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Properties

Label 8-24e8-1.1-c0e4-0-1
Degree 88
Conductor 110075314176110075314176
Sign 11
Analytic cond. 0.006828390.00682839
Root an. cond. 0.5361540.536154
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  + 9-s + 2·25-s − 6·41-s + 2·49-s − 4·73-s − 2·97-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + ⋯
L(s)  = 1  + 9-s + 2·25-s − 6·41-s + 2·49-s − 4·73-s − 2·97-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + ⋯

Functional equation

Λ(s)=((224⋅38)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((224⋅38)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅382^{24} \cdot 3^{8}
Sign: 11
Analytic conductor: 0.006828390.00682839
Root analytic conductor: 0.5361540.536154
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅38, ( :0,0,0,0), 1)(8,\ 2^{24} \cdot 3^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.67324338690.6732433869
L(12)L(\frac12) ≈\approx 0.67324338690.6732433869
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)
bad2 1 1
3C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
good5C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
7C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
11C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
13C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
17C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{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}
29C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
31C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
37C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
41C1C_1×\timesC2C_2 (1+T)4(1+T+T2)2 ( 1 + T )^{4}( 1 + T + T^{2} )^{2}
43C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
47C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
53C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
59C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
61C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
67C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
71C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
73C2C_2 (1+T+T2)4 ( 1 + T + T^{2} )^{4}
79C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
83C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + 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−T+T2)2 ( 1 + T )^{4}( 1 - T + 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

−8.082684795086260868929177818776, −7.49225222265461180192907109696, −7.47440832375795933917676830325, −7.18165579167383673745690429316, −7.06455592311538659600152287601, −6.83401686200139843374107042774, −6.61004794729969118970921584743, −6.33286240630181358207240000623, −6.22700937088384161602121393312, −5.50040927306322818781316997100, −5.48491417883794814218081601630, −5.46310201546624131464893493877, −4.98754688349010874943566305594, −4.66303731426903007899028002723, −4.52399339859595714453183933364, −4.30307733460604796364957181372, −3.84968304718626400168524045296, −3.68277954911902791232040918981, −3.09682451146858744606590839211, −3.07759556465071361272217360998, −2.88552034487942242599068169871, −2.21398352661825444463316834287, −1.72760905627092993267903785277, −1.62958493013945976422657223905, −1.09416233549237762612237209965, 1.09416233549237762612237209965, 1.62958493013945976422657223905, 1.72760905627092993267903785277, 2.21398352661825444463316834287, 2.88552034487942242599068169871, 3.07759556465071361272217360998, 3.09682451146858744606590839211, 3.68277954911902791232040918981, 3.84968304718626400168524045296, 4.30307733460604796364957181372, 4.52399339859595714453183933364, 4.66303731426903007899028002723, 4.98754688349010874943566305594, 5.46310201546624131464893493877, 5.48491417883794814218081601630, 5.50040927306322818781316997100, 6.22700937088384161602121393312, 6.33286240630181358207240000623, 6.61004794729969118970921584743, 6.83401686200139843374107042774, 7.06455592311538659600152287601, 7.18165579167383673745690429316, 7.47440832375795933917676830325, 7.49225222265461180192907109696, 8.082684795086260868929177818776

Graph of the ZZ-function along the critical line