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Properties

Label 8-3549e4-1.1-c0e4-0-0
Degree 88
Conductor 1.586×10141.586\times 10^{14}
Sign 11
Analytic cond. 9.841309.84130
Root an. cond. 1.330851.33085
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-s − 2·7-s + 9-s + 16-s − 25-s + 2·28-s + 2·31-s − 36-s − 2·37-s + 4·43-s + 49-s − 2·63-s − 2·64-s − 2·73-s − 2·79-s − 4·97-s + 100-s − 2·103-s − 2·109-s − 2·112-s − 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + ⋯
L(s)  = 1  − 4-s − 2·7-s + 9-s + 16-s − 25-s + 2·28-s + 2·31-s − 36-s − 2·37-s + 4·43-s + 49-s − 2·63-s − 2·64-s − 2·73-s − 2·79-s − 4·97-s + 100-s − 2·103-s − 2·109-s − 2·112-s − 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 144-s + ⋯

Functional equation

Λ(s)=((34⋅74⋅138)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((34⋅74⋅138)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅74⋅1383^{4} \cdot 7^{4} \cdot 13^{8}
Sign: 11
Analytic conductor: 9.841309.84130
Root analytic conductor: 1.330851.33085
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅74⋅138, ( :0,0,0,0), 1)(8,\ 3^{4} \cdot 7^{4} \cdot 13^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.011658432430.01165843243
L(12)L(\frac12) ≈\approx 0.011658432430.01165843243
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}
13 1 1
good2C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
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}
17C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
19C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
23C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
29C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
31C1C_1×\timesC2C_2 (1−T)4(1+T+T2)2 ( 1 - T )^{4}( 1 + T + T^{2} )^{2}
37C1C_1×\timesC2C_2 (1+T)4(1−T+T2)2 ( 1 + T )^{4}( 1 - T + T^{2} )^{2}
41C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
43C2C_2 (1−T+T2)4 ( 1 - T + T^{2} )^{4}
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} )
59C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
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}
71C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
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}
89C2C_2×\timesC22C_2^2 (1+T2)2(1−T2+T4) ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )
97C2C_2 (1+T+T2)4 ( 1 + T + 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

−6.17991199966726997296602831919, −6.11307998530478322444070964505, −5.87209672797094643549803054868, −5.58207636277526877694270090181, −5.56522204954389770241310280453, −5.23863685721009653849259235095, −5.16822631286312359879695882913, −4.67708045589389552014972631975, −4.64067466894784541431617508837, −4.40418095837583258812050349012, −4.08736645078950007892335390638, −3.98956672634286970689886381270, −3.86820654931012297272886371918, −3.73077619907310873323034468829, −3.40983729907087796069023361136, −3.13063636244558393235943834579, −2.72117364702047314515493220984, −2.62907215917153676817882548423, −2.57439242690243099995285761355, −2.46360901423523538128158775844, −1.58650792027432124703707891705, −1.40031750657258055488451401178, −1.33183346373861693426891108344, −1.04276345699506312281409838878, −0.04382197864941612171331580702, 0.04382197864941612171331580702, 1.04276345699506312281409838878, 1.33183346373861693426891108344, 1.40031750657258055488451401178, 1.58650792027432124703707891705, 2.46360901423523538128158775844, 2.57439242690243099995285761355, 2.62907215917153676817882548423, 2.72117364702047314515493220984, 3.13063636244558393235943834579, 3.40983729907087796069023361136, 3.73077619907310873323034468829, 3.86820654931012297272886371918, 3.98956672634286970689886381270, 4.08736645078950007892335390638, 4.40418095837583258812050349012, 4.64067466894784541431617508837, 4.67708045589389552014972631975, 5.16822631286312359879695882913, 5.23863685721009653849259235095, 5.56522204954389770241310280453, 5.58207636277526877694270090181, 5.87209672797094643549803054868, 6.11307998530478322444070964505, 6.17991199966726997296602831919

Graph of the ZZ-function along the critical line