Properties

Label 2-720-1.1-c3-0-29
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 + 28·7-s − 24·11-s − 70·13-s − 102·17-s − 20·19-s − 72·23-s + 25·25-s − 306·29-s + 136·31-s + 140·35-s − 214·37-s + 150·41-s + 292·43-s − 72·47-s + 441·49-s + 414·53-s − 120·55-s − 744·59-s − 418·61-s − 350·65-s − 188·67-s + 480·71-s + 434·73-s − 672·77-s − 1.35e3·79-s − 612·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s − 0.657·11-s − 1.49·13-s − 1.45·17-s − 0.241·19-s − 0.652·23-s + 1/5·25-s − 1.95·29-s + 0.787·31-s + 0.676·35-s − 0.950·37-s + 0.571·41-s + 1.03·43-s − 0.223·47-s + 9/7·49-s + 1.07·53-s − 0.294·55-s − 1.64·59-s − 0.877·61-s − 0.667·65-s − 0.342·67-s + 0.802·71-s + 0.695·73-s − 0.994·77-s − 1.92·79-s − 0.809·83-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 \( 1 - 4 p T + p^{3} T^{2} \)
11 \( 1 + 24 T + p^{3} T^{2} \)
13 \( 1 + 70 T + p^{3} T^{2} \)
17 \( 1 + 6 p T + p^{3} T^{2} \)
19 \( 1 + 20 T + p^{3} T^{2} \)
23 \( 1 + 72 T + p^{3} T^{2} \)
29 \( 1 + 306 T + p^{3} T^{2} \)
31 \( 1 - 136 T + p^{3} T^{2} \)
37 \( 1 + 214 T + p^{3} T^{2} \)
41 \( 1 - 150 T + p^{3} T^{2} \)
43 \( 1 - 292 T + p^{3} T^{2} \)
47 \( 1 + 72 T + p^{3} T^{2} \)
53 \( 1 - 414 T + p^{3} T^{2} \)
59 \( 1 + 744 T + p^{3} T^{2} \)
61 \( 1 + 418 T + p^{3} T^{2} \)
67 \( 1 + 188 T + p^{3} T^{2} \)
71 \( 1 - 480 T + p^{3} T^{2} \)
73 \( 1 - 434 T + p^{3} T^{2} \)
79 \( 1 + 1352 T + p^{3} T^{2} \)
83 \( 1 + 612 T + p^{3} T^{2} \)
89 \( 1 - 30 T + p^{3} T^{2} \)
97 \( 1 + 286 T + p^{3} T^{2} \)
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

−9.568635028474010394600370619287, −8.716770483921667370122856666651, −7.79155753542801779077323565151, −7.15541737436847758901104467445, −5.84137283806300710110929303645, −4.97100939919980424854376217761, −4.28953158796361739287032141932, −2.49737499628677612533184955654, −1.80620269304406561305488064499, 0, 1.80620269304406561305488064499, 2.49737499628677612533184955654, 4.28953158796361739287032141932, 4.97100939919980424854376217761, 5.84137283806300710110929303645, 7.15541737436847758901104467445, 7.79155753542801779077323565151, 8.716770483921667370122856666651, 9.568635028474010394600370619287

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