Properties

Label 2-720-1.1-c3-0-26
Degree $2$
Conductor $720$
Sign $-1$
Analytic cond. $42.4813$
Root an. cond. $6.51777$
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  + 5·5-s + 4·7-s − 48·11-s + 2·13-s + 114·17-s − 140·19-s + 72·23-s + 25·25-s − 210·29-s − 272·31-s + 20·35-s − 334·37-s + 198·41-s + 268·43-s + 216·47-s − 327·49-s + 78·53-s − 240·55-s + 240·59-s + 302·61-s + 10·65-s − 596·67-s − 768·71-s − 478·73-s − 192·77-s + 640·79-s − 348·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.215·7-s − 1.31·11-s + 0.0426·13-s + 1.62·17-s − 1.69·19-s + 0.652·23-s + 1/5·25-s − 1.34·29-s − 1.57·31-s + 0.0965·35-s − 1.48·37-s + 0.754·41-s + 0.950·43-s + 0.670·47-s − 0.953·49-s + 0.202·53-s − 0.588·55-s + 0.529·59-s + 0.633·61-s + 0.0190·65-s − 1.08·67-s − 1.28·71-s − 0.766·73-s − 0.284·77-s + 0.911·79-s − 0.460·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-1$
Analytic conductor: \(42.4813\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{720} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ -1)\)

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 \( 1 \)
5 \( 1 - p T \)
good7 \( 1 - 4 T + p^{3} T^{2} \)
11 \( 1 + 48 T + p^{3} T^{2} \)
13 \( 1 - 2 T + p^{3} T^{2} \)
17 \( 1 - 114 T + p^{3} T^{2} \)
19 \( 1 + 140 T + p^{3} T^{2} \)
23 \( 1 - 72 T + p^{3} T^{2} \)
29 \( 1 + 210 T + p^{3} T^{2} \)
31 \( 1 + 272 T + p^{3} T^{2} \)
37 \( 1 + 334 T + p^{3} T^{2} \)
41 \( 1 - 198 T + p^{3} T^{2} \)
43 \( 1 - 268 T + p^{3} T^{2} \)
47 \( 1 - 216 T + p^{3} T^{2} \)
53 \( 1 - 78 T + p^{3} T^{2} \)
59 \( 1 - 240 T + p^{3} T^{2} \)
61 \( 1 - 302 T + p^{3} T^{2} \)
67 \( 1 + 596 T + p^{3} T^{2} \)
71 \( 1 + 768 T + p^{3} T^{2} \)
73 \( 1 + 478 T + p^{3} T^{2} \)
79 \( 1 - 640 T + p^{3} T^{2} \)
83 \( 1 + 348 T + p^{3} T^{2} \)
89 \( 1 + 210 T + p^{3} T^{2} \)
97 \( 1 + 1534 T + p^{3} T^{2} \)
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

−9.619491182103650285765017499863, −8.730129701111845440422300778622, −7.82304087992908285675435985494, −7.07599125485653931229090053696, −5.74377227630537620006175915378, −5.29229770631922928062390085809, −3.96918257796561588452124102039, −2.76261014299176375902574960208, −1.64629280193516114942653260711, 0, 1.64629280193516114942653260711, 2.76261014299176375902574960208, 3.96918257796561588452124102039, 5.29229770631922928062390085809, 5.74377227630537620006175915378, 7.07599125485653931229090053696, 7.82304087992908285675435985494, 8.730129701111845440422300778622, 9.619491182103650285765017499863

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