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Properties

Label 4-600e2-1.1-c1e2-0-11
Degree 44
Conductor 360000360000
Sign 11
Analytic cond. 22.953922.9539
Root an. cond. 2.188842.18884
Motivic weight 11
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  − 2·2-s + 2·4-s − 4·7-s − 9-s + 8·14-s − 4·16-s + 12·17-s + 2·18-s + 8·23-s − 8·28-s + 20·31-s + 8·32-s − 24·34-s − 2·36-s + 20·41-s − 16·46-s + 8·47-s − 2·49-s − 40·62-s + 4·63-s − 8·64-s + 24·68-s − 8·71-s − 20·73-s − 28·79-s + 81-s − 40·82-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.51·7-s − 1/3·9-s + 2.13·14-s − 16-s + 2.91·17-s + 0.471·18-s + 1.66·23-s − 1.51·28-s + 3.59·31-s + 1.41·32-s − 4.11·34-s − 1/3·36-s + 3.12·41-s − 2.35·46-s + 1.16·47-s − 2/7·49-s − 5.08·62-s + 0.503·63-s − 64-s + 2.91·68-s − 0.949·71-s − 2.34·73-s − 3.15·79-s + 1/9·81-s − 4.41·82-s + ⋯

Functional equation

Λ(s)=(360000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(360000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 360000360000    =    26⋅32⋅542^{6} \cdot 3^{2} \cdot 5^{4}
Sign: 11
Analytic conductor: 22.953922.9539
Root analytic conductor: 2.188842.18884
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 360000, ( :1/2,1/2), 1)(4,\ 360000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.86275056910.8627505691
L(12)L(\frac12) ≈\approx 0.86275056910.8627505691
L(32)L(\frac{3}{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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C2C_2 1+pT+pT2 1 + p T + p T^{2}
3C2C_2 1+T2 1 + T^{2}
5 1 1
good7C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
11C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
19C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
23C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
29C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
31C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
37C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
53C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
59C22C_2^2 1−54T2+p2T4 1 - 54 T^{2} + p^{2} T^{4}
61C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
67C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
71C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
73C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
79C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
83C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
89C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
97C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{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

−10.46982852197989060389221127174, −10.34519256248327191045397691680, −9.861859387777574106379461685309, −9.760757295120013946296302677340, −9.026677707446116323796825846477, −8.970076493502703952046791255288, −8.263917831551586856894272402927, −7.83727342278446855732740151019, −7.41860431444945416935591347114, −7.15133876144596789457034576020, −6.36429654460564020163414918027, −6.05334046248431055581744507227, −5.69250384819205677507488475638, −4.78342985241194947751533208569, −4.38171239632414891983878406977, −3.41369744806173821743501061303, −2.91095070327648527113839263551, −2.66475744322071252239201683237, −1.15080276500926866166428846606, −0.824933504831208538900601064344, 0.824933504831208538900601064344, 1.15080276500926866166428846606, 2.66475744322071252239201683237, 2.91095070327648527113839263551, 3.41369744806173821743501061303, 4.38171239632414891983878406977, 4.78342985241194947751533208569, 5.69250384819205677507488475638, 6.05334046248431055581744507227, 6.36429654460564020163414918027, 7.15133876144596789457034576020, 7.41860431444945416935591347114, 7.83727342278446855732740151019, 8.263917831551586856894272402927, 8.970076493502703952046791255288, 9.026677707446116323796825846477, 9.760757295120013946296302677340, 9.861859387777574106379461685309, 10.34519256248327191045397691680, 10.46982852197989060389221127174

Graph of the ZZ-function along the critical line