2} \)
53 \( 1 - 572 T + p^{3} T^{2} \)
59 \( 1 + p^{3} T^{2} \)
61 \( 1 + 830 T + p^{3} T^{2} \)
67 \( 1 + p^{3} T^{2} \)
71 \( 1 + p^{3} T^{2} \)
73 \( 1 + 1098 T + p^{3} T^{2} \)
79 \( 1 + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 - 176 T + p^{3} T^{2} \)
97 \( 1 + 594 T + p^{3} T^{2} \)
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

−9.961298787262456769621518672452, −8.969302674038261400762958269182, −8.239074330365357145982852365237, −7.03811931015362881887821006700, −6.32814677262040118027637646148, −5.16894222994826421885905171500, −4.24813172295872249015471040529, −2.86984184630707362193903160492, −1.71918663622732346875492628809, 0, 1.71918663622732346875492628809, 2.86984184630707362193903160492, 4.24813172295872249015471040529, 5.16894222994826421885905171500, 6.32814677262040118027637646148, 7.03811931015362881887821006700, 8.239074330365357145982852365237, 8.969302674038261400762958269182, 9.961298787262456769621518672452

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