Properties

Label 2-720-1.1-c3-0-4
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s − 20·7-s − 56·11-s − 86·13-s + 106·17-s − 4·19-s + 136·23-s + 25·25-s + 206·29-s + 152·31-s − 100·35-s + 282·37-s + 246·41-s − 412·43-s + 40·47-s + 57·49-s + 126·53-s − 280·55-s + 56·59-s − 2·61-s − 430·65-s + 388·67-s − 672·71-s + 1.17e3·73-s + 1.12e3·77-s − 408·79-s + 668·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.07·7-s − 1.53·11-s − 1.83·13-s + 1.51·17-s − 0.0482·19-s + 1.23·23-s + 1/5·25-s + 1.31·29-s + 0.880·31-s − 0.482·35-s + 1.25·37-s + 0.937·41-s − 1.46·43-s + 0.124·47-s + 0.166·49-s + 0.326·53-s − 0.686·55-s + 0.123·59-s − 0.00419·61-s − 0.820·65-s + 0.707·67-s − 1.12·71-s + 1.87·73-s + 1.65·77-s − 0.581·79-s + 0.883·83-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

\(L(2)\) \(\approx\) \(1.445930648\)
\(L(\frac12)\) \(\approx\) \(1.445930648\)
\(L(\frac{5}{2})\) 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 \( 1 + 20 T + p^{3} T^{2} \)
11 \( 1 + 56 T + p^{3} T^{2} \)
13 \( 1 + 86 T + p^{3} T^{2} \)
17 \( 1 - 106 T + p^{3} T^{2} \)
19 \( 1 + 4 T + p^{3} T^{2} \)
23 \( 1 - 136 T + p^{3} T^{2} \)
29 \( 1 - 206 T + p^{3} T^{2} \)
31 \( 1 - 152 T + p^{3} T^{2} \)
37 \( 1 - 282 T + p^{3} T^{2} \)
41 \( 1 - 6 p T + p^{3} T^{2} \)
43 \( 1 + 412 T + p^{3} T^{2} \)
47 \( 1 - 40 T + p^{3} T^{2} \)
53 \( 1 - 126 T + p^{3} T^{2} \)
59 \( 1 - 56 T + p^{3} T^{2} \)
61 \( 1 + 2 T + p^{3} T^{2} \)
67 \( 1 - 388 T + p^{3} T^{2} \)
71 \( 1 + 672 T + p^{3} T^{2} \)
73 \( 1 - 1170 T + p^{3} T^{2} \)
79 \( 1 + 408 T + p^{3} T^{2} \)
83 \( 1 - 668 T + p^{3} T^{2} \)
89 \( 1 + 66 T + p^{3} T^{2} \)
97 \( 1 + 926 T + p^{3} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.966096515770255838195583686179, −9.493864387188800645835462542262, −8.166099746388934360114435975072, −7.41528957462770992748127007713, −6.50942037724942214409297562249, −5.42853774452513999255338041522, −4.77375497356854754717661725198, −3.05806569117810373200245554217, −2.56337424129377760391774000581, −0.66180652198307057085647703255, 0.66180652198307057085647703255, 2.56337424129377760391774000581, 3.05806569117810373200245554217, 4.77375497356854754717661725198, 5.42853774452513999255338041522, 6.50942037724942214409297562249, 7.41528957462770992748127007713, 8.166099746388934360114435975072, 9.493864387188800645835462542262, 9.966096515770255838195583686179

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