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Properties

Label 8-2160e4-1.1-c1e4-0-16
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 88495.988495.9
Root an. cond. 4.153034.15303
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  − 12·11-s + 4·13-s + 24·23-s − 2·25-s + 8·37-s − 24·47-s + 4·49-s − 12·59-s − 4·61-s − 12·71-s + 4·73-s − 32·97-s + 48·107-s + 4·109-s + 52·121-s + 127-s + 131-s + 137-s + 139-s − 48·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + ⋯
L(s)  = 1  − 3.61·11-s + 1.10·13-s + 5.00·23-s − 2/5·25-s + 1.31·37-s − 3.50·47-s + 4/7·49-s − 1.56·59-s − 0.512·61-s − 1.42·71-s + 0.468·73-s − 3.24·97-s + 4.64·107-s + 0.383·109-s + 4.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.01·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅312⋅542^{16} \cdot 3^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 88495.988495.9
Root analytic conductor: 4.153034.15303
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.8435146212.843514621
L(12)L(\frac12) ≈\approx 2.8435146212.843514621
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)
bad2 1 1
3 1 1
5C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
good7D4×C2D_4\times C_2 1−4T2−6T4−4p2T6+p4T8 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}
11D4D_{4} (1+6T+28T2+6pT3+p2T4)2 ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
13D4D_{4} (1−2T−2pT3+p2T4)2 ( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} )^{2}
17C22C_2^2 (1−7T2+p2T4)2 ( 1 - 7 T^{2} + p^{2} T^{4} )^{2}
19D4×C2D_4\times C_2 1−34T2+579T4−34p2T6+p4T8 1 - 34 T^{2} + 579 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8}
23D4D_{4} (1−12T+79T2−12pT3+p2T4)2 ( 1 - 12 T + 79 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
29D4×C2D_4\times C_2 1−44T2+1194T4−44p2T6+p4T8 1 - 44 T^{2} + 1194 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8}
31D4×C2D_4\times C_2 1−82T2+3171T4−82p2T6+p4T8 1 - 82 T^{2} + 3171 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8}
37C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
41D4×C2D_4\times C_2 1−92T2+4506T4−92p2T6+p4T8 1 - 92 T^{2} + 4506 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8}
43D4×C2D_4\times C_2 1−4T2+1002T4−4p2T6+p4T8 1 - 4 T^{2} + 1002 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}
47D4D_{4} (1+12T+118T2+12pT3+p2T4)2 ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
53D4×C2D_4\times C_2 1−86T2+3579T4−86p2T6+p4T8 1 - 86 T^{2} + 3579 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1+6T+52T2+6pT3+p2T4)2 ( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
61C2C_2 (1+T+pT2)4 ( 1 + T + p T^{2} )^{4}
67C22C_2^2 (1−26T2+p2T4)2 ( 1 - 26 T^{2} + p^{2} T^{4} )^{2}
71D4D_{4} (1+6T+148T2+6pT3+p2T4)2 ( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
73D4D_{4} (1−2T+120T2−2pT3+p2T4)2 ( 1 - 2 T + 120 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
79D4×C2D_4\times C_2 1−202T2+20955T4−202p2T6+p4T8 1 - 202 T^{2} + 20955 T^{4} - 202 p^{2} T^{6} + p^{4} T^{8}
83C22C_2^2 (1+139T2+p2T4)2 ( 1 + 139 T^{2} + p^{2} T^{4} )^{2}
89D4×C2D_4\times C_2 1−140T2+11994T4−140p2T6+p4T8 1 - 140 T^{2} + 11994 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}
97C2C_2 (1+8T+pT2)4 ( 1 + 8 T + p 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

−6.54801771529231945686448002656, −6.21719957552703635874923542516, −5.88089786954278546467640145991, −5.87922755984119348496344402269, −5.53847089177282888754189162777, −5.37626723346144320415770465442, −5.13870464228729663608012801838, −4.97232893074457751734717892370, −4.72271820736228343799453897253, −4.69909309616188821483787809321, −4.58802398679959866710061836122, −4.07932918198831638818367437533, −3.84678764616181897334961248691, −3.28590817834723512813050154802, −3.24554581397598036876368584785, −3.18986345766012403224163924120, −2.83995562186468823654378980564, −2.67201338497142055084619081595, −2.62688782480522439807356017849, −2.03031818208954932468058188193, −1.80484765349897089130126594125, −1.31398808466998804631050492899, −1.23032000481983733540545183292, −0.52576368104977398623015708558, −0.43631070961811189632071913433, 0.43631070961811189632071913433, 0.52576368104977398623015708558, 1.23032000481983733540545183292, 1.31398808466998804631050492899, 1.80484765349897089130126594125, 2.03031818208954932468058188193, 2.62688782480522439807356017849, 2.67201338497142055084619081595, 2.83995562186468823654378980564, 3.18986345766012403224163924120, 3.24554581397598036876368584785, 3.28590817834723512813050154802, 3.84678764616181897334961248691, 4.07932918198831638818367437533, 4.58802398679959866710061836122, 4.69909309616188821483787809321, 4.72271820736228343799453897253, 4.97232893074457751734717892370, 5.13870464228729663608012801838, 5.37626723346144320415770465442, 5.53847089177282888754189162777, 5.87922755984119348496344402269, 5.88089786954278546467640145991, 6.21719957552703635874923542516, 6.54801771529231945686448002656

Graph of the ZZ-function along the critical line