Properties

Label 2-720-1.1-c3-0-27
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

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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: \(1\)
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 \( 1 + 2 T + p^{3} T^{2} \)
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 \( 1 - 514 T + p^{3} T^{2} \)
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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

−9.698554711068980100534474093859, −8.836536231239757968855699558389, −7.83749257028573088129008820398, −6.84481843013476577981970739627, −6.12512338147538544757501891691, −4.98740401536024880923986762492, −4.08023009761609070257363231589, −2.73347584249626969031515742757, −1.65086830137977796718777133939, 0, 1.65086830137977796718777133939, 2.73347584249626969031515742757, 4.08023009761609070257363231589, 4.98740401536024880923986762492, 6.12512338147538544757501891691, 6.84481843013476577981970739627, 7.83749257028573088129008820398, 8.836536231239757968855699558389, 9.698554711068980100534474093859

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