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Properties

Label 8-96e8-1.1-c1e4-0-18
Degree 88
Conductor 7.214×10157.214\times 10^{15}
Sign 11
Analytic cond. 2.93277×1072.93277\times 10^{7}
Root an. cond. 8.578468.57846
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  + 8·7-s − 8·13-s − 16·23-s − 8·25-s + 8·31-s − 8·37-s − 16·47-s + 20·49-s − 16·59-s − 24·61-s − 16·67-s − 16·71-s − 8·73-s + 24·79-s + 8·89-s − 64·91-s − 16·97-s − 16·101-s + 24·103-s − 16·107-s − 24·109-s + 8·113-s − 28·121-s − 16·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 3.02·7-s − 2.21·13-s − 3.33·23-s − 8/5·25-s + 1.43·31-s − 1.31·37-s − 2.33·47-s + 20/7·49-s − 2.08·59-s − 3.07·61-s − 1.95·67-s − 1.89·71-s − 0.936·73-s + 2.70·79-s + 0.847·89-s − 6.70·91-s − 1.62·97-s − 1.59·101-s + 2.36·103-s − 1.54·107-s − 2.29·109-s + 0.752·113-s − 2.54·121-s − 1.43·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

Λ(s)=((240⋅38)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((240⋅38)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 240⋅382^{40} \cdot 3^{8}
Sign: 11
Analytic conductor: 2.93277×1072.93277\times 10^{7}
Root analytic conductor: 8.578468.57846
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 240⋅38, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{40} \cdot 3^{8} ,\ ( \ : 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)
bad2 1 1
3 1 1
good5((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+8T2+16T3+26T4+16pT5+8p2T6+p4T8 1 + 8 T^{2} + 16 T^{3} + 26 T^{4} + 16 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8}
7((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1−8T+44T2−24pT3+510T4−24p2T5+44p2T6−8p3T7+p4T8 1 - 8 T + 44 T^{2} - 24 p T^{3} + 510 T^{4} - 24 p^{2} T^{5} + 44 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
11C22:C4C_2^2:C_4 1+28T2+406T4+28p2T6+p4T8 1 + 28 T^{2} + 406 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8}
13((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+8T+40T2+136T3+514T4+136pT5+40p2T6+8p3T7+p4T8 1 + 8 T + 40 T^{2} + 136 T^{3} + 514 T^{4} + 136 p T^{5} + 40 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
17((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+36T2+64T3+614T4+64pT5+36p2T6+p4T8 1 + 36 T^{2} + 64 T^{3} + 614 T^{4} + 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8}
19((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+44T2+64T3+918T4+64pT5+44p2T6+p4T8 1 + 44 T^{2} + 64 T^{3} + 918 T^{4} + 64 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8}
23C2C_2 (1+4T+pT2)4 ( 1 + 4 T + p T^{2} )^{4}
29((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+72T2+112T3+2426T4+112pT5+72p2T6+p4T8 1 + 72 T^{2} + 112 T^{3} + 2426 T^{4} + 112 p T^{5} + 72 p^{2} T^{6} + p^{4} T^{8}
31((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1−8T+108T2−616T3+162pT4−616pT5+108p2T6−8p3T7+p4T8 1 - 8 T + 108 T^{2} - 616 T^{3} + 162 p T^{4} - 616 p T^{5} + 108 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
37((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+8T+104T2+776T3+5346T4+776pT5+104p2T6+8p3T7+p4T8 1 + 8 T + 104 T^{2} + 776 T^{3} + 5346 T^{4} + 776 p T^{5} + 104 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
41((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+68T2−64T3+3206T4−64pT5+68p2T6+p4T8 1 + 68 T^{2} - 64 T^{3} + 3206 T^{4} - 64 p T^{5} + 68 p^{2} T^{6} + p^{4} T^{8}
43((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+76T2−64T3+3830T4−64pT5+76p2T6+p4T8 1 + 76 T^{2} - 64 T^{3} + 3830 T^{4} - 64 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8}
47D4D_{4} (1+8T+78T2+8pT3+p2T4)2 ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
53((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+104T2+272T3+5594T4+272pT5+104p2T6+p4T8 1 + 104 T^{2} + 272 T^{3} + 5594 T^{4} + 272 p T^{5} + 104 p^{2} T^{6} + p^{4} T^{8}
59((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+16T+204T2+1552T3+12758T4+1552pT5+204p2T6+16p3T7+p4T8 1 + 16 T + 204 T^{2} + 1552 T^{3} + 12758 T^{4} + 1552 p T^{5} + 204 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
61((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+24T+392T2+4568T3+40386T4+4568pT5+392p2T6+24p3T7+p4T8 1 + 24 T + 392 T^{2} + 4568 T^{3} + 40386 T^{4} + 4568 p T^{5} + 392 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}
67((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+16T+172T2+912T3+6134T4+912pT5+172p2T6+16p3T7+p4T8 1 + 16 T + 172 T^{2} + 912 T^{3} + 6134 T^{4} + 912 p T^{5} + 172 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
71((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+16T+4pT2+2640T3+28070T4+2640pT5+4p3T6+16p3T7+p4T8 1 + 16 T + 4 p T^{2} + 2640 T^{3} + 28070 T^{4} + 2640 p T^{5} + 4 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
73((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+8T+172T2+1720T3+16006T4+1720pT5+172p2T6+8p3T7+p4T8 1 + 8 T + 172 T^{2} + 1720 T^{3} + 16006 T^{4} + 1720 p T^{5} + 172 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
79((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1−24T+396T2−4344T3+42398T4−4344pT5+396p2T6−24p3T7+p4T8 1 - 24 T + 396 T^{2} - 4344 T^{3} + 42398 T^{4} - 4344 p T^{5} + 396 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}
83((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+252T2−128T3+28598T4−128pT5+252p2T6+p4T8 1 + 252 T^{2} - 128 T^{3} + 28598 T^{4} - 128 p T^{5} + 252 p^{2} T^{6} + p^{4} T^{8}
89((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1−8T+188T2−2424T3+17894T4−2424pT5+188p2T6−8p3T7+p4T8 1 - 8 T + 188 T^{2} - 2424 T^{3} + 17894 T^{4} - 2424 p T^{5} + 188 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
97((C8:C2):C2):C2((C_8 : C_2):C_2):C_2 1+16T+356T2+4400T3+49990T4+4400pT5+356p2T6+16p3T7+p4T8 1 + 16 T + 356 T^{2} + 4400 T^{3} + 49990 T^{4} + 4400 p T^{5} + 356 p^{2} T^{6} + 16 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

−5.86350100999126382601908699032, −5.53872470122145378045055269994, −5.24564869021528211073373915546, −5.17929021222829505231545324668, −4.92753807539620679524477719051, −4.78028828485817050373696020885, −4.73546622840489348481698698930, −4.66872830113867590151973803393, −4.41005038419679448614121496755, −4.13557026154887687313426407861, −3.98339193367130843221693356100, −3.96867063721958662060878579118, −3.61888793136333424851096845837, −3.20972975765980126726721990679, −3.09561240874680195913430193185, −2.95494517854485938188432057147, −2.80296239696175813493460893133, −2.23073696435743740634222633343, −2.10084358229669943227593659841, −2.08783694473472480641157552017, −2.05054507156803413785585931403, −1.49887224021910274400914658027, −1.39770654515595401705486031378, −1.38354991528801901104336646350, −1.10320239994262060691076389474, 0, 0, 0, 0, 1.10320239994262060691076389474, 1.38354991528801901104336646350, 1.39770654515595401705486031378, 1.49887224021910274400914658027, 2.05054507156803413785585931403, 2.08783694473472480641157552017, 2.10084358229669943227593659841, 2.23073696435743740634222633343, 2.80296239696175813493460893133, 2.95494517854485938188432057147, 3.09561240874680195913430193185, 3.20972975765980126726721990679, 3.61888793136333424851096845837, 3.96867063721958662060878579118, 3.98339193367130843221693356100, 4.13557026154887687313426407861, 4.41005038419679448614121496755, 4.66872830113867590151973803393, 4.73546622840489348481698698930, 4.78028828485817050373696020885, 4.92753807539620679524477719051, 5.17929021222829505231545324668, 5.24564869021528211073373915546, 5.53872470122145378045055269994, 5.86350100999126382601908699032

Graph of the ZZ-function along the critical line