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Properties

Label 8-6480e4-1.1-c1e4-0-1
Degree $8$
Conductor $1.763\times 10^{15}$
Sign $1$
Analytic cond. $7.16817\times 10^{6}$
Root an. cond. $7.19326$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(7.16817\times 10^{6}\)
Root analytic conductor: \(7.19326\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{16} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(19.01573662\)
\(L(\frac12)\) \(\approx\) \(19.01573662\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
good7$S_4\times C_2$ \( 1 - T + 13 T^{2} - 4 p T^{3} + 88 T^{4} - 4 p^{2} T^{5} + 13 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + T + 2 T^{2} + 13 T^{3} + 238 T^{4} + 13 p T^{5} + 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 4 T + 28 T^{2} - 136 T^{3} + 406 T^{4} - 136 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 5 T + 38 T^{2} - 215 T^{3} + 886 T^{4} - 215 p T^{5} + 38 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + T - 8 T^{2} + 25 T^{3} + 322 T^{4} + 25 p T^{5} - 8 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 7 T + 95 T^{2} + 436 T^{3} + 3250 T^{4} + 436 p T^{5} + 95 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 7 T + 101 T^{2} - 454 T^{3} + 3970 T^{4} - 454 p T^{5} + 101 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 2 T + 64 T^{2} - 206 T^{3} + 2470 T^{4} - 206 p T^{5} + 64 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 6 T + 100 T^{2} - 414 T^{3} + 4806 T^{4} - 414 p T^{5} + 100 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 12 T + 134 T^{2} - 684 T^{3} + 5259 T^{4} - 684 p T^{5} + 134 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 11 T + 178 T^{2} - 1235 T^{3} + 11398 T^{4} - 1235 p T^{5} + 178 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 7 T + 83 T^{2} + 346 T^{3} + 4456 T^{4} + 346 p T^{5} + 83 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 12 T + 164 T^{2} - 1620 T^{3} + 11910 T^{4} - 1620 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 11 T + 170 T^{2} + 1367 T^{3} + 230 p T^{4} + 1367 p T^{5} + 170 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 19 T + 337 T^{2} - 3502 T^{3} + 33658 T^{4} - 3502 p T^{5} + 337 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 10 T + 208 T^{2} - 1630 T^{3} + 20347 T^{4} - 1630 p T^{5} + 208 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 12 T + 308 T^{2} + 2520 T^{3} + 33582 T^{4} + 2520 p T^{5} + 308 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 9 T + 244 T^{2} - 1395 T^{3} + 23814 T^{4} - 1395 p T^{5} + 244 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 24 T + 196 T^{2} + 648 T^{3} - 19386 T^{4} + 648 p T^{5} + 196 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 23 T + 425 T^{2} + 4820 T^{3} + 51496 T^{4} + 4820 p T^{5} + 425 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 21 T + 329 T^{2} - 3366 T^{3} + 35214 T^{4} - 3366 p T^{5} + 329 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - T + 178 T^{2} - 1519 T^{3} + 14506 T^{4} - 1519 p T^{5} + 178 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line