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Properties

Label 4-2100e2-1.1-c1e2-0-15
Degree 44
Conductor 44100004410000
Sign 11
Analytic cond. 281.185281.185
Root an. cond. 4.094944.09494
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  − 3-s − 7-s − 2·11-s + 6·13-s + 8·17-s + 19-s + 21-s + 8·23-s + 27-s + 8·29-s − 3·31-s + 2·33-s − 37-s − 6·39-s + 12·41-s − 22·43-s + 6·47-s − 6·49-s − 8·51-s − 12·53-s − 57-s − 4·59-s + 6·61-s + 13·67-s − 8·69-s − 20·71-s − 11·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s − 0.603·11-s + 1.66·13-s + 1.94·17-s + 0.229·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s + 1.48·29-s − 0.538·31-s + 0.348·33-s − 0.164·37-s − 0.960·39-s + 1.87·41-s − 3.35·43-s + 0.875·47-s − 6/7·49-s − 1.12·51-s − 1.64·53-s − 0.132·57-s − 0.520·59-s + 0.768·61-s + 1.58·67-s − 0.963·69-s − 2.37·71-s − 1.28·73-s + ⋯

Functional equation

Λ(s)=(4410000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4410000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 44100004410000    =    24⋅32⋅54⋅722^{4} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 281.185281.185
Root analytic conductor: 4.094944.09494
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 4410000, ( :1/2,1/2), 1)(4,\ 4410000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0331132012.033113201
L(12)L(\frac12) ≈\approx 2.0331132012.033113201
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)
bad2 1 1
3C2C_2 1+T+T2 1 + T + T^{2}
5 1 1
7C2C_2 1+T+pT2 1 + T + p T^{2}
good11C22C_2^2 1+2T−7T2+2pT3+p2T4 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
17C22C_2^2 1−8T+47T2−8pT3+p2T4 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4}
19C2C_2 (1−8T+pT2)(1+7T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} )
23C22C_2^2 1−8T+41T2−8pT3+p2T4 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4}
29C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
31C22C_2^2 1+3T−22T2+3pT3+p2T4 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4}
37C2C_2 (1−10T+pT2)(1+11T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} )
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C2C_2 (1+11T+pT2)2 ( 1 + 11 T + p T^{2} )^{2}
47C22C_2^2 1−6T−11T2−6pT3+p2T4 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+12T+91T2+12pT3+p2T4 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+4T−43T2+4pT3+p2T4 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−6T−25T2−6pT3+p2T4 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−13T+102T2−13pT3+p2T4 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4}
71C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
73C22C_2^2 1+11T+48T2+11pT3+p2T4 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−3T−70T2−3pT3+p2T4 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4}
83C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
89C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
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

−9.404867300453660491687727090464, −8.756491253430226388308794240206, −8.481586197286060665889451892708, −8.274031303280547287898977168784, −7.75881040428835068151252853549, −7.24994931249509686195318535508, −7.01362081111772859095173615222, −6.48508810262852846364315702894, −5.96596966402527894882673227731, −5.91976027743212472565729749444, −5.35265945924216109852996909503, −4.86976500263577129667346499165, −4.68342841456055938379128876223, −3.86458612992151212977773671285, −3.39885405141085957937069191497, −3.07471891287172468221521771786, −2.76403516598285403654098936602, −1.61882765419548335065548026140, −1.26499302594765782524353377093, −0.59146379977460636388211260175, 0.59146379977460636388211260175, 1.26499302594765782524353377093, 1.61882765419548335065548026140, 2.76403516598285403654098936602, 3.07471891287172468221521771786, 3.39885405141085957937069191497, 3.86458612992151212977773671285, 4.68342841456055938379128876223, 4.86976500263577129667346499165, 5.35265945924216109852996909503, 5.91976027743212472565729749444, 5.96596966402527894882673227731, 6.48508810262852846364315702894, 7.01362081111772859095173615222, 7.24994931249509686195318535508, 7.75881040428835068151252853549, 8.274031303280547287898977168784, 8.481586197286060665889451892708, 8.756491253430226388308794240206, 9.404867300453660491687727090464

Graph of the ZZ-function along the critical line