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Properties

Label 8-507e4-1.1-c1e4-0-4
Degree 88
Conductor 6607418840166074188401
Sign 11
Analytic cond. 268.621268.621
Root an. cond. 2.012062.01206
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4-s + 9-s + 2·12-s + 4·16-s − 12·17-s − 20·25-s − 2·27-s − 12·29-s + 36-s − 8·43-s + 8·48-s + 2·49-s − 24·51-s + 24·53-s + 4·61-s + 11·64-s − 12·68-s − 40·75-s − 32·79-s − 4·81-s − 24·87-s − 20·100-s + 12·101-s − 32·103-s − 24·107-s − 2·108-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s + 1/3·9-s + 0.577·12-s + 16-s − 2.91·17-s − 4·25-s − 0.384·27-s − 2.22·29-s + 1/6·36-s − 1.21·43-s + 1.15·48-s + 2/7·49-s − 3.36·51-s + 3.29·53-s + 0.512·61-s + 11/8·64-s − 1.45·68-s − 4.61·75-s − 3.60·79-s − 4/9·81-s − 2.57·87-s − 2·100-s + 1.19·101-s − 3.15·103-s − 2.32·107-s − 0.192·108-s + ⋯

Functional equation

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

Invariants

Degree: 88
Conductor: 34⋅1383^{4} \cdot 13^{8}
Sign: 11
Analytic conductor: 268.621268.621
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅138, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.98389017230.9838901723
L(12)L(\frac12) ≈\approx 0.98389017230.9838901723
L(32)L(\frac{3}{2}) 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)
bad3C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
13 1 1
good2C23C_2^3 1−T2−3T4−p2T6+p4T8 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8}
5C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
7C22C_2^2×\timesC22C_2^2 (1−13T2+p2T4)(1+11T2+p2T4) ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} )
11C23C_2^3 1−10T2−21T4−10p2T6+p4T8 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8}
17C22C_2^2 (1+6T+19T2+6pT3+p2T4)2 ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
19C22C_2^2×\timesC22C_2^2 (1−37T2+p2T4)(1+11T2+p2T4) ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} )
23C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
29C22C_2^2 (1+6T+7T2+6pT3+p2T4)2 ( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
31C22C_2^2 (1+50T2+p2T4)2 ( 1 + 50 T^{2} + p^{2} T^{4} )^{2}
37C22C_2^2×\timesC22C_2^2 (1−73T2+p2T4)(1+47T2+p2T4) ( 1 - 73 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} )
41C23C_2^3 1−34T2−525T4−34p2T6+p4T8 1 - 34 T^{2} - 525 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8}
43C22C_2^2 (1+4T−27T2+4pT3+p2T4)2 ( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
47C22C_2^2 (1+82T2+p2T4)2 ( 1 + 82 T^{2} + p^{2} T^{4} )^{2}
53C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
59C23C_2^3 1−10T2−3381T4−10p2T6+p4T8 1 - 10 T^{2} - 3381 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8}
61C22C_2^2 (1−2T−57T2−2pT3+p2T4)2 ( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
67C23C_2^3 1−26T2−3813T4−26p2T6+p4T8 1 - 26 T^{2} - 3813 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8}
71C23C_2^3 1−130T2+11859T4−130p2T6+p4T8 1 - 130 T^{2} + 11859 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8}
73C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
79C2C_2 (1+8T+pT2)4 ( 1 + 8 T + p T^{2} )^{4}
83C22C_2^2 (1+154T2+p2T4)2 ( 1 + 154 T^{2} + p^{2} T^{4} )^{2}
89C23C_2^3 1−130T2+8979T4−130p2T6+p4T8 1 - 130 T^{2} + 8979 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8}
97C22C_2^2×\timesC22C_2^2 (1−169T2+p2T4)(1+167T2+p2T4) ( 1 - 169 T^{2} + p^{2} T^{4} )( 1 + 167 T^{2} + p^{2} T^{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

−8.105864610898901813475340139241, −7.54680014348575957132171804626, −7.40846100464523643984076280573, −7.27017958399500349143551862434, −7.06214387103409971680685012792, −6.60014161382903242036073866970, −6.57274609028942619329965000870, −6.21434428146745364696109689731, −5.78741246628291012785137877972, −5.71132870315572181055302443724, −5.47761857973137237889707746894, −5.30381557664612946007544395456, −4.85090603086031049925575186469, −4.33232017353456202698221065040, −4.02311220813120766645505299471, −3.96586037906396517021802940317, −3.88752404187981442698898770881, −3.52649651814239581667528243385, −2.95834841522328463941104787985, −2.58674421072034485304408973566, −2.47213258723872095346986646651, −2.00196344104988349130777804531, −1.82202202495074669595275408454, −1.51288112759546948772989045946, −0.26348764836235493063783754692, 0.26348764836235493063783754692, 1.51288112759546948772989045946, 1.82202202495074669595275408454, 2.00196344104988349130777804531, 2.47213258723872095346986646651, 2.58674421072034485304408973566, 2.95834841522328463941104787985, 3.52649651814239581667528243385, 3.88752404187981442698898770881, 3.96586037906396517021802940317, 4.02311220813120766645505299471, 4.33232017353456202698221065040, 4.85090603086031049925575186469, 5.30381557664612946007544395456, 5.47761857973137237889707746894, 5.71132870315572181055302443724, 5.78741246628291012785137877972, 6.21434428146745364696109689731, 6.57274609028942619329965000870, 6.60014161382903242036073866970, 7.06214387103409971680685012792, 7.27017958399500349143551862434, 7.40846100464523643984076280573, 7.54680014348575957132171804626, 8.105864610898901813475340139241

Graph of the ZZ-function along the critical line