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Properties

Label 8-273e4-1.1-c0e4-0-3
Degree 88
Conductor 55545718415554571841
Sign 11
Analytic cond. 0.0003445710.000344571
Root an. cond. 0.3691130.369113
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 − 2·7-s + 9-s − 4·13-s + 16-s − 25-s − 4·28-s + 2·31-s + 2·36-s + 4·37-s − 2·43-s + 49-s − 8·52-s − 2·63-s − 2·64-s − 2·73-s − 2·79-s + 8·91-s + 2·97-s − 2·100-s − 2·103-s − 2·109-s − 2·112-s − 4·117-s − 2·121-s + 4·124-s + 127-s + ⋯
L(s)  = 1  + 2·4-s − 2·7-s + 9-s − 4·13-s + 16-s − 25-s − 4·28-s + 2·31-s + 2·36-s + 4·37-s − 2·43-s + 49-s − 8·52-s − 2·63-s − 2·64-s − 2·73-s − 2·79-s + 8·91-s + 2·97-s − 2·100-s − 2·103-s − 2·109-s − 2·112-s − 4·117-s − 2·121-s + 4·124-s + 127-s + ⋯

Functional equation

Λ(s)=((34⋅74⋅134)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((34⋅74⋅134)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅74⋅1343^{4} \cdot 7^{4} \cdot 13^{4}
Sign: 11
Analytic conductor: 0.0003445710.000344571
Root analytic conductor: 0.3691130.369113
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅74⋅134, ( :0,0,0,0), 1)(8,\ 3^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.38211552790.3821155279
L(12)L(\frac12) ≈\approx 0.38211552790.3821155279
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}
7C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
13C1C_1 (1+T)4 ( 1 + T )^{4}
good2C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
5C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
11C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
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}
23C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
29C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
31C1C_1×\timesC2C_2 (1−T)4(1+T+T2)2 ( 1 - T )^{4}( 1 + T + T^{2} )^{2}
37C2C_2 (1−T+T2)4 ( 1 - T + T^{2} )^{4}
41C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
43C1C_1×\timesC2C_2 (1+T)4(1−T+T2)2 ( 1 + T )^{4}( 1 - T + T^{2} )^{2}
47C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
53C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
59C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{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}
71C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
73C1C_1×\timesC2C_2 (1+T)4(1−T+T2)2 ( 1 + T )^{4}( 1 - T + T^{2} )^{2}
79C1C_1×\timesC2C_2 (1+T)4(1−T+T2)2 ( 1 + T )^{4}( 1 - T + T^{2} )^{2}
83C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
89C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
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

−9.261984649232099377938399527062, −8.657188688802703417166928955622, −8.231170295604727435905978789256, −8.007607814910001464625463047681, −7.71822027925855096827287524729, −7.51828293431547257653370785905, −7.22864699467596249750084034109, −7.16237759234306319121707143661, −6.79630155492941171133576287977, −6.72194297379546005470977822708, −6.29154362102306211253296752947, −6.11384775433892806220920232724, −6.08316196652020099419979268741, −5.48172561341015488566674130080, −5.07790150804553031667076782138, −4.79555038552527192406901869376, −4.49079745162150097735768487124, −4.21133916261879778572438295852, −3.99770012676609839692866788023, −3.08829971249185133363349303392, −2.92245919067083545283945155759, −2.75991142853910062421931303486, −2.52377090036080620067152435299, −2.09368771117900151686760285036, −1.56870668635183199225822516103, 1.56870668635183199225822516103, 2.09368771117900151686760285036, 2.52377090036080620067152435299, 2.75991142853910062421931303486, 2.92245919067083545283945155759, 3.08829971249185133363349303392, 3.99770012676609839692866788023, 4.21133916261879778572438295852, 4.49079745162150097735768487124, 4.79555038552527192406901869376, 5.07790150804553031667076782138, 5.48172561341015488566674130080, 6.08316196652020099419979268741, 6.11384775433892806220920232724, 6.29154362102306211253296752947, 6.72194297379546005470977822708, 6.79630155492941171133576287977, 7.16237759234306319121707143661, 7.22864699467596249750084034109, 7.51828293431547257653370785905, 7.71822027925855096827287524729, 8.007607814910001464625463047681, 8.231170295604727435905978789256, 8.657188688802703417166928955622, 9.261984649232099377938399527062

Graph of the ZZ-function along the critical line