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Properties

Label 4-3960e2-1.1-c1e2-0-7
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·11-s − 8·19-s − 25-s + 4·29-s + 16·31-s + 12·41-s + 10·49-s + 4·55-s + 8·59-s + 28·61-s − 16·71-s + 16·79-s − 20·89-s − 16·95-s + 20·101-s + 28·109-s + 3·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 32·155-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.603·11-s − 1.83·19-s − 1/5·25-s + 0.742·29-s + 2.87·31-s + 1.87·41-s + 10/7·49-s + 0.539·55-s + 1.04·59-s + 3.58·61-s − 1.89·71-s + 1.80·79-s − 2.11·89-s − 1.64·95-s + 1.99·101-s + 2.68·109-s + 3/11·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 2.57·155-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 4.2626249154.262624915
L(12)L(\frac12) ≈\approx 4.2626249154.262624915
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
5C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
11C1C_1 (1−T)2 ( 1 - T )^{2}
good7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
13C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
29C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
37C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
47C22C_2^2 1−90T2+p2T4 1 - 90 T^{2} + p^{2} T^{4}
53C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
59C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
61C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
67C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
71C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
73C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C22C_2^2 1−162T2+p2T4 1 - 162 T^{2} + p^{2} T^{4}
89C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
97C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 T + p 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

−8.561690076916054853886704956183, −8.509151204991037720525387181518, −7.969278943331290246984175117134, −7.56380120870672226378041199914, −7.07471794438814770083035953668, −6.68899086485885057007120434305, −6.39804456224879840857164182770, −6.17149480136203216358665343108, −5.66882303551193056114112541739, −5.48925641320362647852716259597, −4.67497077699503002441332596121, −4.58221927067700631649930411133, −4.02440603694632319159604924134, −3.86762120371455746166277772484, −3.00571192406293325630737643907, −2.65791307822691878667918729706, −2.15097134920176642783220814059, −1.95780733126578720706026041785, −0.913959038410848158142972276161, −0.76682447496435372864109877362, 0.76682447496435372864109877362, 0.913959038410848158142972276161, 1.95780733126578720706026041785, 2.15097134920176642783220814059, 2.65791307822691878667918729706, 3.00571192406293325630737643907, 3.86762120371455746166277772484, 4.02440603694632319159604924134, 4.58221927067700631649930411133, 4.67497077699503002441332596121, 5.48925641320362647852716259597, 5.66882303551193056114112541739, 6.17149480136203216358665343108, 6.39804456224879840857164182770, 6.68899086485885057007120434305, 7.07471794438814770083035953668, 7.56380120870672226378041199914, 7.969278943331290246984175117134, 8.509151204991037720525387181518, 8.561690076916054853886704956183

Graph of the ZZ-function along the critical line