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Properties

Label 8-1920e4-1.1-c1e4-0-3
Degree 88
Conductor 135895.450×108135895.450\times 10^{8}
Sign 11
Analytic cond. 55247.555247.5
Root an. cond. 3.915513.91551
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  + 8·5-s − 12·13-s + 20·17-s + 38·25-s − 24·29-s − 12·37-s − 32·41-s − 20·53-s − 96·65-s − 20·73-s − 81-s + 160·85-s + 12·97-s − 16·109-s + 28·113-s + 20·121-s + 136·125-s + 127-s + 131-s + 137-s + 139-s − 192·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 3.57·5-s − 3.32·13-s + 4.85·17-s + 38/5·25-s − 4.45·29-s − 1.97·37-s − 4.99·41-s − 2.74·53-s − 11.9·65-s − 2.34·73-s − 1/9·81-s + 17.3·85-s + 1.21·97-s − 1.53·109-s + 2.63·113-s + 1.81·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 15.9·145-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)=((228⋅34⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((228⋅34⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 228⋅34⋅542^{28} \cdot 3^{4} \cdot 5^{4}
Sign: 11
Analytic conductor: 55247.555247.5
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 228⋅34⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{28} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.25000376630.2500037663
L(12)L(\frac12) ≈\approx 0.25000376630.2500037663
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
3C22C_2^2 1+T4 1 + T^{4}
5C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
good7C23C_2^3 1−94T4+p4T8 1 - 94 T^{4} + p^{4} T^{8}
11C22C_2^2 (1−10T2+p2T4)2 ( 1 - 10 T^{2} + p^{2} T^{4} )^{2}
13C22C_2^2 (1+6T+18T2+6pT3+p2T4)2 ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
17C2C_2 (1−8T+pT2)2(1−2T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
23C23C_2^3 1−158T4+p4T8 1 - 158 T^{4} + p^{4} T^{8}
29C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
31C22C_2^2 (1−30T2+p2T4)2 ( 1 - 30 T^{2} + p^{2} T^{4} )^{2}
37C22C_2^2 (1+6T+18T2+6pT3+p2T4)2 ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
41C2C_2 (1+8T+pT2)4 ( 1 + 8 T + p T^{2} )^{4}
43C23C_2^3 1+1202T4+p4T8 1 + 1202 T^{4} + p^{4} T^{8}
47C23C_2^3 1−1918T4+p4T8 1 - 1918 T^{4} + p^{4} T^{8}
53C2C_2 (1−4T+pT2)2(1+14T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2}
59C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
61C2C_2 (1−12T+pT2)2(1+12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2}
67C23C_2^3 1+4946T4+p4T8 1 + 4946 T^{4} + p^{4} T^{8}
71C22C_2^2 (1−110T2+p2T4)2 ( 1 - 110 T^{2} + p^{2} T^{4} )^{2}
73C2C_2 (1−6T+pT2)2(1+16T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2}
79C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
83C23C_2^3 1−13294T4+p4T8 1 - 13294 T^{4} + p^{4} T^{8}
89C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
97C22C_2^2 (1−6T+18T2−6pT3+p2T4)2 ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + 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

−6.61568626765287622981600276759, −6.12216550651505874121068326797, −6.07278344419823668271149195784, −5.73463678624474268412850840589, −5.63946340764097627956386226692, −5.54239247078673681291059626217, −5.30047454010301453138406881305, −5.17215632767617599661915582820, −4.97271564187281920565508706509, −4.92146738121244052351036185268, −4.67515599524343816682096514028, −4.22956014479475581733902520196, −3.64171354712622111871275206164, −3.51518665897926578158381286700, −3.27647525801721334203075121597, −3.19351118537316870345001659675, −2.96028370757907681607453039673, −2.70681415432186837393293066592, −2.07575371651784362521431630525, −2.01937466697292819524265259815, −1.75926320571486989965816483907, −1.75125544801927630172107883135, −1.39330709911579664665272738313, −1.03228913426012835381701440636, −0.06610330703651128282527983553, 0.06610330703651128282527983553, 1.03228913426012835381701440636, 1.39330709911579664665272738313, 1.75125544801927630172107883135, 1.75926320571486989965816483907, 2.01937466697292819524265259815, 2.07575371651784362521431630525, 2.70681415432186837393293066592, 2.96028370757907681607453039673, 3.19351118537316870345001659675, 3.27647525801721334203075121597, 3.51518665897926578158381286700, 3.64171354712622111871275206164, 4.22956014479475581733902520196, 4.67515599524343816682096514028, 4.92146738121244052351036185268, 4.97271564187281920565508706509, 5.17215632767617599661915582820, 5.30047454010301453138406881305, 5.54239247078673681291059626217, 5.63946340764097627956386226692, 5.73463678624474268412850840589, 6.07278344419823668271149195784, 6.12216550651505874121068326797, 6.61568626765287622981600276759

Graph of the ZZ-function along the critical line