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Properties

Label 8-224e4-1.1-c1e4-0-6
Degree 88
Conductor 25176309762517630976
Sign 11
Analytic cond. 10.235210.2352
Root an. cond. 1.337401.33740
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 4·3-s + 8·4-s − 8·5-s + 16·6-s − 8·8-s + 4·9-s + 32·10-s − 12·11-s − 32·12-s + 32·15-s − 4·16-s − 16·18-s − 4·19-s − 64·20-s + 48·22-s − 4·23-s + 32·24-s + 40·25-s + 4·27-s − 16·29-s − 128·30-s − 24·31-s + 32·32-s + 48·33-s + 32·36-s − 16·37-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 4·4-s − 3.57·5-s + 6.53·6-s − 2.82·8-s + 4/3·9-s + 10.1·10-s − 3.61·11-s − 9.23·12-s + 8.26·15-s − 16-s − 3.77·18-s − 0.917·19-s − 14.3·20-s + 10.2·22-s − 0.834·23-s + 6.53·24-s + 8·25-s + 0.769·27-s − 2.97·29-s − 23.3·30-s − 4.31·31-s + 5.65·32-s + 8.35·33-s + 16/3·36-s − 2.63·37-s + ⋯

Functional equation

Λ(s)=((220⋅74)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((220⋅74)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 220⋅742^{20} \cdot 7^{4}
Sign: 11
Analytic conductor: 10.235210.2352
Root analytic conductor: 1.337401.33740
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 220⋅74, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{20} \cdot 7^{4} ,\ ( \ : 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)
bad2C2C_2 (1+pT+pT2)2 ( 1 + p T + p T^{2} )^{2}
7C22C_2^2 1+T4 1 + T^{4}
good3D4×C2D_4\times C_2 1+4T+4pT2+28T3+56T4+28pT5+4p3T6+4p3T7+p4T8 1 + 4 T + 4 p T^{2} + 28 T^{3} + 56 T^{4} + 28 p T^{5} + 4 p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
5D4×C2D_4\times C_2 1+8T+24T2+32T3+32T4+32pT5+24p2T6+8p3T7+p4T8 1 + 8 T + 24 T^{2} + 32 T^{3} + 32 T^{4} + 32 p T^{5} + 24 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
11D4×C2D_4\times C_2 1+12T+86T2+428T3+1634T4+428pT5+86p2T6+12p3T7+p4T8 1 + 12 T + 86 T^{2} + 428 T^{3} + 1634 T^{4} + 428 p T^{5} + 86 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
13D4×C2D_4\times C_2 1+8T2+72T3+32T4+72pT5+8p2T6+p4T8 1 + 8 T^{2} + 72 T^{3} + 32 T^{4} + 72 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8}
17C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
19D4×C2D_4\times C_2 1+4T+36T2+196T3+920T4+196pT5+36p2T6+4p3T7+p4T8 1 + 4 T + 36 T^{2} + 196 T^{3} + 920 T^{4} + 196 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
23C22C_2^2 (1+2T+2T2+2pT3+p2T4)2 ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
29D4×C2D_4\times C_2 1+16T+162T2+1216T3+7490T4+1216pT5+162p2T6+16p3T7+p4T8 1 + 16 T + 162 T^{2} + 1216 T^{3} + 7490 T^{4} + 1216 p T^{5} + 162 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
31D4D_{4} (1+12T+90T2+12pT3+p2T4)2 ( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1+16T+162T2+1312T3+9026T4+1312pT5+162p2T6+16p3T7+p4T8 1 + 16 T + 162 T^{2} + 1312 T^{3} + 9026 T^{4} + 1312 p T^{5} + 162 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
41D4×C2D_4\times C_2 1+8T+32T2+360T3+4034T4+360pT5+32p2T6+8p3T7+p4T8 1 + 8 T + 32 T^{2} + 360 T^{3} + 4034 T^{4} + 360 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
43D4×C2D_4\times C_2 1+16T+114T2+632T3+3810T4+632pT5+114p2T6+16p3T7+p4T8 1 + 16 T + 114 T^{2} + 632 T^{3} + 3810 T^{4} + 632 p T^{5} + 114 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
47D4×C2D_4\times C_2 1−100T2+5766T4−100p2T6+p4T8 1 - 100 T^{2} + 5766 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8}
53D4×C2D_4\times C_2 1−20T+118T2+412T3−8478T4+412pT5+118p2T6−20p3T7+p4T8 1 - 20 T + 118 T^{2} + 412 T^{3} - 8478 T^{4} + 412 p T^{5} + 118 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}
59D4×C2D_4\times C_2 1+20T+300T2+3180T3+28600T4+3180pT5+300p2T6+20p3T7+p4T8 1 + 20 T + 300 T^{2} + 3180 T^{3} + 28600 T^{4} + 3180 p T^{5} + 300 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8}
61D4×C2D_4\times C_2 1+16T+96T2+256T3+512T4+256pT5+96p2T6+16p3T7+p4T8 1 + 16 T + 96 T^{2} + 256 T^{3} + 512 T^{4} + 256 p T^{5} + 96 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
67D4×C2D_4\times C_2 1−16T+226T2−2456T3+23362T4−2456pT5+226p2T6−16p3T7+p4T8 1 - 16 T + 226 T^{2} - 2456 T^{3} + 23362 T^{4} - 2456 p T^{5} + 226 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}
71C23C_2^3 1+1154T4+p4T8 1 + 1154 T^{4} + p^{4} T^{8}
73D4×C2D_4\times C_2 1−24T+288T2−3096T3+30146T4−3096pT5+288p2T6−24p3T7+p4T8 1 - 24 T + 288 T^{2} - 3096 T^{3} + 30146 T^{4} - 3096 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}
79C22C_2^2 (1−60T2+p2T4)2 ( 1 - 60 T^{2} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1+4T+132T2+1236T3+11608T4+1236pT5+132p2T6+4p3T7+p4T8 1 + 4 T + 132 T^{2} + 1236 T^{3} + 11608 T^{4} + 1236 p T^{5} + 132 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
89D4×C2D_4\times C_2 1+32T+512T2+6432T3+68258T4+6432pT5+512p2T6+32p3T7+p4T8 1 + 32 T + 512 T^{2} + 6432 T^{3} + 68258 T^{4} + 6432 p T^{5} + 512 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8}
97D4D_{4} (1−8T+202T2−8pT3+p2T4)2 ( 1 - 8 T + 202 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
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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

−9.476213480338880563508409335678, −8.957861858034234570810244812229, −8.846134160901408277104581993178, −8.773808095506330094418033374357, −8.203221857541121846376713502020, −8.153882298323288257339066305161, −7.83958791277833236794267892183, −7.81945012662596335823302185270, −7.66736408639700318581125906583, −7.06858680240611138633294870478, −7.00839530588096599093699724654, −6.97711291655205444164352179646, −6.64860267540627262234622253763, −5.74970298381492036367853470277, −5.65424998802923106598365758339, −5.25860399477540022808461512775, −5.23259133295239489655345610239, −4.94061853535647076738824538356, −4.52918745644796550103106267128, −3.81927197185510651802505405975, −3.79388063117807347582304027346, −3.40735467730031703534106433827, −2.82810455494575849020957114256, −2.09573197158845129667852700435, −1.73567312311397554560939721570, 0, 0, 0, 0, 1.73567312311397554560939721570, 2.09573197158845129667852700435, 2.82810455494575849020957114256, 3.40735467730031703534106433827, 3.79388063117807347582304027346, 3.81927197185510651802505405975, 4.52918745644796550103106267128, 4.94061853535647076738824538356, 5.23259133295239489655345610239, 5.25860399477540022808461512775, 5.65424998802923106598365758339, 5.74970298381492036367853470277, 6.64860267540627262234622253763, 6.97711291655205444164352179646, 7.00839530588096599093699724654, 7.06858680240611138633294870478, 7.66736408639700318581125906583, 7.81945012662596335823302185270, 7.83958791277833236794267892183, 8.153882298323288257339066305161, 8.203221857541121846376713502020, 8.773808095506330094418033374357, 8.846134160901408277104581993178, 8.957861858034234570810244812229, 9.476213480338880563508409335678

Graph of the ZZ-function along the critical line