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Properties

Label 2-720-1.1-c3-0-29
Degree 22
Conductor 720720
Sign −1-1
Analytic cond. 42.481342.4813
Root an. cond. 6.517776.51777
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s + 28·7-s − 24·11-s − 70·13-s − 102·17-s − 20·19-s − 72·23-s + 25·25-s − 306·29-s + 136·31-s + 140·35-s − 214·37-s + 150·41-s + 292·43-s − 72·47-s + 441·49-s + 414·53-s − 120·55-s − 744·59-s − 418·61-s − 350·65-s − 188·67-s + 480·71-s + 434·73-s − 672·77-s − 1.35e3·79-s − 612·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s − 0.657·11-s − 1.49·13-s − 1.45·17-s − 0.241·19-s − 0.652·23-s + 1/5·25-s − 1.95·29-s + 0.787·31-s + 0.676·35-s − 0.950·37-s + 0.571·41-s + 1.03·43-s − 0.223·47-s + 9/7·49-s + 1.07·53-s − 0.294·55-s − 1.64·59-s − 0.877·61-s − 0.667·65-s − 0.342·67-s + 0.802·71-s + 0.695·73-s − 0.994·77-s − 1.92·79-s − 0.809·83-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: −1-1
Analytic conductor: 42.481342.4813
Root analytic conductor: 6.517776.51777
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 720, ( :3/2), −1)(2,\ 720,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
5 1−pT 1 - p T
good7 1−4pT+p3T2 1 - 4 p T + p^{3} T^{2}
11 1+24T+p3T2 1 + 24 T + p^{3} T^{2}
13 1+70T+p3T2 1 + 70 T + p^{3} T^{2}
17 1+6pT+p3T2 1 + 6 p T + p^{3} T^{2}
19 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
23 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
29 1+306T+p3T2 1 + 306 T + p^{3} T^{2}
31 1−136T+p3T2 1 - 136 T + p^{3} T^{2}
37 1+214T+p3T2 1 + 214 T + p^{3} T^{2}
41 1−150T+p3T2 1 - 150 T + p^{3} T^{2}
43 1−292T+p3T2 1 - 292 T + p^{3} T^{2}
47 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
53 1−414T+p3T2 1 - 414 T + p^{3} T^{2}
59 1+744T+p3T2 1 + 744 T + p^{3} T^{2}
61 1+418T+p3T2 1 + 418 T + p^{3} T^{2}
67 1+188T+p3T2 1 + 188 T + p^{3} T^{2}
71 1−480T+p3T2 1 - 480 T + p^{3} T^{2}
73 1−434T+p3T2 1 - 434 T + p^{3} T^{2}
79 1+1352T+p3T2 1 + 1352 T + p^{3} T^{2}
83 1+612T+p3T2 1 + 612 T + p^{3} T^{2}
89 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
97 1+286T+p3T2 1 + 286 T + p^{3} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−9.568635028474010394600370619287, −8.716770483921667370122856666651, −7.79155753542801779077323565151, −7.15541737436847758901104467445, −5.84137283806300710110929303645, −4.97100939919980424854376217761, −4.28953158796361739287032141932, −2.49737499628677612533184955654, −1.80620269304406561305488064499, 0, 1.80620269304406561305488064499, 2.49737499628677612533184955654, 4.28953158796361739287032141932, 4.97100939919980424854376217761, 5.84137283806300710110929303645, 7.15541737436847758901104467445, 7.79155753542801779077323565151, 8.716770483921667370122856666651, 9.568635028474010394600370619287

Graph of the ZZ-function along the critical line