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Properties

Label 8-2160e4-1.1-c2e4-0-13
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 1.19992×1071.19992\times 10^{7}
Root an. cond. 7.671747.67174
Motivic weight 22
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  − 8·7-s + 6·11-s + 20·13-s + 76·19-s + 24·23-s + 5·25-s + 60·29-s + 4·31-s + 56·37-s + 90·41-s + 22·43-s + 204·47-s + 54·49-s − 174·59-s + 44·61-s − 14·67-s − 148·73-s − 48·77-s + 160·79-s − 312·83-s − 160·91-s + 2·97-s + 444·101-s + 172·103-s − 400·109-s − 24·113-s − 41·121-s + ⋯
L(s)  = 1  − 8/7·7-s + 6/11·11-s + 1.53·13-s + 4·19-s + 1.04·23-s + 1/5·25-s + 2.06·29-s + 4/31·31-s + 1.51·37-s + 2.19·41-s + 0.511·43-s + 4.34·47-s + 1.10·49-s − 2.94·59-s + 0.721·61-s − 0.208·67-s − 2.02·73-s − 0.623·77-s + 2.02·79-s − 3.75·83-s − 1.75·91-s + 2/97·97-s + 4.39·101-s + 1.66·103-s − 3.66·109-s − 0.212·113-s − 0.338·121-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(3−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(3-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s+1)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)^{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: 1.19992×1071.19992\times 10^{7}
Root analytic conductor: 7.671747.67174
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :1,1,1,1), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 14.8214471214.82144712
L(12)L(\frac12) ≈\approx 14.8214471214.82144712
L(2)L(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
5C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
good7D4×C2D_4\times C_2 1+8T+10T2−352T3−2621T4−352p2T5+10p4T6+8p6T7+p8T8 1 + 8 T + 10 T^{2} - 352 T^{3} - 2621 T^{4} - 352 p^{2} T^{5} + 10 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8}
11D4×C2D_4\times C_2 1−6T+7pT2−390T3−8964T4−390p2T5+7p5T6−6p6T7+p8T8 1 - 6 T + 7 p T^{2} - 390 T^{3} - 8964 T^{4} - 390 p^{2} T^{5} + 7 p^{5} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8}
13D4×C2D_4\times C_2 1−20T+22T2−800T3+46723T4−800p2T5+22p4T6−20p6T7+p8T8 1 - 20 T + 22 T^{2} - 800 T^{3} + 46723 T^{4} - 800 p^{2} T^{5} + 22 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8}
17D4×C2D_4\times C_2 1−790T2+320907T4−790p4T6+p8T8 1 - 790 T^{2} + 320907 T^{4} - 790 p^{4} T^{6} + p^{8} T^{8}
19D4D_{4} (1−2pT+1023T2−2p3T3+p4T4)2 ( 1 - 2 p T + 1023 T^{2} - 2 p^{3} T^{3} + p^{4} T^{4} )^{2}
23D4×C2D_4\times C_2 1−24T+578T2−9264T3−29277T4−9264p2T5+578p4T6−24p6T7+p8T8 1 - 24 T + 578 T^{2} - 9264 T^{3} - 29277 T^{4} - 9264 p^{2} T^{5} + 578 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8}
29D4×C2D_4\times C_2 1−60T+2462T2−75720T3+1894563T4−75720p2T5+2462p4T6−60p6T7+p8T8 1 - 60 T + 2462 T^{2} - 75720 T^{3} + 1894563 T^{4} - 75720 p^{2} T^{5} + 2462 p^{4} T^{6} - 60 p^{6} T^{7} + p^{8} T^{8}
31D4×C2D_4\times C_2 1−4T+1030T2+11744T3+89299T4+11744p2T5+1030p4T6−4p6T7+p8T8 1 - 4 T + 1030 T^{2} + 11744 T^{3} + 89299 T^{4} + 11744 p^{2} T^{5} + 1030 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8}
37C2C_2 (1−14T+p2T2)4 ( 1 - 14 T + p^{2} T^{2} )^{4}
41C22C_2^2 (1−45T+2356T2−45p2T3+p4T4)2 ( 1 - 45 T + 2356 T^{2} - 45 p^{2} T^{3} + p^{4} T^{4} )^{2}
43D4×C2D_4\times C_2 1−22T−2375T2+18458T3+3860164T4+18458p2T5−2375p4T6−22p6T7+p8T8 1 - 22 T - 2375 T^{2} + 18458 T^{3} + 3860164 T^{4} + 18458 p^{2} T^{5} - 2375 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8}
47D4×C2D_4\times C_2 1−204T+21578T2−1572024T3+85146003T4−1572024p2T5+21578p4T6−204p6T7+p8T8 1 - 204 T + 21578 T^{2} - 1572024 T^{3} + 85146003 T^{4} - 1572024 p^{2} T^{5} + 21578 p^{4} T^{6} - 204 p^{6} T^{7} + p^{8} T^{8}
53D4×C2D_4\times C_2 1−9700T2+38880102T4−9700p4T6+p8T8 1 - 9700 T^{2} + 38880102 T^{4} - 9700 p^{4} T^{6} + p^{8} T^{8}
59D4×C2D_4\times C_2 1+174T+19397T2+1619070T3+109595916T4+1619070p2T5+19397p4T6+174p6T7+p8T8 1 + 174 T + 19397 T^{2} + 1619070 T^{3} + 109595916 T^{4} + 1619070 p^{2} T^{5} + 19397 p^{4} T^{6} + 174 p^{6} T^{7} + p^{8} T^{8}
61D4×C2D_4\times C_2 1−44T−5450T2+2464T3+33503299T4+2464p2T5−5450p4T6−44p6T7+p8T8 1 - 44 T - 5450 T^{2} + 2464 T^{3} + 33503299 T^{4} + 2464 p^{2} T^{5} - 5450 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8}
67D4×C2D_4\times C_2 1+14T−191T2−120274T3−20881196T4−120274p2T5−191p4T6+14p6T7+p8T8 1 + 14 T - 191 T^{2} - 120274 T^{3} - 20881196 T^{4} - 120274 p^{2} T^{5} - 191 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8}
71D4×C2D_4\times C_2 1−9148T2+46550598T4−9148p4T6+p8T8 1 - 9148 T^{2} + 46550598 T^{4} - 9148 p^{4} T^{6} + p^{8} T^{8}
73D4D_{4} (1+74T+11487T2+74p2T3+p4T4)2 ( 1 + 74 T + 11487 T^{2} + 74 p^{2} T^{3} + p^{4} T^{4} )^{2}
79D4×C2D_4\times C_2 1−160T+7258T2−937600T3+137709283T4−937600p2T5+7258p4T6−160p6T7+p8T8 1 - 160 T + 7258 T^{2} - 937600 T^{3} + 137709283 T^{4} - 937600 p^{2} T^{5} + 7258 p^{4} T^{6} - 160 p^{6} T^{7} + p^{8} T^{8}
83C22C_2^2 (1+156T+15001T2+156p2T3+p4T4)2 ( 1 + 156 T + 15001 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2}
89D4×C2D_4\times C_2 1−22468T2+231780678T4−22468p4T6+p8T8 1 - 22468 T^{2} + 231780678 T^{4} - 22468 p^{4} T^{6} + p^{8} T^{8}
97D4×C2D_4\times C_2 1−2T−8675T2+20278T3−13241876T4+20278p2T5−8675p4T6−2p6T7+p8T8 1 - 2 T - 8675 T^{2} + 20278 T^{3} - 13241876 T^{4} + 20278 p^{2} T^{5} - 8675 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} 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

−6.23815531205974755219894868323, −6.19059065796740037700908590705, −5.70671187419513817706726543896, −5.66108020847162495790946867432, −5.64331975652631695699335635138, −5.21458186493688662759400567399, −5.10938578662391252640132466635, −4.77185458418120338753507080011, −4.32392219935286735565107297109, −4.28087313414391594029167260304, −4.26676860046853419802175643567, −3.81902219545204855343213895485, −3.68675188433559511462929770764, −3.22910799283999924091780783616, −3.11702518747926252298455180037, −2.95566761813608632407561024938, −2.83693312245337143217560867850, −2.46351873647662955124014954851, −2.37554572481287018688972237516, −1.70210117489154571244647799554, −1.18156326639398668372176884846, −1.17707553305712283958066325555, −1.05029815419755575348635292947, −0.60777361732385959559167709955, −0.56399570451897981737189089946, 0.56399570451897981737189089946, 0.60777361732385959559167709955, 1.05029815419755575348635292947, 1.17707553305712283958066325555, 1.18156326639398668372176884846, 1.70210117489154571244647799554, 2.37554572481287018688972237516, 2.46351873647662955124014954851, 2.83693312245337143217560867850, 2.95566761813608632407561024938, 3.11702518747926252298455180037, 3.22910799283999924091780783616, 3.68675188433559511462929770764, 3.81902219545204855343213895485, 4.26676860046853419802175643567, 4.28087313414391594029167260304, 4.32392219935286735565107297109, 4.77185458418120338753507080011, 5.10938578662391252640132466635, 5.21458186493688662759400567399, 5.64331975652631695699335635138, 5.66108020847162495790946867432, 5.70671187419513817706726543896, 6.19059065796740037700908590705, 6.23815531205974755219894868323

Graph of the ZZ-function along the critical line