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Properties

Label 8-3808e4-1.1-c0e4-0-4
Degree 88
Conductor 2.103×10142.103\times 10^{14}
Sign 11
Analytic cond. 13.044113.0441
Root an. cond. 1.378561.37856
Motivic weight 00
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  + 4·7-s − 9-s + 4·17-s − 25-s + 2·31-s + 2·41-s + 10·49-s − 4·63-s − 2·73-s + 2·97-s + 16·119-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s − 4·169-s + 173-s − 4·175-s + 179-s + 181-s + ⋯
L(s)  = 1  + 4·7-s − 9-s + 4·17-s − 25-s + 2·31-s + 2·41-s + 10·49-s − 4·63-s − 2·73-s + 2·97-s + 16·119-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s − 4·169-s + 173-s − 4·175-s + 179-s + 181-s + ⋯

Functional equation

Λ(s)=((220⋅74⋅174)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((220⋅74⋅174)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 220⋅74⋅1742^{20} \cdot 7^{4} \cdot 17^{4}
Sign: 11
Analytic conductor: 13.044113.0441
Root analytic conductor: 1.378561.37856
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅74⋅174, ( :0,0,0,0), 1)(8,\ 2^{20} \cdot 7^{4} \cdot 17^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 4.6419261224.641926122
L(12)L(\frac12) ≈\approx 4.6419261224.641926122
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)
bad2 1 1
7C1C_1 (1−T)4 ( 1 - T )^{4}
17C1C_1 (1−T)4 ( 1 - T )^{4}
good3C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
5C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
11C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
13C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
19C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
23C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
29C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
31C4C_4 (1−T+T2−T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}
37C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
41C4C_4 (1−T+T2−T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}
43C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
47C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
53C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
59C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
61C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
67C4C_4×\timesC4C_4 (1−T+T2−T3+T4)(1+T+T2+T3+T4) ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )
71C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
73C4C_4 (1+T+T2+T3+T4)2 ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}
79C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
83C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
89C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
97C4C_4 (1−T+T2−T3+T4)2 ( 1 - T + T^{2} - T^{3} + 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

−5.93513208391249524060185296544, −5.85186660711243372905135614546, −5.80113768708420282360123574107, −5.57893420695897229207622306939, −5.44680303014830919589535263719, −5.18469459265082166997242566710, −5.03216854716428668642060173521, −4.84430981138075124449622775563, −4.60164786023931492249245285627, −4.52203560917423071567810775307, −4.27887544337917475954071656420, −4.02326968539852381565238900161, −3.64511799659171553158835914152, −3.57189688292854495768567703790, −3.55185959981753435916359257989, −2.88624268658424691385516941650, −2.69590975403555762289568677486, −2.68638259072815263849897155521, −2.42015973939543157762047423402, −2.01237503475484045843156791844, −1.80917207467162596516210792736, −1.44197080686420177861433626896, −1.16836248226138173994248206264, −1.01955770609237974815582454792, −0.984455087445693683622800858141, 0.984455087445693683622800858141, 1.01955770609237974815582454792, 1.16836248226138173994248206264, 1.44197080686420177861433626896, 1.80917207467162596516210792736, 2.01237503475484045843156791844, 2.42015973939543157762047423402, 2.68638259072815263849897155521, 2.69590975403555762289568677486, 2.88624268658424691385516941650, 3.55185959981753435916359257989, 3.57189688292854495768567703790, 3.64511799659171553158835914152, 4.02326968539852381565238900161, 4.27887544337917475954071656420, 4.52203560917423071567810775307, 4.60164786023931492249245285627, 4.84430981138075124449622775563, 5.03216854716428668642060173521, 5.18469459265082166997242566710, 5.44680303014830919589535263719, 5.57893420695897229207622306939, 5.80113768708420282360123574107, 5.85186660711243372905135614546, 5.93513208391249524060185296544

Graph of the ZZ-function along the critical line