Properties

Label 2-720-1.1-c3-0-28
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 + 18·7-s − 34·11-s + 12·13-s − 102·17-s − 164·19-s − 48·23-s + 25·25-s + 146·29-s − 100·31-s + 90·35-s + 328·37-s − 288·41-s − 120·43-s − 16·47-s − 19·49-s − 126·53-s − 170·55-s − 642·59-s + 602·61-s + 60·65-s − 436·67-s − 652·71-s + 1.06e3·73-s − 612·77-s − 388·79-s + 444·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.971·7-s − 0.931·11-s + 0.256·13-s − 1.45·17-s − 1.98·19-s − 0.435·23-s + 1/5·25-s + 0.934·29-s − 0.579·31-s + 0.434·35-s + 1.45·37-s − 1.09·41-s − 0.425·43-s − 0.0496·47-s − 0.0553·49-s − 0.326·53-s − 0.416·55-s − 1.41·59-s + 1.26·61-s + 0.114·65-s − 0.795·67-s − 1.08·71-s + 1.70·73-s − 0.905·77-s − 0.552·79-s + 0.587·83-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 \( 1 - 18 T + p^{3} T^{2} \)
11 \( 1 + 34 T + p^{3} T^{2} \)
13 \( 1 - 12 T + p^{3} T^{2} \)
17 \( 1 + 6 p T + p^{3} T^{2} \)
19 \( 1 + 164 T + p^{3} T^{2} \)
23 \( 1 + 48 T + p^{3} T^{2} \)
29 \( 1 - 146 T + p^{3} T^{2} \)
31 \( 1 + 100 T + p^{3} T^{2} \)
37 \( 1 - 328 T + p^{3} T^{2} \)
41 \( 1 + 288 T + p^{3} T^{2} \)
43 \( 1 + 120 T + p^{3} T^{2} \)
47 \( 1 + 16 T + p^{3} T^{2} \)
53 \( 1 + 126 T + p^{3} T^{2} \)
59 \( 1 + 642 T + p^{3} T^{2} \)
61 \( 1 - 602 T + p^{3} T^{2} \)
67 \( 1 + 436 T + p^{3} T^{2} \)
71 \( 1 + 652 T + p^{3} T^{2} \)
73 \( 1 - 1062 T + p^{3} T^{2} \)
79 \( 1 + 388 T + p^{3} T^{2} \)
83 \( 1 - 444 T + p^{3} T^{2} \)
89 \( 1 + 820 T + p^{3} T^{2} \)
97 \( 1 + 766 T + p^{3} T^{2} \)
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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

−9.598023949803419899905638252894, −8.499274051364086291368336254455, −8.140087082031022352012151639789, −6.84643026874068490471649050787, −6.05164972946561926212109785925, −4.91871015722656249551079093458, −4.23803846914387519133137654675, −2.58216215805839475919722932872, −1.74510056691010278427860866614, 0, 1.74510056691010278427860866614, 2.58216215805839475919722932872, 4.23803846914387519133137654675, 4.91871015722656249551079093458, 6.05164972946561926212109785925, 6.84643026874068490471649050787, 8.140087082031022352012151639789, 8.499274051364086291368336254455, 9.598023949803419899905638252894

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