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Properties

Label 8-2160e4-1.1-c0e4-0-3
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 1.350341.35034
Root an. cond. 1.038251.03825
Motivic weight 00
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  − 2·4-s + 3·16-s − 4·17-s + 4·19-s + 4·31-s + 4·47-s − 4·53-s − 4·64-s + 8·68-s − 8·76-s + 4·79-s − 4·109-s + 4·113-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 8·188-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s − 4·17-s + 4·19-s + 4·31-s + 4·47-s − 4·53-s − 4·64-s + 8·68-s − 8·76-s + 4·79-s − 4·109-s + 4·113-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 8·188-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅312⋅542^{16} \cdot 3^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 1.350341.35034
Root analytic conductor: 1.038251.03825
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :0,0,0,0), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.79416126960.7941612696
L(12)L(\frac12) ≈\approx 0.79416126960.7941612696
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)
bad2C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
3 1 1
5C22C_2^2 1+T4 1 + T^{4}
good7C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
11C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
13C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
17C2C_2 (1+T+T2)4 ( 1 + T + T^{2} )^{4}
19C1C_1×\timesC2C_2 (1−T)4(1+T2)2 ( 1 - T )^{4}( 1 + T^{2} )^{2}
23C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
29C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
31C2C_2 (1−T+T2)4 ( 1 - T + T^{2} )^{4}
37C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
41C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
43C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
47C2C_2 (1−T+T2)4 ( 1 - T + T^{2} )^{4}
53C1C_1×\timesC2C_2 (1+T)4(1+T2)2 ( 1 + T )^{4}( 1 + T^{2} )^{2}
59C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
61C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
67C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
71C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
73C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
79C2C_2 (1−T+T2)4 ( 1 - T + T^{2} )^{4}
83C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
89C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
97C22C_2^2 (1+T4)2 ( 1 + 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

−6.55090295747870129447713192329, −6.37868536208954240287024296718, −6.33625715295151641563856967775, −6.07356121775251998160959923331, −5.76346771413092523889889977936, −5.55978910583392285385163396250, −5.21723281758893574486414839805, −4.99390085124004374958723872221, −4.99282014844883649316989246006, −4.74793045520512264184654670387, −4.59898512648147778312873068240, −4.40886999868329333792093800180, −4.05802977341906386499302823398, −3.85749179312171973910118910423, −3.83492817778156374844949669927, −3.28956062057192296939018032324, −3.20377485212611317776624811218, −2.78365700511872524726253274281, −2.75183510207255080940771329695, −2.37350128787113147952784948725, −2.16327285279202566813070696185, −1.57335299944060337354564257844, −1.15776006786732501871274067496, −1.01710712023317427930031504358, −0.58725968488981043727340413373, 0.58725968488981043727340413373, 1.01710712023317427930031504358, 1.15776006786732501871274067496, 1.57335299944060337354564257844, 2.16327285279202566813070696185, 2.37350128787113147952784948725, 2.75183510207255080940771329695, 2.78365700511872524726253274281, 3.20377485212611317776624811218, 3.28956062057192296939018032324, 3.83492817778156374844949669927, 3.85749179312171973910118910423, 4.05802977341906386499302823398, 4.40886999868329333792093800180, 4.59898512648147778312873068240, 4.74793045520512264184654670387, 4.99282014844883649316989246006, 4.99390085124004374958723872221, 5.21723281758893574486414839805, 5.55978910583392285385163396250, 5.76346771413092523889889977936, 6.07356121775251998160959923331, 6.33625715295151641563856967775, 6.37868536208954240287024296718, 6.55090295747870129447713192329

Graph of the ZZ-function along the critical line