Properties

Label 2-720-1.1-c3-0-25
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 − 6·7-s + 32·11-s − 38·13-s − 26·17-s − 100·19-s − 78·23-s + 25·25-s + 50·29-s + 108·31-s − 30·35-s + 266·37-s − 22·41-s − 442·43-s − 514·47-s − 307·49-s − 2·53-s + 160·55-s + 500·59-s − 518·61-s − 190·65-s − 126·67-s + 412·71-s − 878·73-s − 192·77-s − 600·79-s + 282·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.323·7-s + 0.877·11-s − 0.810·13-s − 0.370·17-s − 1.20·19-s − 0.707·23-s + 1/5·25-s + 0.320·29-s + 0.625·31-s − 0.144·35-s + 1.18·37-s − 0.0838·41-s − 1.56·43-s − 1.59·47-s − 0.895·49-s − 0.00518·53-s + 0.392·55-s + 1.10·59-s − 1.08·61-s − 0.362·65-s − 0.229·67-s + 0.688·71-s − 1.40·73-s − 0.284·77-s − 0.854·79-s + 0.372·83-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 \( 1 + 6 T + p^{3} T^{2} \)
11 \( 1 - 32 T + p^{3} T^{2} \)
13 \( 1 + 38 T + p^{3} T^{2} \)
17 \( 1 + 26 T + p^{3} T^{2} \)
19 \( 1 + 100 T + p^{3} T^{2} \)
23 \( 1 + 78 T + p^{3} T^{2} \)
29 \( 1 - 50 T + p^{3} T^{2} \)
31 \( 1 - 108 T + p^{3} T^{2} \)
37 \( 1 - 266 T + p^{3} T^{2} \)
41 \( 1 + 22 T + p^{3} T^{2} \)
43 \( 1 + 442 T + p^{3} T^{2} \)
47 \( 1 + 514 T + p^{3} T^{2} \)
53 \( 1 + 2 T + p^{3} T^{2} \)
59 \( 1 - 500 T + p^{3} T^{2} \)
61 \( 1 + 518 T + p^{3} T^{2} \)
67 \( 1 + 126 T + p^{3} T^{2} \)
71 \( 1 - 412 T + p^{3} T^{2} \)
73 \( 1 + 878 T + p^{3} T^{2} \)
79 \( 1 + 600 T + p^{3} T^{2} \)
83 \( 1 - 282 T + p^{3} T^{2} \)
89 \( 1 - 150 T + p^{3} T^{2} \)
97 \( 1 - 386 T + p^{3} T^{2} \)
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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

−9.700125703914960327548469542768, −8.788632635770644325053411477283, −7.928900760729304727783551766282, −6.66711280201578740803293966275, −6.27431653263672002410284253390, −4.96437391335913047294346096408, −4.06116841666061147546014838913, −2.76198394169540751676854251925, −1.63411637176480508555443978470, 0, 1.63411637176480508555443978470, 2.76198394169540751676854251925, 4.06116841666061147546014838913, 4.96437391335913047294346096408, 6.27431653263672002410284253390, 6.66711280201578740803293966275, 7.928900760729304727783551766282, 8.788632635770644325053411477283, 9.700125703914960327548469542768

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