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Properties

Label 8-2160e4-1.1-c0e4-0-4
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·13-s − 16-s + 4·43-s − 4·61-s − 8·67-s + 4·79-s + 4·97-s + 4·103-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 4·208-s + ⋯
L(s)  = 1  + 4·13-s − 16-s + 4·43-s − 4·61-s − 8·67-s + 4·79-s + 4·97-s + 4·103-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 4·208-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 1.8660103951.866010395
L(12)L(\frac12) ≈\approx 1.8660103951.866010395
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)
bad2C22C_2^2 1+T4 1 + T^{4}
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}
13C2C_2 (1−T+T2)4 ( 1 - T + T^{2} )^{4}
17C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
19C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
23C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
29C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
31C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
37C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
41C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
43C2C_2 (1−T+T2)4 ( 1 - T + T^{2} )^{4}
47C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
53C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
59C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
61C1C_1×\timesC2C_2 (1+T)4(1+T2)2 ( 1 + T )^{4}( 1 + T^{2} )^{2}
67C1C_1 (1+T)8 ( 1 + T )^{8}
71C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
73C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
79C2C_2 (1−T+T2)4 ( 1 - T + T^{2} )^{4}
83C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
89C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
97C1C_1×\timesC2C_2 (1−T)4(1+T2)2 ( 1 - T )^{4}( 1 + T^{2} )^{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

−6.34395911253079863992755588639, −6.28804878744290457656649636560, −6.18977513684971683335564709606, −5.92128939355772233324332526796, −5.87122011656231905380225966440, −5.87005217889429243749792127137, −5.86014901763654521353025591785, −4.87647789558870307351524115810, −4.80200654485976270368628064404, −4.75940237151608450439752727147, −4.67505154122390280143155279133, −4.31460274816729762327262066864, −4.02217649545060334502972794743, −3.78030257639767920730206854393, −3.66466909963963894600455414189, −3.22748587578898096533824690291, −3.22721566942045914690209991482, −3.02859785345208158434072111812, −2.69936718627400046150890702690, −2.05799946024511865788737037551, −2.03708726662288201654305360948, −1.84698139170589525640033417564, −1.24981379391915894066428144000, −1.11715492894546494497931521665, −0.78519822039765553627000189864, 0.78519822039765553627000189864, 1.11715492894546494497931521665, 1.24981379391915894066428144000, 1.84698139170589525640033417564, 2.03708726662288201654305360948, 2.05799946024511865788737037551, 2.69936718627400046150890702690, 3.02859785345208158434072111812, 3.22721566942045914690209991482, 3.22748587578898096533824690291, 3.66466909963963894600455414189, 3.78030257639767920730206854393, 4.02217649545060334502972794743, 4.31460274816729762327262066864, 4.67505154122390280143155279133, 4.75940237151608450439752727147, 4.80200654485976270368628064404, 4.87647789558870307351524115810, 5.86014901763654521353025591785, 5.87005217889429243749792127137, 5.87122011656231905380225966440, 5.92128939355772233324332526796, 6.18977513684971683335564709606, 6.28804878744290457656649636560, 6.34395911253079863992755588639

Graph of the ZZ-function along the critical line