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Properties

Label 4-180e2-1.1-c0e2-0-2
Degree 44
Conductor 3240032400
Sign 11
Analytic cond. 0.008069730.00806973
Root an. cond. 0.2997190.299719
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s − 14-s − 15-s − 16-s − 21-s − 23-s − 24-s − 27-s + 29-s − 30-s + 35-s + 40-s + 41-s − 42-s + 2·43-s − 46-s − 47-s − 48-s + 49-s − 54-s + 56-s + ⋯
L(s)  = 1  + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s − 14-s − 15-s − 16-s − 21-s − 23-s − 24-s − 27-s + 29-s − 30-s + 35-s + 40-s + 41-s − 42-s + 2·43-s − 46-s − 47-s − 48-s + 49-s − 54-s + 56-s + ⋯

Functional equation

Λ(s)=(32400s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(32400s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3240032400    =    24⋅34⋅522^{4} \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 0.008069730.00806973
Root analytic conductor: 0.2997190.299719
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 32400, ( :0,0), 1)(4,\ 32400,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.66747776570.6674777657
L(12)L(\frac12) ≈\approx 0.66747776570.6674777657
L(1)L(1) 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C2C_2 1−T+T2 1 - T + T^{2}
3C2C_2 1−T+T2 1 - T + T^{2}
5C2C_2 1+T+T2 1 + T + T^{2}
good7C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
11C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
13C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
17C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
19C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
23C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
29C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
31C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
37C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
41C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
43C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
47C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
53C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
59C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
61C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
67C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
71C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
73C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
79C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
83C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
89C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
97C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.25266003726459056934228925380, −12.63618427465138715563032354413, −12.34233518978514723443508067396, −11.93483636824218202361140453035, −11.32059066798460248258732799065, −10.87563652432827529726951676893, −9.877856881384228442551797346298, −9.755306957471470115556646605831, −9.052683058898347171661999093782, −8.548958346639384954526037770529, −8.175583683876575953362057897275, −7.48473382694494253624140028139, −6.99564495359185224065559830264, −5.96797693585416330644713210577, −5.96541110648486450872095668393, −4.86178503697550767939988741389, −4.02505199161513633549089042307, −3.87921310680456161343166166782, −3.01554335708622184459064059850, −2.52796415098278775375283093093, 2.52796415098278775375283093093, 3.01554335708622184459064059850, 3.87921310680456161343166166782, 4.02505199161513633549089042307, 4.86178503697550767939988741389, 5.96541110648486450872095668393, 5.96797693585416330644713210577, 6.99564495359185224065559830264, 7.48473382694494253624140028139, 8.175583683876575953362057897275, 8.548958346639384954526037770529, 9.052683058898347171661999093782, 9.755306957471470115556646605831, 9.877856881384228442551797346298, 10.87563652432827529726951676893, 11.32059066798460248258732799065, 11.93483636824218202361140453035, 12.34233518978514723443508067396, 12.63618427465138715563032354413, 13.25266003726459056934228925380

Graph of the ZZ-function along the critical line