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Properties

Label 8-525e4-1.1-c1e4-0-19
Degree 88
Conductor 7596914062575969140625
Sign 11
Analytic cond. 308.848308.848
Root an. cond. 2.047472.04747
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  + 9-s + 12·11-s + 4·16-s + 2·19-s + 32·29-s − 2·31-s − 24·41-s + 13·49-s − 16·59-s + 28·61-s + 24·71-s − 2·79-s − 24·89-s + 12·99-s + 20·101-s − 30·109-s + 58·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1/3·9-s + 3.61·11-s + 16-s + 0.458·19-s + 5.94·29-s − 0.359·31-s − 3.74·41-s + 13/7·49-s − 2.08·59-s + 3.58·61-s + 2.84·71-s − 0.225·79-s − 2.54·89-s + 1.20·99-s + 1.99·101-s − 2.87·109-s + 5.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

Λ(s)=((34⋅58⋅74)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((34⋅58⋅74)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅58⋅743^{4} \cdot 5^{8} \cdot 7^{4}
Sign: 11
Analytic conductor: 308.848308.848
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅58⋅74, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 4.8949861554.894986155
L(12)L(\frac12) ≈\approx 4.8949861554.894986155
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)
bad3C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
5 1 1
7C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
good2C22C_2^2×\timesC22C_2^2 (1−pT+pT2−p2T3+p2T4)(1+pT+pT2+p2T3+p2T4) ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )
11C22C_2^2 (1−6T+25T2−6pT3+p2T4)2 ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
13C22C_2^2 (1−17T2+p2T4)2 ( 1 - 17 T^{2} + p^{2} T^{4} )^{2}
17C23C_2^3 1+18T2+35T4+18p2T6+p4T8 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8}
19C2C_2 (1−8T+pT2)2(1+7T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2}
23C23C_2^3 1+30T2+371T4+30p2T6+p4T8 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8}
29C2C_2 (1−8T+pT2)4 ( 1 - 8 T + p T^{2} )^{4}
31C22C_2^2 (1+T−30T2+pT3+p2T4)2 ( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2}
37C23C_2^3 1+25T2−744T4+25p2T6+p4T8 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8}
41C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
43C22C_2^2 (1−85T2+p2T4)2 ( 1 - 85 T^{2} + p^{2} T^{4} )^{2}
47C23C_2^3 1+90T2+5891T4+90p2T6+p4T8 1 + 90 T^{2} + 5891 T^{4} + 90 p^{2} T^{6} + p^{4} T^{8}
53C22C_2^2×\timesC22C_2^2 (1−14T+143T2−14pT3+p2T4)(1+14T+143T2+14pT3+p2T4) ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} )
59C22C_2^2 (1+8T+5T2+8pT3+p2T4)2 ( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
61C2C_2 (1−13T+pT2)2(1−T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2}
67C23C_2^3 1+85T2+2736T4+85p2T6+p4T8 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8}
71C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
73C23C_2^3 1+145T2+15696T4+145p2T6+p4T8 1 + 145 T^{2} + 15696 T^{4} + 145 p^{2} T^{6} + p^{4} T^{8}
79C22C_2^2 (1+T−78T2+pT3+p2T4)2 ( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2}
83C22C_2^2 (1−162T2+p2T4)2 ( 1 - 162 T^{2} + p^{2} T^{4} )^{2}
89C22C_2^2 (1+12T+55T2+12pT3+p2T4)2 ( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
97C22C_2^2 (1−158T2+p2T4)2 ( 1 - 158 T^{2} + p^{2} T^{4} )^{2}
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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

−8.102608414592475861344752028736, −7.49603697203859800003880980761, −7.38525620926607830956039300932, −6.82489351120759028108738776922, −6.61615645401728381683934064004, −6.61266813666980583828202704548, −6.58735462151392219376968625295, −6.54135486601632489052566134698, −5.91821278944175291492919135726, −5.70350134481724424894284273379, −5.18944761079774619361655332962, −5.02329368734033427578974993846, −4.98882105457079440425963807600, −4.51683677201848150417456923096, −4.05806247224254642935848743224, −3.95049740998953673640855107249, −3.91233971269841431758581248533, −3.40106121564142083238630934460, −3.13302237763329420687032780019, −2.78879427067648028942137348310, −2.45005423925989771272156664703, −1.86455872379287335964664445562, −1.33411197160609318755312097842, −1.05841596916921040601553479351, −1.00549029580644746002583426987, 1.00549029580644746002583426987, 1.05841596916921040601553479351, 1.33411197160609318755312097842, 1.86455872379287335964664445562, 2.45005423925989771272156664703, 2.78879427067648028942137348310, 3.13302237763329420687032780019, 3.40106121564142083238630934460, 3.91233971269841431758581248533, 3.95049740998953673640855107249, 4.05806247224254642935848743224, 4.51683677201848150417456923096, 4.98882105457079440425963807600, 5.02329368734033427578974993846, 5.18944761079774619361655332962, 5.70350134481724424894284273379, 5.91821278944175291492919135726, 6.54135486601632489052566134698, 6.58735462151392219376968625295, 6.61266813666980583828202704548, 6.61615645401728381683934064004, 6.82489351120759028108738776922, 7.38525620926607830956039300932, 7.49603697203859800003880980761, 8.102608414592475861344752028736

Graph of the ZZ-function along the critical line