↑

Properties

Label 8-87e8-1.1-c1e4-0-3
Degree 88
Conductor 3.282×10153.282\times 10^{15}
Sign 11
Analytic cond. 1.33432×1071.33432\times 10^{7}
Root an. cond. 7.774237.77423
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s + 7-s − 2·13-s + 40·16-s − 19-s − 20·25-s − 8·28-s − 4·31-s + 11·37-s + 5·43-s + 7·49-s + 16·52-s − 13·61-s − 160·64-s − 11·67-s − 10·73-s + 8·76-s − 13·79-s − 2·91-s + 14·97-s + 160·100-s − 20·103-s − 17·109-s + 40·112-s − 44·121-s + 32·124-s + 127-s + ⋯
L(s)  = 1  − 4·4-s + 0.377·7-s − 0.554·13-s + 10·16-s − 0.229·19-s − 4·25-s − 1.51·28-s − 0.718·31-s + 1.80·37-s + 0.762·43-s + 49-s + 2.21·52-s − 1.66·61-s − 20·64-s − 1.34·67-s − 1.17·73-s + 0.917·76-s − 1.46·79-s − 0.209·91-s + 1.42·97-s + 16·100-s − 1.97·103-s − 1.62·109-s + 3.77·112-s − 4·121-s + 2.87·124-s + 0.0887·127-s + ⋯

Functional equation

Λ(s)=((38⋅298)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((38⋅298)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 38⋅2983^{8} \cdot 29^{8}
Sign: 11
Analytic conductor: 1.33432×1071.33432\times 10^{7}
Root analytic conductor: 7.774237.77423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 38⋅298, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{8} \cdot 29^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad3 1 1
29 1 1
good2C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
5C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
7C4×C2C_4\times C_2 1−T−6T2+13T3+29T4+13pT5−6p2T6−p3T7+p4T8 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 13 p T^{5} - 6 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}
11C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
13C4×C2C_4\times C_2 1+2T−9T2−44T3+29T4−44pT5−9p2T6+2p3T7+p4T8 1 + 2 T - 9 T^{2} - 44 T^{3} + 29 T^{4} - 44 p T^{5} - 9 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
17C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
19C4×C2C_4\times C_2 1+T−18T2−37T3+305T4−37pT5−18p2T6+p3T7+p4T8 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} - 37 p T^{5} - 18 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}
23C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
31C4×C2C_4\times C_2 1+4T−15T2−184T3−271T4−184pT5−15p2T6+4p3T7+p4T8 1 + 4 T - 15 T^{2} - 184 T^{3} - 271 T^{4} - 184 p T^{5} - 15 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
37C4×C2C_4\times C_2 1−11T+84T2−517T3+2579T4−517pT5+84p2T6−11p3T7+p4T8 1 - 11 T + 84 T^{2} - 517 T^{3} + 2579 T^{4} - 517 p T^{5} + 84 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8}
41C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
43C4×C2C_4\times C_2 1−5T−18T2+305T3−751T4+305pT5−18p2T6−5p3T7+p4T8 1 - 5 T - 18 T^{2} + 305 T^{3} - 751 T^{4} + 305 p T^{5} - 18 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}
47C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
53C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
59C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
61C4×C2C_4\times C_2 1+13T+108T2+611T3+1355T4+611pT5+108p2T6+13p3T7+p4T8 1 + 13 T + 108 T^{2} + 611 T^{3} + 1355 T^{4} + 611 p T^{5} + 108 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8}
67C4×C2C_4\times C_2 1+11T+54T2−143T3−5191T4−143pT5+54p2T6+11p3T7+p4T8 1 + 11 T + 54 T^{2} - 143 T^{3} - 5191 T^{4} - 143 p T^{5} + 54 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8}
71C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
73C4×C2C_4\times C_2 1+10T+27T2−460T3−6571T4−460pT5+27p2T6+10p3T7+p4T8 1 + 10 T + 27 T^{2} - 460 T^{3} - 6571 T^{4} - 460 p T^{5} + 27 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}
79C4×C2C_4\times C_2 1+13T+90T2+143T3−5251T4+143pT5+90p2T6+13p3T7+p4T8 1 + 13 T + 90 T^{2} + 143 T^{3} - 5251 T^{4} + 143 p T^{5} + 90 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8}
83C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
89C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
97C4×C2C_4\times C_2 1−14T+99T2−28T3−9211T4−28pT5+99p2T6−14p3T7+p4T8 1 - 14 T + 99 T^{2} - 28 T^{3} - 9211 T^{4} - 28 p T^{5} + 99 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}
show more
show less
   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

−5.80301203246656688854737903269, −5.67591118116113773684498184987, −5.44478910106616633733563200060, −5.34654806383379519053839353412, −5.30432906245248856915194150942, −4.77144654773534155492871763363, −4.69190740101543008274899999648, −4.58560557516145415607200296221, −4.37379820046605395706154618229, −4.22779291434716124932379545367, −4.06444897393581446474919392252, −4.02758453305681070578143329897, −3.92808481731856575411189618124, −3.44127997653737385813060965211, −3.20849347332780585268741605872, −3.19118721596011276153274623370, −3.18458874315738660889833243891, −2.47341979823624841074061692368, −2.41913107615580637458210270704, −2.10436212484739088754172804614, −1.91164743924746212893676156274, −1.39912475947805497258548745717, −1.28486526853184290613784953983, −1.00376856468987677843214071953, −0.932877276837166783780316361544, 0, 0, 0, 0, 0.932877276837166783780316361544, 1.00376856468987677843214071953, 1.28486526853184290613784953983, 1.39912475947805497258548745717, 1.91164743924746212893676156274, 2.10436212484739088754172804614, 2.41913107615580637458210270704, 2.47341979823624841074061692368, 3.18458874315738660889833243891, 3.19118721596011276153274623370, 3.20849347332780585268741605872, 3.44127997653737385813060965211, 3.92808481731856575411189618124, 4.02758453305681070578143329897, 4.06444897393581446474919392252, 4.22779291434716124932379545367, 4.37379820046605395706154618229, 4.58560557516145415607200296221, 4.69190740101543008274899999648, 4.77144654773534155492871763363, 5.30432906245248856915194150942, 5.34654806383379519053839353412, 5.44478910106616633733563200060, 5.67591118116113773684498184987, 5.80301203246656688854737903269

Graph of the ZZ-function along the critical line