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Properties

Label 8-2160e4-1.1-c0e4-0-5
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 1.4444296841.444429684
L(12)L(\frac12) ≈\approx 1.4444296841.444429684
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

−6.51365430204369063828084806564, −6.34033660299625330191893121681, −6.32492904311922242424183096741, −5.90596665185178497415205119033, −5.50067882348676699322112194661, −5.48244397069680638827321513553, −5.42331498553022058175997726393, −5.24465263767617297644196765222, −4.91850566462571358267776070422, −4.85402560668873980919629450803, −4.80282883277623440770059015313, −4.26955174502167684606224047712, −3.82491265488193320850354273197, −3.81055102782744502031573539565, −3.74460014154716811243422677034, −3.40287507272241670787520641834, −3.12546505449989127700786031103, −2.94775284821295146102062262321, −2.82505653744155938864322342252, −2.57825240248484960503014651720, −1.94436799820489478805099549287, −1.23837974652002235132161079893, −1.11347230893064976083318470057, −1.08810651104447672285511050984, −0.970317249283850218835519861589, 0.970317249283850218835519861589, 1.08810651104447672285511050984, 1.11347230893064976083318470057, 1.23837974652002235132161079893, 1.94436799820489478805099549287, 2.57825240248484960503014651720, 2.82505653744155938864322342252, 2.94775284821295146102062262321, 3.12546505449989127700786031103, 3.40287507272241670787520641834, 3.74460014154716811243422677034, 3.81055102782744502031573539565, 3.82491265488193320850354273197, 4.26955174502167684606224047712, 4.80282883277623440770059015313, 4.85402560668873980919629450803, 4.91850566462571358267776070422, 5.24465263767617297644196765222, 5.42331498553022058175997726393, 5.48244397069680638827321513553, 5.50067882348676699322112194661, 5.90596665185178497415205119033, 6.32492904311922242424183096741, 6.34033660299625330191893121681, 6.51365430204369063828084806564

Graph of the ZZ-function along the critical line