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Properties

Label 4-40e4-1.1-c1e2-0-21
Degree 44
Conductor 25600002560000
Sign 11
Analytic cond. 163.227163.227
Root an. cond. 3.574363.57436
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  − 2·3-s + 6·7-s + 2·9-s − 12·21-s + 2·23-s − 6·27-s + 12·29-s − 18·43-s + 14·47-s + 18·49-s + 12·63-s − 6·67-s − 4·69-s + 11·81-s + 22·83-s − 24·87-s + 12·89-s + 36·101-s + 18·103-s + 26·107-s − 22·121-s + 127-s + 36·129-s + 131-s + 137-s + 139-s − 28·141-s + ⋯
L(s)  = 1  − 1.15·3-s + 2.26·7-s + 2/3·9-s − 2.61·21-s + 0.417·23-s − 1.15·27-s + 2.22·29-s − 2.74·43-s + 2.04·47-s + 18/7·49-s + 1.51·63-s − 0.733·67-s − 0.481·69-s + 11/9·81-s + 2.41·83-s − 2.57·87-s + 1.27·89-s + 3.58·101-s + 1.77·103-s + 2.51·107-s − 2·121-s + 0.0887·127-s + 3.16·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.35·141-s + ⋯

Functional equation

Λ(s)=(2560000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2560000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 25600002560000    =    212⋅542^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 163.227163.227
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2560000, ( :1/2,1/2), 1)(4,\ 2560000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1598375562.159837556
L(12)L(\frac12) ≈\approx 2.1598375562.159837556
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
5 1 1
good3C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
7C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
41C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
43C22C_2^2 1+18T+162T2+18pT3+p2T4 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−14T+98T2−14pT3+p2T4 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4}
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
67C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1−22T+242T2−22pT3+p2T4 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C2C_2 (1+pT2)2 ( 1 + 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

−9.808634071172554708725211779728, −8.981021072051790332711909896389, −8.715682100582172526057577956387, −8.553190499172138405910183511041, −7.86922279576880230218534033104, −7.62561266042530916678711360244, −7.35729692236798593523829620330, −6.72016697862793868423748699054, −6.17364264952357770932618056764, −6.10430167146131886553183041943, −5.29369115774265646522695955691, −5.06187783798438436590623467379, −4.69630400834468146692411967977, −4.55449588525619354671206126566, −3.72895650149875186242656031563, −3.27386988931851935633473820789, −2.31422058439203483863311661110, −1.96314886680260062258064664354, −1.24744484253273168861608210071, −0.69687543080154082339848196756, 0.69687543080154082339848196756, 1.24744484253273168861608210071, 1.96314886680260062258064664354, 2.31422058439203483863311661110, 3.27386988931851935633473820789, 3.72895650149875186242656031563, 4.55449588525619354671206126566, 4.69630400834468146692411967977, 5.06187783798438436590623467379, 5.29369115774265646522695955691, 6.10430167146131886553183041943, 6.17364264952357770932618056764, 6.72016697862793868423748699054, 7.35729692236798593523829620330, 7.62561266042530916678711360244, 7.86922279576880230218534033104, 8.553190499172138405910183511041, 8.715682100582172526057577956387, 8.981021072051790332711909896389, 9.808634071172554708725211779728

Graph of the ZZ-function along the critical line