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Properties

Label 8-2160e4-1.1-c1e4-0-11
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 4.1965817964.196581796
L(12)L(\frac12) ≈\approx 4.1965817964.196581796
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

−6.58419375687555026679701157218, −6.08848902854573756044185132745, −6.06680025212631090811517403248, −5.82657391349391843484439766218, −5.69630346055068226369794198039, −5.67740047924299222287591487497, −5.36834288999675612483169787054, −4.90332606115584730836988250686, −4.59701210179266857549394732317, −4.20655347708816041475239438060, −4.17986634769664571409090512376, −4.00356594034588411006981372264, −3.98952132474515535591517576670, −3.79616696703649373667506068245, −3.58413243263072182514323428625, −3.33149808263759768275105894124, −2.77039976744070424873159568624, −2.38417746702589173654462851348, −2.32550684349787033502632185703, −2.19874616451576287335400237850, −1.48999441325242354282972106993, −1.48164652490249028041693153842, −1.30780022258140895828684577990, −0.813451255177655805913773751107, −0.35413234837232041830278530207, 0.35413234837232041830278530207, 0.813451255177655805913773751107, 1.30780022258140895828684577990, 1.48164652490249028041693153842, 1.48999441325242354282972106993, 2.19874616451576287335400237850, 2.32550684349787033502632185703, 2.38417746702589173654462851348, 2.77039976744070424873159568624, 3.33149808263759768275105894124, 3.58413243263072182514323428625, 3.79616696703649373667506068245, 3.98952132474515535591517576670, 4.00356594034588411006981372264, 4.17986634769664571409090512376, 4.20655347708816041475239438060, 4.59701210179266857549394732317, 4.90332606115584730836988250686, 5.36834288999675612483169787054, 5.67740047924299222287591487497, 5.69630346055068226369794198039, 5.82657391349391843484439766218, 6.06680025212631090811517403248, 6.08848902854573756044185132745, 6.58419375687555026679701157218

Graph of the ZZ-function along the critical line