Properties

Label 2-720-1.1-c3-0-23
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 + 16·7-s − 28·11-s − 26·13-s + 62·17-s + 68·19-s − 208·23-s + 25·25-s + 58·29-s − 160·31-s − 80·35-s + 270·37-s − 282·41-s − 76·43-s − 280·47-s − 87·49-s + 210·53-s + 140·55-s + 196·59-s + 742·61-s + 130·65-s − 836·67-s − 504·71-s − 1.06e3·73-s − 448·77-s − 768·79-s − 1.05e3·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.863·7-s − 0.767·11-s − 0.554·13-s + 0.884·17-s + 0.821·19-s − 1.88·23-s + 1/5·25-s + 0.371·29-s − 0.926·31-s − 0.386·35-s + 1.19·37-s − 1.07·41-s − 0.269·43-s − 0.868·47-s − 0.253·49-s + 0.544·53-s + 0.343·55-s + 0.432·59-s + 1.55·61-s + 0.248·65-s − 1.52·67-s − 0.842·71-s − 1.70·73-s − 0.663·77-s − 1.09·79-s − 1.39·83-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 \( 1 - 16 T + p^{3} T^{2} \)
11 \( 1 + 28 T + p^{3} T^{2} \)
13 \( 1 + 2 p T + p^{3} T^{2} \)
17 \( 1 - 62 T + p^{3} T^{2} \)
19 \( 1 - 68 T + p^{3} T^{2} \)
23 \( 1 + 208 T + p^{3} T^{2} \)
29 \( 1 - 2 p T + p^{3} T^{2} \)
31 \( 1 + 160 T + p^{3} T^{2} \)
37 \( 1 - 270 T + p^{3} T^{2} \)
41 \( 1 + 282 T + p^{3} T^{2} \)
43 \( 1 + 76 T + p^{3} T^{2} \)
47 \( 1 + 280 T + p^{3} T^{2} \)
53 \( 1 - 210 T + p^{3} T^{2} \)
59 \( 1 - 196 T + p^{3} T^{2} \)
61 \( 1 - 742 T + p^{3} T^{2} \)
67 \( 1 + 836 T + p^{3} T^{2} \)
71 \( 1 + 504 T + p^{3} T^{2} \)
73 \( 1 + 1062 T + p^{3} T^{2} \)
79 \( 1 + 768 T + p^{3} T^{2} \)
83 \( 1 + 1052 T + p^{3} T^{2} \)
89 \( 1 - 726 T + p^{3} T^{2} \)
97 \( 1 + 1406 T + p^{3} T^{2} \)
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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

−9.800095994046997545597161350359, −8.478744813245837004359988632783, −7.86523313374082870419126362774, −7.22828625858627094936085097158, −5.82590851836972082420520483793, −5.05455036767600041321342701724, −4.05367443821206399974325006276, −2.84483562175671081789187281342, −1.55331718631072820004742209192, 0, 1.55331718631072820004742209192, 2.84483562175671081789187281342, 4.05367443821206399974325006276, 5.05455036767600041321342701724, 5.82590851836972082420520483793, 7.22828625858627094936085097158, 7.86523313374082870419126362774, 8.478744813245837004359988632783, 9.800095994046997545597161350359

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