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Properties

Label 8-832e4-1.1-c1e4-0-17
Degree 88
Conductor 479174066176479174066176
Sign 11
Analytic cond. 1948.051948.05
Root an. cond. 2.577502.57750
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  + 6·9-s − 6·13-s − 2·17-s + 6·25-s − 10·29-s − 6·37-s + 30·41-s − 14·49-s + 28·53-s + 10·61-s + 9·81-s + 2·101-s + 14·113-s − 36·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + 163-s + 167-s + 13·169-s + 173-s + ⋯
L(s)  = 1  + 2·9-s − 1.66·13-s − 0.485·17-s + 6/5·25-s − 1.85·29-s − 0.986·37-s + 4.68·41-s − 2·49-s + 3.84·53-s + 1.28·61-s + 81-s + 0.199·101-s + 1.31·113-s − 3.32·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 169-s + 0.0760·173-s + ⋯

Functional equation

Λ(s)=((224⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅1342^{24} \cdot 13^{4}
Sign: 11
Analytic conductor: 1948.051948.05
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{24} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 3.6360138483.636013848
L(12)L(\frac12) ≈\approx 3.6360138483.636013848
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
13C22C_2^2 1+6T+23T2+6pT3+p2T4 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4}
good3C2C_2 (1−pT+pT2)2(1+pT+pT2)2 ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2}
5C22C_2^2×\timesC22C_2^2 (1−2T−T2−2pT3+p2T4)(1+2T−T2+2pT3+p2T4) ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} )
7C22C_2^2 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
11C22C_2^2 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
17C2C_2×\timesC22C_2^2 (1+2T+pT2)2(1−2T−13T2−2pT3+p2T4) ( 1 + 2 T + p T^{2} )^{2}( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )
19C22C_2^2 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
23C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
29C2C_2×\timesC22C_2^2 (1+10T+pT2)2(1−10T+71T2−10pT3+p2T4) ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} )
31C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
37C2C_2×\timesC22C_2^2 (1+2T+pT2)2(1+2T−33T2+2pT3+p2T4) ( 1 + 2 T + p T^{2} )^{2}( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )
41C2C_2×\timesC22C_2^2 (1−10T+pT2)2(1−10T+59T2−10pT3+p2T4) ( 1 - 10 T + p T^{2} )^{2}( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} )
43C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
47C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
53C22C_2^2 (1−14T+143T2−14pT3+p2T4)2 ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}
59C22C_2^2 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
61C2C_2×\timesC22C_2^2 (1−10T+pT2)2(1+10T+39T2+10pT3+p2T4) ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )
67C22C_2^2 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
71C22C_2^2 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
73C22C_2^2×\timesC22C_2^2 (1−6T−37T2−6pT3+p2T4)(1+6T−37T2+6pT3+p2T4) ( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} )
79C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
83C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
89C22C_2^2×\timesC22C_2^2 (1−10T+11T2−10pT3+p2T4)(1+10T+11T2+10pT3+p2T4) ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} )
97C22C_2^2×\timesC22C_2^2 (1−18T+227T2−18pT3+p2T4)(1+18T+227T2+18pT3+p2T4) ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{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

−7.35980518253909685575193985582, −7.11942912006529193396149136100, −7.03220731796157655742113515926, −6.80320491925983670412259406086, −6.44367094403460190356525907121, −6.23153950091752575337230252525, −5.93545879801345982654360249689, −5.62098404848355457271241564505, −5.52683977412504862741298081037, −5.16317656016863442312993040933, −4.93609722390651832568221771329, −4.73809689660409866311526501350, −4.40555019126567613805182183371, −4.12669727066899849429281092934, −4.11707234918729206463761626035, −3.74094497163267420718224581248, −3.56469633631798894588431908035, −2.90842341105189175552671471525, −2.73975812479733410530884682120, −2.53224780402548652811020075225, −2.03640995641928478005437404317, −1.88963296536091567284510408188, −1.49618214413520375105363943287, −0.78531418781299928607607743514, −0.63023673440877619986476727710, 0.63023673440877619986476727710, 0.78531418781299928607607743514, 1.49618214413520375105363943287, 1.88963296536091567284510408188, 2.03640995641928478005437404317, 2.53224780402548652811020075225, 2.73975812479733410530884682120, 2.90842341105189175552671471525, 3.56469633631798894588431908035, 3.74094497163267420718224581248, 4.11707234918729206463761626035, 4.12669727066899849429281092934, 4.40555019126567613805182183371, 4.73809689660409866311526501350, 4.93609722390651832568221771329, 5.16317656016863442312993040933, 5.52683977412504862741298081037, 5.62098404848355457271241564505, 5.93545879801345982654360249689, 6.23153950091752575337230252525, 6.44367094403460190356525907121, 6.80320491925983670412259406086, 7.03220731796157655742113515926, 7.11942912006529193396149136100, 7.35980518253909685575193985582

Graph of the ZZ-function along the critical line