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Properties

Label 4-3960e2-1.1-c1e2-0-1
Degree 44
Conductor 1568160015681600
Sign 11
Analytic cond. 999.872999.872
Root an. cond. 5.623235.62323
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·7-s − 2·11-s − 8·19-s + 3·25-s + 4·29-s − 4·31-s + 8·35-s + 4·37-s + 4·41-s − 4·43-s + 8·47-s + 4·53-s + 4·55-s + 12·59-s + 12·61-s + 4·71-s + 8·77-s − 16·79-s + 12·83-s + 4·89-s + 16·95-s + 12·97-s + 20·101-s + 12·107-s − 4·109-s + 4·113-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s − 0.603·11-s − 1.83·19-s + 3/5·25-s + 0.742·29-s − 0.718·31-s + 1.35·35-s + 0.657·37-s + 0.624·41-s − 0.609·43-s + 1.16·47-s + 0.549·53-s + 0.539·55-s + 1.56·59-s + 1.53·61-s + 0.474·71-s + 0.911·77-s − 1.80·79-s + 1.31·83-s + 0.423·89-s + 1.64·95-s + 1.21·97-s + 1.99·101-s + 1.16·107-s − 0.383·109-s + 0.376·113-s + ⋯

Functional equation

Λ(s)=(15681600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(15681600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1568160015681600    =    26⋅34⋅52⋅1122^{6} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 999.872999.872
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 15681600, ( :1/2,1/2), 1)(4,\ 15681600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2175618551.217561855
L(12)L(\frac12) ≈\approx 1.2175618551.217561855
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
3 1 1
5C1C_1 (1+T)2 ( 1 + T )^{2}
11C1C_1 (1+T)2 ( 1 + T )^{2}
good7D4D_{4} 1+4T+16T2+4pT3+p2T4 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4}
13C22C_2^2 1+24T2+p2T4 1 + 24 T^{2} + p^{2} T^{4}
17C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
19D4D_{4} 1+8T+46T2+8pT3+p2T4 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
29D4D_{4} 1−4T+30T2−4pT3+p2T4 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+4T+58T2+4pT3+p2T4 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−4T+46T2−4pT3+p2T4 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−4T+54T2−4pT3+p2T4 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+4T+40T2+4pT3+p2T4 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−8T+38T2−8pT3+p2T4 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−4T+38T2−4pT3+p2T4 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−12T+146T2−12pT3+p2T4 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−12T+150T2−12pT3+p2T4 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
71D4D_{4} 1−4T+74T2−4pT3+p2T4 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+96T2+p2T4 1 + 96 T^{2} + p^{2} T^{4}
79D4D_{4} 1+16T+150T2+16pT3+p2T4 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−12T+184T2−12pT3+p2T4 1 - 12 T + 184 T^{2} - 12 p T^{3} + p^{2} T^{4}
89C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
97D4D_{4} 1−12T+222T2−12pT3+p2T4 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4}
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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

−8.606088853871438762727263899983, −8.315436329354925363242536196633, −7.897752765158683490692135554973, −7.59123582022504381012651933355, −7.05007267462379839584269795809, −6.89069243100507081497991434046, −6.42726327278666507722018433274, −6.20028629326285780889600834027, −5.72418659637057702562248655727, −5.32239700098825477563651214801, −4.82908027777640229947502383782, −4.31200300994431665749310696553, −4.10045471059502583758509609818, −3.62941404009603366868147852602, −3.19897247149752880576649682272, −2.84705963764479627571533574096, −2.23213888462537797815943033338, −1.95474066511156490144334047948, −0.72567876661307557380803438624, −0.47043807554754514671534769534, 0.47043807554754514671534769534, 0.72567876661307557380803438624, 1.95474066511156490144334047948, 2.23213888462537797815943033338, 2.84705963764479627571533574096, 3.19897247149752880576649682272, 3.62941404009603366868147852602, 4.10045471059502583758509609818, 4.31200300994431665749310696553, 4.82908027777640229947502383782, 5.32239700098825477563651214801, 5.72418659637057702562248655727, 6.20028629326285780889600834027, 6.42726327278666507722018433274, 6.89069243100507081497991434046, 7.05007267462379839584269795809, 7.59123582022504381012651933355, 7.897752765158683490692135554973, 8.315436329354925363242536196633, 8.606088853871438762727263899983

Graph of the ZZ-function along the critical line