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Properties

Label 8-1575e4-1.1-c0e4-0-3
Degree 88
Conductor 6.154×10126.154\times 10^{12}
Sign 11
Analytic cond. 0.3817250.381725
Root an. cond. 0.8865810.886581
Motivic weight 00
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  + 2·4-s − 2·9-s + 2·11-s + 16-s + 4·29-s − 4·36-s + 4·44-s + 49-s − 2·64-s − 4·71-s − 2·79-s + 3·81-s − 4·99-s + 4·109-s + 8·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯
L(s)  = 1  + 2·4-s − 2·9-s + 2·11-s + 16-s + 4·29-s − 4·36-s + 4·44-s + 49-s − 2·64-s − 4·71-s − 2·79-s + 3·81-s − 4·99-s + 4·109-s + 8·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯

Functional equation

Λ(s)=((38⋅58⋅74)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((38⋅58⋅74)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 38⋅58⋅743^{8} \cdot 5^{8} \cdot 7^{4}
Sign: 11
Analytic conductor: 0.3817250.381725
Root analytic conductor: 0.8865810.886581
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 38⋅58⋅74, ( :0,0,0,0), 1)(8,\ 3^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.0636545342.063654534
L(12)L(\frac12) ≈\approx 2.0636545342.063654534
L(1)L(1) 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)
bad3C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
5 1 1
7C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
good2C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
11C1C_1×\timesC2C_2 (1−T)4(1+T+T2)2 ( 1 - T )^{4}( 1 + T + T^{2} )^{2}
13C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
17C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
19C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
23C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
29C2C_2 (1−T+T2)4 ( 1 - T + T^{2} )^{4}
31C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
37C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
41C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
43C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
47C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
53C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
59C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
61C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
67C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
71C2C_2 (1+T+T2)4 ( 1 + T + T^{2} )^{4}
73C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
79C1C_1×\timesC2C_2 (1+T)4(1−T+T2)2 ( 1 + T )^{4}( 1 - T + T^{2} )^{2}
83C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
89C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
97C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.04617732380829806092168981110, −6.51033319227279921106186720480, −6.37228678742432010156662628566, −6.36043914413746361929217544352, −6.30533529410754720474081293563, −5.79949499433153945011159035881, −5.79071754921210242733575081470, −5.77847644038558579992083573489, −5.19468295254110238710123648436, −4.82148606476732153881546481254, −4.72639573927937808472050199215, −4.67433901265314038003178145429, −4.14384563040902869233923008956, −4.02025632214192708359342399432, −3.88473803549106906982580107473, −3.22163759071610718594362779710, −3.01694497752613453450060376560, −2.98671574054719098737459256306, −2.89592097041120463039240858315, −2.57117304024433075536279854819, −2.05669390229901320017423026929, −1.98492320336306636557745181058, −1.65675978560010566305981127842, −1.07673924168719496661651400743, −0.933100740722115110091882544548, 0.933100740722115110091882544548, 1.07673924168719496661651400743, 1.65675978560010566305981127842, 1.98492320336306636557745181058, 2.05669390229901320017423026929, 2.57117304024433075536279854819, 2.89592097041120463039240858315, 2.98671574054719098737459256306, 3.01694497752613453450060376560, 3.22163759071610718594362779710, 3.88473803549106906982580107473, 4.02025632214192708359342399432, 4.14384563040902869233923008956, 4.67433901265314038003178145429, 4.72639573927937808472050199215, 4.82148606476732153881546481254, 5.19468295254110238710123648436, 5.77847644038558579992083573489, 5.79071754921210242733575081470, 5.79949499433153945011159035881, 6.30533529410754720474081293563, 6.36043914413746361929217544352, 6.37228678742432010156662628566, 6.51033319227279921106186720480, 7.04617732380829806092168981110

Graph of the ZZ-function along the critical line