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Properties

Label 8-3675e4-1.1-c0e4-0-1
Degree 88
Conductor 1.824×10141.824\times 10^{14}
Sign 11
Analytic cond. 11.315011.3150
Root an. cond. 1.354271.35427
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  − 2·4-s + 9-s + 16-s − 2·36-s + 2·64-s − 4·79-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2·4-s + 9-s + 16-s − 2·36-s + 2·64-s − 4·79-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

Λ(s)=((34⋅58⋅78)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((34⋅58⋅78)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅58⋅783^{4} \cdot 5^{8} \cdot 7^{8}
Sign: 11
Analytic conductor: 11.315011.3150
Root analytic conductor: 1.354271.35427
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅58⋅78, ( :0,0,0,0), 1)(8,\ 3^{4} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.63029671710.6302967171
L(12)L(\frac12) ≈\approx 0.63029671710.6302967171
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)
bad3C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
5 1 1
7 1 1
good2C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
11C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
13C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
17C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
19C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
23C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
29C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
31C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
37C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
41C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
43C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
47C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
53C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
59C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
61C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
67C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
71C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
73C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
79C2C_2 (1+T+T2)4 ( 1 + T + T^{2} )^{4}
83C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
89C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
97C2C_2 (1+T2)4 ( 1 + T^{2} )^{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

−6.32181177316750204734320762764, −5.90451412257362294042994342512, −5.79672324053475848840061591292, −5.49494456103752256754041698531, −5.49483563171297673495130064646, −5.11463551762738628865296784253, −4.92543970175945260740955435605, −4.77580967830460677458052028817, −4.67969709517985621770026115620, −4.36988783741455222517272017635, −4.15967593893003488904474067750, −4.14509172404348767709188493748, −3.85837619821449973749196039083, −3.73752328725421659897846026453, −3.43273239751555506297278398309, −2.96176616992779338001518320308, −2.92387641823813717208499251324, −2.86706591115734972858905404922, −2.30439102216675380266862881058, −2.01247329896486790040110599672, −1.83573580210360279505733256308, −1.58854842116750095714050760257, −1.11557228854828374353590489623, −0.892901649175799724836015149968, −0.37646360222908505327163146353, 0.37646360222908505327163146353, 0.892901649175799724836015149968, 1.11557228854828374353590489623, 1.58854842116750095714050760257, 1.83573580210360279505733256308, 2.01247329896486790040110599672, 2.30439102216675380266862881058, 2.86706591115734972858905404922, 2.92387641823813717208499251324, 2.96176616992779338001518320308, 3.43273239751555506297278398309, 3.73752328725421659897846026453, 3.85837619821449973749196039083, 4.14509172404348767709188493748, 4.15967593893003488904474067750, 4.36988783741455222517272017635, 4.67969709517985621770026115620, 4.77580967830460677458052028817, 4.92543970175945260740955435605, 5.11463551762738628865296784253, 5.49483563171297673495130064646, 5.49494456103752256754041698531, 5.79672324053475848840061591292, 5.90451412257362294042994342512, 6.32181177316750204734320762764

Graph of the ZZ-function along the critical line