Properties

Label 2-720-1.1-c3-0-3
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s − 2·7-s − 34·11-s − 68·13-s + 38·17-s − 4·19-s + 152·23-s + 25·25-s + 46·29-s + 260·31-s + 10·35-s − 312·37-s − 48·41-s + 200·43-s + 104·47-s − 339·49-s + 414·53-s + 170·55-s − 2·59-s − 38·61-s + 340·65-s + 244·67-s + 708·71-s − 378·73-s + 68·77-s + 852·79-s + 844·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.107·7-s − 0.931·11-s − 1.45·13-s + 0.542·17-s − 0.0482·19-s + 1.37·23-s + 1/5·25-s + 0.294·29-s + 1.50·31-s + 0.0482·35-s − 1.38·37-s − 0.182·41-s + 0.709·43-s + 0.322·47-s − 0.988·49-s + 1.07·53-s + 0.416·55-s − 0.00441·59-s − 0.0797·61-s + 0.648·65-s + 0.444·67-s + 1.18·71-s − 0.606·73-s + 0.100·77-s + 1.21·79-s + 1.11·83-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: \(0\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.378835983\)
\(L(\frac12)\) \(\approx\) \(1.378835983\)
\(L(\frac{5}{2})\) 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 \( 1 + 2 T + p^{3} T^{2} \)
11 \( 1 + 34 T + p^{3} T^{2} \)
13 \( 1 + 68 T + p^{3} T^{2} \)
17 \( 1 - 38 T + p^{3} T^{2} \)
19 \( 1 + 4 T + p^{3} T^{2} \)
23 \( 1 - 152 T + p^{3} T^{2} \)
29 \( 1 - 46 T + p^{3} T^{2} \)
31 \( 1 - 260 T + p^{3} T^{2} \)
37 \( 1 + 312 T + p^{3} T^{2} \)
41 \( 1 + 48 T + p^{3} T^{2} \)
43 \( 1 - 200 T + p^{3} T^{2} \)
47 \( 1 - 104 T + p^{3} T^{2} \)
53 \( 1 - 414 T + p^{3} T^{2} \)
59 \( 1 + 2 T + p^{3} T^{2} \)
61 \( 1 + 38 T + p^{3} T^{2} \)
67 \( 1 - 244 T + p^{3} T^{2} \)
71 \( 1 - 708 T + p^{3} T^{2} \)
73 \( 1 + 378 T + p^{3} T^{2} \)
79 \( 1 - 852 T + p^{3} T^{2} \)
83 \( 1 - 844 T + p^{3} T^{2} \)
89 \( 1 - 1380 T + p^{3} T^{2} \)
97 \( 1 - 514 T + p^{3} T^{2} \)
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

−10.11187782250396200274996933909, −9.189121205191921281272103721032, −8.170146821112863584525485937680, −7.45764529737290360366412027284, −6.65000600802405336255556184818, −5.29201333312129347876388304177, −4.70975174807131255435401986548, −3.31070307408484721161973864201, −2.39165222229586030186304490517, −0.65328079464842706794257121661, 0.65328079464842706794257121661, 2.39165222229586030186304490517, 3.31070307408484721161973864201, 4.70975174807131255435401986548, 5.29201333312129347876388304177, 6.65000600802405336255556184818, 7.45764529737290360366412027284, 8.170146821112863584525485937680, 9.189121205191921281272103721032, 10.11187782250396200274996933909

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