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Properties

Label 8-2106e4-1.1-c1e4-0-2
Degree 88
Conductor 1.967×10131.967\times 10^{13}
Sign 11
Analytic cond. 79972.779972.7
Root an. cond. 4.100794.10079
Motivic weight 11
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  − 2·2-s + 4-s + 2·5-s − 2·7-s + 2·8-s − 4·10-s − 8·11-s + 2·13-s + 4·14-s − 4·16-s + 2·20-s + 16·22-s + 2·23-s + 8·25-s − 4·26-s − 2·28-s − 16·29-s − 8·31-s + 2·32-s − 4·35-s − 24·37-s + 4·40-s − 6·41-s + 4·43-s − 8·44-s − 4·46-s − 8·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 0.707·8-s − 1.26·10-s − 2.41·11-s + 0.554·13-s + 1.06·14-s − 16-s + 0.447·20-s + 3.41·22-s + 0.417·23-s + 8/5·25-s − 0.784·26-s − 0.377·28-s − 2.97·29-s − 1.43·31-s + 0.353·32-s − 0.676·35-s − 3.94·37-s + 0.632·40-s − 0.937·41-s + 0.609·43-s − 1.20·44-s − 0.589·46-s − 1.16·47-s + ⋯

Functional equation

Λ(s)=((24⋅316⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((24⋅316⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 24⋅316⋅1342^{4} \cdot 3^{16} \cdot 13^{4}
Sign: 11
Analytic conductor: 79972.779972.7
Root analytic conductor: 4.100794.10079
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅316⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{4} \cdot 3^{16} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.12616883930.1261688393
L(12)L(\frac12) ≈\approx 0.12616883930.1261688393
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)
bad2C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
3 1 1
13C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
good5D4×C2D_4\times C_2 1−2T−4T2+4T3+19T4+4pT5−4p2T6−2p3T7+p4T8 1 - 2 T - 4 T^{2} + 4 T^{3} + 19 T^{4} + 4 p T^{5} - 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
7D4×C2D_4\times C_2 1+2T+T2−22T3−68T4−22pT5+p2T6+2p3T7+p4T8 1 + 2 T + T^{2} - 22 T^{3} - 68 T^{4} - 22 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
11D4×C2D_4\times C_2 1+8T+29T2+104T3+400T4+104pT5+29p2T6+8p3T7+p4T8 1 + 8 T + 29 T^{2} + 104 T^{3} + 400 T^{4} + 104 p T^{5} + 29 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
17C22C_2^2 (1−14T2+p2T4)2 ( 1 - 14 T^{2} + p^{2} T^{4} )^{2}
19C22C_2^2 (1+35T2+p2T4)2 ( 1 + 35 T^{2} + p^{2} T^{4} )^{2}
23D4×C2D_4\times C_2 1−2T−16T2+52T3−221T4+52pT5−16p2T6−2p3T7+p4T8 1 - 2 T - 16 T^{2} + 52 T^{3} - 221 T^{4} + 52 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1+16T+137T2+976T3+5896T4+976pT5+137p2T6+16p3T7+p4T8 1 + 16 T + 137 T^{2} + 976 T^{3} + 5896 T^{4} + 976 p T^{5} + 137 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
31D4×C2D_4\times C_2 1+8T−2T2+32T3+1411T4+32pT5−2p2T6+8p3T7+p4T8 1 + 8 T - 2 T^{2} + 32 T^{3} + 1411 T^{4} + 32 p T^{5} - 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
37D4D_{4} (1+12T+98T2+12pT3+p2T4)2 ( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
41D4×C2D_4\times C_2 1+6T−52T2+36T3+4587T4+36pT5−52p2T6+6p3T7+p4T8 1 + 6 T - 52 T^{2} + 36 T^{3} + 4587 T^{4} + 36 p T^{5} - 52 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
43D4×C2D_4\times C_2 1−4T−26T2+176T3−773T4+176pT5−26p2T6−4p3T7+p4T8 1 - 4 T - 26 T^{2} + 176 T^{3} - 773 T^{4} + 176 p T^{5} - 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
47D4×C2D_4\times C_2 1+8T−34T2+32T3+4387T4+32pT5−34p2T6+8p3T7+p4T8 1 + 8 T - 34 T^{2} + 32 T^{3} + 4387 T^{4} + 32 p T^{5} - 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
53C22C_2^2 (1+31T2+p2T4)2 ( 1 + 31 T^{2} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1+4T+41T2−572T3−3800T4−572pT5+41p2T6+4p3T7+p4T8 1 + 4 T + 41 T^{2} - 572 T^{3} - 3800 T^{4} - 572 p T^{5} + 41 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
61D4×C2D_4\times C_2 1−4T−83T2+92T3+5104T4+92pT5−83p2T6−4p3T7+p4T8 1 - 4 T - 83 T^{2} + 92 T^{3} + 5104 T^{4} + 92 p T^{5} - 83 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
67D4×C2D_4\times C_2 1−4T−14T2+416T3−4661T4+416pT5−14p2T6−4p3T7+p4T8 1 - 4 T - 14 T^{2} + 416 T^{3} - 4661 T^{4} + 416 p T^{5} - 14 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
71D4D_{4} (1−14T+179T2−14pT3+p2T4)2 ( 1 - 14 T + 179 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}
73D4D_{4} (1+20T+234T2+20pT3+p2T4)2 ( 1 + 20 T + 234 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2}
79D4×C2D_4\times C_2 1+22T+232T2+2068T3+19027T4+2068pT5+232p2T6+22p3T7+p4T8 1 + 22 T + 232 T^{2} + 2068 T^{3} + 19027 T^{4} + 2068 p T^{5} + 232 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8}
83D4×C2D_4\times C_2 1+12T+17T2−468T3−2712T4−468pT5+17p2T6+12p3T7+p4T8 1 + 12 T + 17 T^{2} - 468 T^{3} - 2712 T^{4} - 468 p T^{5} + 17 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
89D4D_{4} (1+18T+232T2+18pT3+p2T4)2 ( 1 + 18 T + 232 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1+22T+172T2+2596T3+40987T4+2596pT5+172p2T6+22p3T7+p4T8 1 + 22 T + 172 T^{2} + 2596 T^{3} + 40987 T^{4} + 2596 p T^{5} + 172 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8}
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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

−6.50343023347658297004144691260, −6.45067596669846458275625739340, −5.94739075853214764131726926404, −5.68940830979196286181444393927, −5.49933495030901630931239670510, −5.44405449424575371312589533902, −5.41890917998285990853110665142, −5.24596935984901056438058120416, −4.77445412032991310138170276528, −4.70079091388146782023133003278, −4.31383190499627625342435877829, −4.02794798507960119317426867884, −3.76119310597636983660811651881, −3.53610977633730726269567666634, −3.41479546194235440042794125489, −2.98283677983013092562436453214, −2.80516177645499214022127405530, −2.52979534827778185615574596786, −2.44433801294175895821029272154, −1.76899579406743106525331365008, −1.61189512844226034472100702988, −1.55348797070545485839239480369, −1.29793889288221724067249575558, −0.24224900280989825176447733712, −0.23210311593268611110266612020, 0.23210311593268611110266612020, 0.24224900280989825176447733712, 1.29793889288221724067249575558, 1.55348797070545485839239480369, 1.61189512844226034472100702988, 1.76899579406743106525331365008, 2.44433801294175895821029272154, 2.52979534827778185615574596786, 2.80516177645499214022127405530, 2.98283677983013092562436453214, 3.41479546194235440042794125489, 3.53610977633730726269567666634, 3.76119310597636983660811651881, 4.02794798507960119317426867884, 4.31383190499627625342435877829, 4.70079091388146782023133003278, 4.77445412032991310138170276528, 5.24596935984901056438058120416, 5.41890917998285990853110665142, 5.44405449424575371312589533902, 5.49933495030901630931239670510, 5.68940830979196286181444393927, 5.94739075853214764131726926404, 6.45067596669846458275625739340, 6.50343023347658297004144691260

Graph of the ZZ-function along the critical line