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Properties

Label 8-6480e4-1.1-c1e4-0-1
Degree 88
Conductor 1.763×10151.763\times 10^{15}
Sign 11
Analytic cond. 7.16817×1067.16817\times 10^{6}
Root an. cond. 7.193267.19326
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  + 4·5-s + 7-s − 11-s + 4·13-s + 5·17-s − 19-s − 7·23-s + 10·25-s + 7·29-s + 2·31-s + 4·35-s + 6·37-s + 12·41-s + 11·43-s − 7·47-s − 12·49-s + 12·53-s − 4·55-s − 11·59-s + 19·61-s + 16·65-s + 10·67-s − 12·71-s + 9·73-s − 77-s + 24·79-s − 23·83-s + ⋯
L(s)  = 1  + 1.78·5-s + 0.377·7-s − 0.301·11-s + 1.10·13-s + 1.21·17-s − 0.229·19-s − 1.45·23-s + 2·25-s + 1.29·29-s + 0.359·31-s + 0.676·35-s + 0.986·37-s + 1.87·41-s + 1.67·43-s − 1.02·47-s − 1.71·49-s + 1.64·53-s − 0.539·55-s − 1.43·59-s + 2.43·61-s + 1.98·65-s + 1.22·67-s − 1.42·71-s + 1.05·73-s − 0.113·77-s + 2.70·79-s − 2.52·83-s + ⋯

Functional equation

Λ(s)=((216⋅316⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅316⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \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⋅316⋅542^{16} \cdot 3^{16} \cdot 5^{4}
Sign: 11
Analytic conductor: 7.16817×1067.16817\times 10^{6}
Root analytic conductor: 7.193267.19326
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅316⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 3^{16} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 19.0157366219.01573662
L(12)L(\frac12) ≈\approx 19.0157366219.01573662
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
5C1C_1 (1−T)4 ( 1 - T )^{4}
good7S4×C2S_4\times C_2 1−T+13T2−4pT3+88T4−4p2T5+13p2T6−p3T7+p4T8 1 - T + 13 T^{2} - 4 p T^{3} + 88 T^{4} - 4 p^{2} T^{5} + 13 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}
11C2≀S4C_2 \wr S_4 1+T+2T2+13T3+238T4+13pT5+2p2T6+p3T7+p4T8 1 + T + 2 T^{2} + 13 T^{3} + 238 T^{4} + 13 p T^{5} + 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}
13C2≀S4C_2 \wr S_4 1−4T+28T2−136T3+406T4−136pT5+28p2T6−4p3T7+p4T8 1 - 4 T + 28 T^{2} - 136 T^{3} + 406 T^{4} - 136 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
17C2≀S4C_2 \wr S_4 1−5T+38T2−215T3+886T4−215pT5+38p2T6−5p3T7+p4T8 1 - 5 T + 38 T^{2} - 215 T^{3} + 886 T^{4} - 215 p T^{5} + 38 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}
19C2≀S4C_2 \wr S_4 1+T−8T2+25T3+322T4+25pT5−8p2T6+p3T7+p4T8 1 + T - 8 T^{2} + 25 T^{3} + 322 T^{4} + 25 p T^{5} - 8 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}
23C2≀S4C_2 \wr S_4 1+7T+95T2+436T3+3250T4+436pT5+95p2T6+7p3T7+p4T8 1 + 7 T + 95 T^{2} + 436 T^{3} + 3250 T^{4} + 436 p T^{5} + 95 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}
29C2≀S4C_2 \wr S_4 1−7T+101T2−454T3+3970T4−454pT5+101p2T6−7p3T7+p4T8 1 - 7 T + 101 T^{2} - 454 T^{3} + 3970 T^{4} - 454 p T^{5} + 101 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}
31C2≀S4C_2 \wr S_4 1−2T+64T2−206T3+2470T4−206pT5+64p2T6−2p3T7+p4T8 1 - 2 T + 64 T^{2} - 206 T^{3} + 2470 T^{4} - 206 p T^{5} + 64 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
37C2≀S4C_2 \wr S_4 1−6T+100T2−414T3+4806T4−414pT5+100p2T6−6p3T7+p4T8 1 - 6 T + 100 T^{2} - 414 T^{3} + 4806 T^{4} - 414 p T^{5} + 100 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
41C2≀S4C_2 \wr S_4 1−12T+134T2−684T3+5259T4−684pT5+134p2T6−12p3T7+p4T8 1 - 12 T + 134 T^{2} - 684 T^{3} + 5259 T^{4} - 684 p T^{5} + 134 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}
43C2≀S4C_2 \wr S_4 1−11T+178T2−1235T3+11398T4−1235pT5+178p2T6−11p3T7+p4T8 1 - 11 T + 178 T^{2} - 1235 T^{3} + 11398 T^{4} - 1235 p T^{5} + 178 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8}
47C2≀S4C_2 \wr S_4 1+7T+83T2+346T3+4456T4+346pT5+83p2T6+7p3T7+p4T8 1 + 7 T + 83 T^{2} + 346 T^{3} + 4456 T^{4} + 346 p T^{5} + 83 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}
53C2≀S4C_2 \wr S_4 1−12T+164T2−1620T3+11910T4−1620pT5+164p2T6−12p3T7+p4T8 1 - 12 T + 164 T^{2} - 1620 T^{3} + 11910 T^{4} - 1620 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}
59C2≀S4C_2 \wr S_4 1+11T+170T2+1367T3+230pT4+1367pT5+170p2T6+11p3T7+p4T8 1 + 11 T + 170 T^{2} + 1367 T^{3} + 230 p T^{4} + 1367 p T^{5} + 170 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8}
61C2≀S4C_2 \wr S_4 1−19T+337T2−3502T3+33658T4−3502pT5+337p2T6−19p3T7+p4T8 1 - 19 T + 337 T^{2} - 3502 T^{3} + 33658 T^{4} - 3502 p T^{5} + 337 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8}
67C2≀S4C_2 \wr S_4 1−10T+208T2−1630T3+20347T4−1630pT5+208p2T6−10p3T7+p4T8 1 - 10 T + 208 T^{2} - 1630 T^{3} + 20347 T^{4} - 1630 p T^{5} + 208 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}
71C2≀S4C_2 \wr S_4 1+12T+308T2+2520T3+33582T4+2520pT5+308p2T6+12p3T7+p4T8 1 + 12 T + 308 T^{2} + 2520 T^{3} + 33582 T^{4} + 2520 p T^{5} + 308 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
73C2≀S4C_2 \wr S_4 1−9T+244T2−1395T3+23814T4−1395pT5+244p2T6−9p3T7+p4T8 1 - 9 T + 244 T^{2} - 1395 T^{3} + 23814 T^{4} - 1395 p T^{5} + 244 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}
79C2≀S4C_2 \wr S_4 1−24T+196T2+648T3−19386T4+648pT5+196p2T6−24p3T7+p4T8 1 - 24 T + 196 T^{2} + 648 T^{3} - 19386 T^{4} + 648 p T^{5} + 196 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}
83C2≀S4C_2 \wr S_4 1+23T+425T2+4820T3+51496T4+4820pT5+425p2T6+23p3T7+p4T8 1 + 23 T + 425 T^{2} + 4820 T^{3} + 51496 T^{4} + 4820 p T^{5} + 425 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8}
89C2≀S4C_2 \wr S_4 1−21T+329T2−3366T3+35214T4−3366pT5+329p2T6−21p3T7+p4T8 1 - 21 T + 329 T^{2} - 3366 T^{3} + 35214 T^{4} - 3366 p T^{5} + 329 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8}
97C2≀S4C_2 \wr S_4 1−T+178T2−1519T3+14506T4−1519pT5+178p2T6−p3T7+p4T8 1 - T + 178 T^{2} - 1519 T^{3} + 14506 T^{4} - 1519 p T^{5} + 178 p^{2} T^{6} - 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

−5.84472318689221125699222374686, −5.35085665056741555096697644476, −5.29728476669164893721153059566, −5.25187645704199186982455270215, −4.93875945065464647333230265862, −4.57914688918907683285568537238, −4.47353728382967180642034849876, −4.38460558315350216167745181507, −4.34212944116838976093605573840, −3.80566764476569587991713530921, −3.55197066434517151432097471000, −3.49119432541602348092443955134, −3.49058999990764495960044904378, −2.94351761527776211780913887488, −2.80886101622709067625655333676, −2.61771070816209598691420058924, −2.43261700231109649831869141726, −2.11166606127574900519902338700, −1.89983096557817494817584214441, −1.79694556849377185811927061473, −1.62387000382622096167288019199, −0.942204769781160033092270366829, −0.933904778781747995355308999752, −0.811831955513461632413192716327, −0.48845638341821851160548601542, 0.48845638341821851160548601542, 0.811831955513461632413192716327, 0.933904778781747995355308999752, 0.942204769781160033092270366829, 1.62387000382622096167288019199, 1.79694556849377185811927061473, 1.89983096557817494817584214441, 2.11166606127574900519902338700, 2.43261700231109649831869141726, 2.61771070816209598691420058924, 2.80886101622709067625655333676, 2.94351761527776211780913887488, 3.49058999990764495960044904378, 3.49119432541602348092443955134, 3.55197066434517151432097471000, 3.80566764476569587991713530921, 4.34212944116838976093605573840, 4.38460558315350216167745181507, 4.47353728382967180642034849876, 4.57914688918907683285568537238, 4.93875945065464647333230265862, 5.25187645704199186982455270215, 5.29728476669164893721153059566, 5.35085665056741555096697644476, 5.84472318689221125699222374686

Graph of the ZZ-function along the critical line