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Properties

Label 8-1920e4-1.1-c1e4-0-5
Degree 88
Conductor 1.359×10131.359\times 10^{13}
Sign 11
Analytic cond. 55247.555247.5
Root an. cond. 3.915513.91551
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  − 2·3-s − 4·5-s − 4·7-s + 2·9-s + 8·13-s + 8·15-s − 8·17-s − 8·19-s + 8·21-s + 2·25-s − 6·27-s + 16·35-s − 24·37-s − 16·39-s − 8·45-s − 8·49-s + 16·51-s + 16·57-s − 8·63-s − 32·65-s − 4·75-s + 11·81-s + 12·83-s + 32·85-s − 32·91-s + 32·95-s + 32·101-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s − 1.51·7-s + 2/3·9-s + 2.21·13-s + 2.06·15-s − 1.94·17-s − 1.83·19-s + 1.74·21-s + 2/5·25-s − 1.15·27-s + 2.70·35-s − 3.94·37-s − 2.56·39-s − 1.19·45-s − 8/7·49-s + 2.24·51-s + 2.11·57-s − 1.00·63-s − 3.96·65-s − 0.461·75-s + 11/9·81-s + 1.31·83-s + 3.47·85-s − 3.35·91-s + 3.28·95-s + 3.18·101-s + ⋯

Functional equation

Λ(s)=((228⋅34⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((228⋅34⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 228⋅34⋅542^{28} \cdot 3^{4} \cdot 5^{4}
Sign: 11
Analytic conductor: 55247.555247.5
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 228⋅34⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{28} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.075621801720.07562180172
L(12)L(\frac12) ≈\approx 0.075621801720.07562180172
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
3C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
5C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
good7D4D_{4} (1+2T+10T2+2pT3+p2T4)2 ( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
11C22C_2^2 (1−6T2+p2T4)2 ( 1 - 6 T^{2} + p^{2} T^{4} )^{2}
13D4D_{4} (1−4T+10T2−4pT3+p2T4)2 ( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
17D4D_{4} (1+4T+18T2+4pT3+p2T4)2 ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
19D4D_{4} (1+4T+22T2+4pT3+p2T4)2 ( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
23D4×C2D_4\times C_2 1−80T2+2638T4−80p2T6+p4T8 1 - 80 T^{2} + 2638 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8}
29C22C_2^2 (1+38T2+p2T4)2 ( 1 + 38 T^{2} + p^{2} T^{4} )^{2}
31C4×C2C_4\times C_2 1−76T2+3046T4−76p2T6+p4T8 1 - 76 T^{2} + 3046 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8}
37D4D_{4} (1+12T+90T2+12pT3+p2T4)2 ( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
41C22C_2^2 (1−66T2+p2T4)2 ( 1 - 66 T^{2} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1−64T2+3742T4−64p2T6+p4T8 1 - 64 T^{2} + 3742 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8}
47D4×C2D_4\times C_2 1−80T2+5038T4−80p2T6+p4T8 1 - 80 T^{2} + 5038 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8}
53C22C_2^2 (1−26T2+p2T4)2 ( 1 - 26 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1−38T2+p2T4)2 ( 1 - 38 T^{2} + p^{2} T^{4} )^{2}
61D4×C2D_4\times C_2 1−52T2+2998T4−52p2T6+p4T8 1 - 52 T^{2} + 2998 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8}
67D4×C2D_4\times C_2 1−128T2+8574T4−128p2T6+p4T8 1 - 128 T^{2} + 8574 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8}
71C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
73D4×C2D_4\times C_2 1−100T2+8038T4−100p2T6+p4T8 1 - 100 T^{2} + 8038 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8}
79D4×C2D_4\times C_2 1−204T2+20006T4−204p2T6+p4T8 1 - 204 T^{2} + 20006 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8}
83D4D_{4} (1−6T+50T2−6pT3+p2T4)2 ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
89C22C_2^2 (1−114T2+p2T4)2 ( 1 - 114 T^{2} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−196T2+23302T4−196p2T6+p4T8 1 - 196 T^{2} + 23302 T^{4} - 196 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

−6.52211982496569215610738779668, −6.50405224968986190626471867064, −6.37568268827427035038902799427, −5.80660121952352394802535089679, −5.80213177917306486313438171353, −5.56493272807924006166457090284, −5.43821716337003280411871969304, −4.90076460901012061773843165093, −4.75621365085638684658952130187, −4.62855873528324580411610889894, −4.37142055357000575672811179246, −4.13271124916776380689088033413, −3.77822380228752773342998721276, −3.68038995722779402458139277682, −3.53417922306261054969677734272, −3.42624119283966729587300551481, −3.19993331502472225196442097619, −2.66806962355898017103246054473, −2.33119268148321605330572802028, −2.05576500429050122653053643896, −1.65209907119912943721267385622, −1.62477496664293442295210721671, −0.956765282153551266667044222471, −0.32758658200646278590944924021, −0.14537017013971243800965563307, 0.14537017013971243800965563307, 0.32758658200646278590944924021, 0.956765282153551266667044222471, 1.62477496664293442295210721671, 1.65209907119912943721267385622, 2.05576500429050122653053643896, 2.33119268148321605330572802028, 2.66806962355898017103246054473, 3.19993331502472225196442097619, 3.42624119283966729587300551481, 3.53417922306261054969677734272, 3.68038995722779402458139277682, 3.77822380228752773342998721276, 4.13271124916776380689088033413, 4.37142055357000575672811179246, 4.62855873528324580411610889894, 4.75621365085638684658952130187, 4.90076460901012061773843165093, 5.43821716337003280411871969304, 5.56493272807924006166457090284, 5.80213177917306486313438171353, 5.80660121952352394802535089679, 6.37568268827427035038902799427, 6.50405224968986190626471867064, 6.52211982496569215610738779668

Graph of the ZZ-function along the critical line