↑

Properties

Label 8-43e8-1.1-c3e4-0-0
Degree $8$
Conductor $1.169\times 10^{13}$
Sign $1$
Analytic cond. $1.41648\times 10^{8}$
Root an. cond. $10.4448$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 11·3-s − 7·4-s + 27·5-s + 44·6-s + 20·7-s − 38·8-s + 2·9-s + 108·10-s − 62·11-s − 77·12-s − 2·13-s + 80·14-s + 297·15-s + 95·16-s − 207·17-s + 8·18-s − 99·19-s − 189·20-s + 220·21-s − 248·22-s − 103·23-s − 418·24-s + 64·25-s − 8·26-s − 448·27-s − 140·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.11·3-s − 7/8·4-s + 2.41·5-s + 2.99·6-s + 1.07·7-s − 1.67·8-s + 2/27·9-s + 3.41·10-s − 1.69·11-s − 1.85·12-s − 0.0426·13-s + 1.52·14-s + 5.11·15-s + 1.48·16-s − 2.95·17-s + 0.104·18-s − 1.19·19-s − 2.11·20-s + 2.28·21-s − 2.40·22-s − 0.933·23-s − 3.55·24-s + 0.511·25-s − 0.0603·26-s − 3.19·27-s − 0.944·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(43^{8}\)
Sign: $1$
Analytic conductor: \(1.41648\times 10^{8}\)
Root analytic conductor: \(10.4448\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 43^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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)$
bad43 \( 1 \)
good2$C_2 \wr S_4$ \( 1 - p^{2} T + 23 T^{2} - 41 p T^{3} + 121 p T^{4} - 41 p^{4} T^{5} + 23 p^{6} T^{6} - p^{11} T^{7} + p^{12} T^{8} \)
3$C_2 \wr S_4$ \( 1 - 11 T + 119 T^{2} - 839 T^{3} + 4996 T^{4} - 839 p^{3} T^{5} + 119 p^{6} T^{6} - 11 p^{9} T^{7} + p^{12} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 27 T + 133 p T^{2} - 9849 T^{3} + 134272 T^{4} - 9849 p^{3} T^{5} + 133 p^{7} T^{6} - 27 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 20 T + 768 T^{2} - 13108 T^{3} + 314942 T^{4} - 13108 p^{3} T^{5} + 768 p^{6} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 62 T + 3077 T^{2} + 39962 T^{3} + 1317884 T^{4} + 39962 p^{3} T^{5} + 3077 p^{6} T^{6} + 62 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 2 T + 8273 T^{2} + 102 p^{2} T^{3} + 26690692 T^{4} + 102 p^{5} T^{5} + 8273 p^{6} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 207 T + 14258 T^{2} - 190032 T^{3} - 71042783 T^{4} - 190032 p^{3} T^{5} + 14258 p^{6} T^{6} + 207 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 99 T + 22201 T^{2} + 1662075 T^{3} + 222917608 T^{4} + 1662075 p^{3} T^{5} + 22201 p^{6} T^{6} + 99 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 103 T + 1466 p T^{2} + 2793222 T^{3} + 586593953 T^{4} + 2793222 p^{3} T^{5} + 1466 p^{7} T^{6} + 103 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 99 T - 13567 T^{2} + 167967 T^{3} + 888213804 T^{4} + 167967 p^{3} T^{5} - 13567 p^{6} T^{6} - 99 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 131 T + 64436 T^{2} - 3696474 T^{3} + 1950279047 T^{4} - 3696474 p^{3} T^{5} + 64436 p^{6} T^{6} - 131 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 449 T + 266817 T^{2} - 71392239 T^{3} + 21942685768 T^{4} - 71392239 p^{3} T^{5} + 266817 p^{6} T^{6} - 449 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 491 T + 187430 T^{2} + 55413200 T^{3} + 16109516117 T^{4} + 55413200 p^{3} T^{5} + 187430 p^{6} T^{6} + 491 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 19 T + 108577 T^{2} - 15073647 T^{3} + 11126438636 T^{4} - 15073647 p^{3} T^{5} + 108577 p^{6} T^{6} - 19 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 1220 T + 1002413 T^{2} + 551938550 T^{3} + 246387730196 T^{4} + 551938550 p^{3} T^{5} + 1002413 p^{6} T^{6} + 1220 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 816 T + 987944 T^{2} - 513131184 T^{3} + 321570111646 T^{4} - 513131184 p^{3} T^{5} + 987944 p^{6} T^{6} - 816 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 372 T + 843468 T^{2} + 226528412 T^{3} + 278949483798 T^{4} + 226528412 p^{3} T^{5} + 843468 p^{6} T^{6} + 372 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 110 T + 855641 T^{2} - 172379298 T^{3} + 331907899304 T^{4} - 172379298 p^{3} T^{5} + 855641 p^{6} T^{6} - 110 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 468 T + 925804 T^{2} + 494793108 T^{3} + 407494264950 T^{4} + 494793108 p^{3} T^{5} + 925804 p^{6} T^{6} + 468 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 628 T + 1127772 T^{2} + 544172556 T^{3} + 627880103782 T^{4} + 544172556 p^{3} T^{5} + 1127772 p^{6} T^{6} + 628 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 1095 T + 1663851 T^{2} - 1399726691 T^{3} + 1195813590120 T^{4} - 1399726691 p^{3} T^{5} + 1663851 p^{6} T^{6} - 1095 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 980 T + 1568765 T^{2} + 705601262 T^{3} + 896947637720 T^{4} + 705601262 p^{3} T^{5} + 1568765 p^{6} T^{6} + 980 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 738 T + 1788008 T^{2} - 1031145174 T^{3} + 1620376904526 T^{4} - 1031145174 p^{3} T^{5} + 1788008 p^{6} T^{6} - 738 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 1765 T + 3796458 T^{2} + 4179032168 T^{3} + 5180997869897 T^{4} + 4179032168 p^{3} T^{5} + 3796458 p^{6} T^{6} + 1765 p^{9} T^{7} + p^{12} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.43278629542226099927337497379, −6.39936172454143977212715704580, −6.03332594107210824730946912775, −5.92080775936802332092686642347, −5.76710597517692916306109811708, −5.41907921225187491283556226191, −5.20817361765475841045077368017, −5.13627787627830132268956236966, −4.91347637913940669845163695835, −4.65389818227367782583751966213, −4.40188784204464604067928807587, −4.37705733816484982842556604846, −4.15720666455233696320225560799, −3.71870875181774969895387455388, −3.65091904397105701899754918364, −3.20103895402032805496519269994, −3.08243661656579386576290621145, −2.72288313146457568352568758262, −2.38001727738204969431985730139, −2.31976667057094236769476417396, −2.27537234084120989222980366677, −2.16365727124874158773849373196, −1.50341701671064959257445275026, −1.39060556511566641332602570905, −1.23360119240767554854678263735, 0, 0, 0, 0, 1.23360119240767554854678263735, 1.39060556511566641332602570905, 1.50341701671064959257445275026, 2.16365727124874158773849373196, 2.27537234084120989222980366677, 2.31976667057094236769476417396, 2.38001727738204969431985730139, 2.72288313146457568352568758262, 3.08243661656579386576290621145, 3.20103895402032805496519269994, 3.65091904397105701899754918364, 3.71870875181774969895387455388, 4.15720666455233696320225560799, 4.37705733816484982842556604846, 4.40188784204464604067928807587, 4.65389818227367782583751966213, 4.91347637913940669845163695835, 5.13627787627830132268956236966, 5.20817361765475841045077368017, 5.41907921225187491283556226191, 5.76710597517692916306109811708, 5.92080775936802332092686642347, 6.03332594107210824730946912775, 6.39936172454143977212715704580, 6.43278629542226099927337497379

Graph of the $Z$-function along the critical line