Properties

Label 2-720-1.1-c3-0-19
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 − 34·7-s + 18·11-s + 12·13-s + 106·17-s + 44·19-s + 56·23-s + 25·25-s − 270·29-s − 204·31-s − 170·35-s + 120·37-s − 80·41-s − 536·43-s − 536·47-s + 813·49-s − 542·53-s + 90·55-s − 174·59-s + 186·61-s + 60·65-s − 332·67-s − 132·71-s − 602·73-s − 612·77-s + 548·79-s − 492·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.83·7-s + 0.493·11-s + 0.256·13-s + 1.51·17-s + 0.531·19-s + 0.507·23-s + 1/5·25-s − 1.72·29-s − 1.18·31-s − 0.821·35-s + 0.533·37-s − 0.304·41-s − 1.90·43-s − 1.66·47-s + 2.37·49-s − 1.40·53-s + 0.220·55-s − 0.383·59-s + 0.390·61-s + 0.114·65-s − 0.605·67-s − 0.220·71-s − 0.965·73-s − 0.905·77-s + 0.780·79-s − 0.650·83-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 \( 1 + 34 T + p^{3} T^{2} \)
11 \( 1 - 18 T + p^{3} T^{2} \)
13 \( 1 - 12 T + p^{3} T^{2} \)
17 \( 1 - 106 T + p^{3} T^{2} \)
19 \( 1 - 44 T + p^{3} T^{2} \)
23 \( 1 - 56 T + p^{3} T^{2} \)
29 \( 1 + 270 T + p^{3} T^{2} \)
31 \( 1 + 204 T + p^{3} T^{2} \)
37 \( 1 - 120 T + p^{3} T^{2} \)
41 \( 1 + 80 T + p^{3} T^{2} \)
43 \( 1 + 536 T + p^{3} T^{2} \)
47 \( 1 + 536 T + p^{3} T^{2} \)
53 \( 1 + 542 T + p^{3} T^{2} \)
59 \( 1 + 174 T + p^{3} T^{2} \)
61 \( 1 - 186 T + p^{3} T^{2} \)
67 \( 1 + 332 T + p^{3} T^{2} \)
71 \( 1 + 132 T + p^{3} T^{2} \)
73 \( 1 + 602 T + p^{3} T^{2} \)
79 \( 1 - 548 T + p^{3} T^{2} \)
83 \( 1 + 492 T + p^{3} T^{2} \)
89 \( 1 - 1052 T + p^{3} T^{2} \)
97 \( 1 - 482 T + p^{3} T^{2} \)
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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

−9.608953731299650841890066401627, −9.057427983709098036792655771257, −7.73323846350184087475417457616, −6.81847892209639060270204661909, −6.06954826095111377752373881004, −5.26048846395843449438972536185, −3.61113749066029937377856775328, −3.14091650863687332822324609739, −1.50134958372147065178214757112, 0, 1.50134958372147065178214757112, 3.14091650863687332822324609739, 3.61113749066029937377856775328, 5.26048846395843449438972536185, 6.06954826095111377752373881004, 6.81847892209639060270204661909, 7.73323846350184087475417457616, 9.057427983709098036792655771257, 9.608953731299650841890066401627

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