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Properties

Label 8-4080e4-1.1-c1e4-0-1
Degree 88
Conductor 2.771×10142.771\times 10^{14}
Sign 11
Analytic cond. 1.12654×1061.12654\times 10^{6}
Root an. cond. 5.707795.70779
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  + 4·5-s − 2·9-s + 8·19-s + 8·25-s + 8·29-s + 16·31-s − 24·41-s − 8·45-s + 16·49-s − 16·59-s + 24·61-s − 16·79-s + 3·81-s − 32·89-s + 32·95-s − 24·101-s − 24·109-s + 4·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + 64·155-s + ⋯
L(s)  = 1  + 1.78·5-s − 2/3·9-s + 1.83·19-s + 8/5·25-s + 1.48·29-s + 2.87·31-s − 3.74·41-s − 1.19·45-s + 16/7·49-s − 2.08·59-s + 3.07·61-s − 1.80·79-s + 1/3·81-s − 3.39·89-s + 3.28·95-s − 2.38·101-s − 2.29·109-s + 4/11·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 5.14·155-s + ⋯

Functional equation

Λ(s)=((216⋅34⋅54⋅174)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅34⋅54⋅174)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅34⋅54⋅1742^{16} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4}
Sign: 11
Analytic conductor: 1.12654×1061.12654\times 10^{6}
Root analytic conductor: 5.707795.70779
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅34⋅54⋅174, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.17183611090.1718361109
L(12)L(\frac12) ≈\approx 0.17183611090.1718361109
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
3C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
5C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
17C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
good7C22C_2^2 (1−8T2+p2T4)2 ( 1 - 8 T^{2} + p^{2} T^{4} )^{2}
11C22C_2^2 (1−2T2+p2T4)2 ( 1 - 2 T^{2} + p^{2} T^{4} )^{2}
13C2C_2 (1−4T+pT2)2(1+4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}
19D4D_{4} (1−4T+18T2−4pT3+p2T4)2 ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
23D4×C2D_4\times C_2 1−48T2+1250T4−48p2T6+p4T8 1 - 48 T^{2} + 1250 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8}
29C22C_2^2 (1−4T+8T2−4pT3+p2T4)2 ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
31D4D_{4} (1−8T+72T2−8pT3+p2T4)2 ( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1−128T2+6738T4−128p2T6+p4T8 1 - 128 T^{2} + 6738 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8}
41D4D_{4} (1+12T+94T2+12pT3+p2T4)2 ( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1−116T2+6678T4−116p2T6+p4T8 1 - 116 T^{2} + 6678 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8}
47C22C_2^2 (1−70T2+p2T4)2 ( 1 - 70 T^{2} + p^{2} T^{4} )^{2}
53D4×C2D_4\times C_2 1−92T2+4278T4−92p2T6+p4T8 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1+8T+38T2+8pT3+p2T4)2 ( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
61D4D_{4} (1−12T+104T2−12pT3+p2T4)2 ( 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
67C22C_2^2 (1−118T2+p2T4)2 ( 1 - 118 T^{2} + p^{2} T^{4} )^{2}
71C22C_2^2 (1+136T2+p2T4)2 ( 1 + 136 T^{2} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1−44T2+1542T4−44p2T6+p4T8 1 - 44 T^{2} + 1542 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8}
79D4D_{4} (1+8T+168T2+8pT3+p2T4)2 ( 1 + 8 T + 168 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1−276T2+32438T4−276p2T6+p4T8 1 - 276 T^{2} + 32438 T^{4} - 276 p^{2} T^{6} + p^{4} T^{8}
89D4D_{4} (1+16T+146T2+16pT3+p2T4)2 ( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−124T2+8838T4−124p2T6+p4T8 1 - 124 T^{2} + 8838 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8}
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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

−5.71615033917889365051271680355, −5.69981809756008738847383342237, −5.66168493836214635226380192685, −5.37436247625350306179736631687, −5.13937244784000361665436975107, −5.13561255662015351067837857613, −4.99450179456941052393743825975, −4.42298613922671301505983586913, −4.39163963409752062781800915277, −4.36172922595048116706620417208, −3.92617311660089698721507968146, −3.74444109631654097981509059312, −3.25779132716824230853886671808, −3.25429604018941805484311855901, −3.06531669101422666275542285758, −2.69200624115798459327207886097, −2.61269823971012573913601631014, −2.49643948346853361644558694936, −2.16291168133642042922214327702, −1.85245462365445737482962103645, −1.41248449617268462384734977436, −1.29585817545417155703813123686, −1.05082387167400933654185443094, −0.928726437205907433432490542839, −0.04578628185703732185850633570, 0.04578628185703732185850633570, 0.928726437205907433432490542839, 1.05082387167400933654185443094, 1.29585817545417155703813123686, 1.41248449617268462384734977436, 1.85245462365445737482962103645, 2.16291168133642042922214327702, 2.49643948346853361644558694936, 2.61269823971012573913601631014, 2.69200624115798459327207886097, 3.06531669101422666275542285758, 3.25429604018941805484311855901, 3.25779132716824230853886671808, 3.74444109631654097981509059312, 3.92617311660089698721507968146, 4.36172922595048116706620417208, 4.39163963409752062781800915277, 4.42298613922671301505983586913, 4.99450179456941052393743825975, 5.13561255662015351067837857613, 5.13937244784000361665436975107, 5.37436247625350306179736631687, 5.66168493836214635226380192685, 5.69981809756008738847383342237, 5.71615033917889365051271680355

Graph of the ZZ-function along the critical line