Properties

Label 8-55e4-1.1-c3e4-0-1
Degree $8$
Conductor $9150625$
Sign $1$
Analytic cond. $110.895$
Root an. cond. $1.80141$
Motivic weight $3$
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  + 2-s + 9·3-s − 6·4-s + 20·5-s + 9·6-s + 9·7-s + 2·8-s + 11·9-s + 20·10-s + 44·11-s − 54·12-s + 70·13-s + 9·14-s + 180·15-s − 21·16-s + 103·17-s + 11·18-s − 205·19-s − 120·20-s + 81·21-s + 44·22-s − 56·23-s + 18·24-s + 250·25-s + 70·26-s − 90·27-s − 54·28-s + ⋯
L(s)  = 1  + 0.353·2-s + 1.73·3-s − 3/4·4-s + 1.78·5-s + 0.612·6-s + 0.485·7-s + 0.0883·8-s + 0.407·9-s + 0.632·10-s + 1.20·11-s − 1.29·12-s + 1.49·13-s + 0.171·14-s + 3.09·15-s − 0.328·16-s + 1.46·17-s + 0.144·18-s − 2.47·19-s − 1.34·20-s + 0.841·21-s + 0.426·22-s − 0.507·23-s + 0.153·24-s + 2·25-s + 0.528·26-s − 0.641·27-s − 0.364·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(9150625\)    =    \(5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(110.895\)
Root analytic conductor: \(1.80141\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 9150625,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(7.274868421\)
\(L(\frac12)\) \(\approx\) \(7.274868421\)
\(L(\frac{5}{2})\) 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)$
bad5$C_1$ \( ( 1 - p T )^{4} \)
11$C_1$ \( ( 1 - p T )^{4} \)
good2$C_2 \wr S_4$ \( 1 - T + 7 T^{2} - 15 T^{3} + 5 p^{4} T^{4} - 15 p^{3} T^{5} + 7 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} \)
3$C_2 \wr S_4$ \( 1 - p^{2} T + 70 T^{2} - 49 p^{2} T^{3} + 2674 T^{4} - 49 p^{5} T^{5} + 70 p^{6} T^{6} - p^{11} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 9 T + 708 T^{2} - 7885 T^{3} + 337686 T^{4} - 7885 p^{3} T^{5} + 708 p^{6} T^{6} - 9 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 70 T + 6048 T^{2} - 213922 T^{3} + 14034062 T^{4} - 213922 p^{3} T^{5} + 6048 p^{6} T^{6} - 70 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 103 T + 13792 T^{2} - 887937 T^{3} + 87444206 T^{4} - 887937 p^{3} T^{5} + 13792 p^{6} T^{6} - 103 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 205 T + 34976 T^{2} + 3873885 T^{3} + 380485006 T^{4} + 3873885 p^{3} T^{5} + 34976 p^{6} T^{6} + 205 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 56 T + 2040 p T^{2} + 1999704 T^{3} + 845457934 T^{4} + 1999704 p^{3} T^{5} + 2040 p^{7} T^{6} + 56 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 79 T + 26818 T^{2} + 3964293 T^{3} + 1224657242 T^{4} + 3964293 p^{3} T^{5} + 26818 p^{6} T^{6} + 79 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 49 T + 94508 T^{2} - 3937677 T^{3} + 3982227494 T^{4} - 3937677 p^{3} T^{5} + 94508 p^{6} T^{6} - 49 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 289 T + 76410 T^{2} + 5380301 T^{3} - 1483114390 T^{4} + 5380301 p^{3} T^{5} + 76410 p^{6} T^{6} - 289 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 736 T + 441052 T^{2} - 165296160 T^{3} + 51625016486 T^{4} - 165296160 p^{3} T^{5} + 441052 p^{6} T^{6} - 736 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 152 T + 158892 T^{2} + 27361880 T^{3} + 15212838326 T^{4} + 27361880 p^{3} T^{5} + 158892 p^{6} T^{6} + 152 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 412 T + 138336 T^{2} - 73140204 T^{3} + 27137622526 T^{4} - 73140204 p^{3} T^{5} + 138336 p^{6} T^{6} - 412 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 1685 T + 1590202 T^{2} - 996012543 T^{3} + 449858930474 T^{4} - 996012543 p^{3} T^{5} + 1590202 p^{6} T^{6} - 1685 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 842 T + 352020 T^{2} - 46490886 T^{3} - 63133979162 T^{4} - 46490886 p^{3} T^{5} + 352020 p^{6} T^{6} + 842 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 1097 T + 1346310 T^{2} + 822011171 T^{3} + 512711118962 T^{4} + 822011171 p^{3} T^{5} + 1346310 p^{6} T^{6} + 1097 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 122 T - 57444 T^{2} - 8936710 T^{3} + 101266983734 T^{4} - 8936710 p^{3} T^{5} - 57444 p^{6} T^{6} + 122 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 521 T + 927280 T^{2} + 492331245 T^{3} + 408706607486 T^{4} + 492331245 p^{3} T^{5} + 927280 p^{6} T^{6} + 521 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 590 T + 565128 T^{2} + 6063986 p T^{3} + 275503933262 T^{4} + 6063986 p^{4} T^{5} + 565128 p^{6} T^{6} + 590 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 1118 T + 1188812 T^{2} + 493112646 T^{3} + 396666815654 T^{4} + 493112646 p^{3} T^{5} + 1188812 p^{6} T^{6} + 1118 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 122 T + 876132 T^{2} + 323762010 T^{3} + 688436592166 T^{4} + 323762010 p^{3} T^{5} + 876132 p^{6} T^{6} + 122 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 181 T + 2279242 T^{2} + 437587107 T^{3} + 2247142370474 T^{4} + 437587107 p^{3} T^{5} + 2279242 p^{6} T^{6} + 181 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 1474 T + 3318960 T^{2} - 3496771774 T^{3} + 4536975597470 T^{4} - 3496771774 p^{3} T^{5} + 3318960 p^{6} T^{6} - 1474 p^{9} T^{7} + p^{12} T^{8} \)
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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

−10.67756232988677772205142636585, −10.62674435111164435369475841147, −10.25601475548394904271188041891, −9.787207065563630190217342159878, −9.642911815724621114655167451281, −8.895482721571723595682031093014, −8.886191116102974503470666933340, −8.833735353102634924544331122145, −8.805942642734660003602883950522, −8.045360654252855019094626248255, −7.69201545213449970226854442205, −7.60494472825424610602861201375, −6.71958737707093313517450815223, −6.46357810473545174548790919213, −6.00844911252045306958664154283, −5.79321802723758171688565278358, −5.58621097731803228636301926928, −4.66350039491162786629787560024, −4.27696240740388187614970845924, −4.14678549143656247135259834678, −3.50787378425920602948612573599, −2.75750791098450057495496417973, −2.51251147950086715389097290488, −1.75507532413122864052179972519, −1.17378418719408757037588761930, 1.17378418719408757037588761930, 1.75507532413122864052179972519, 2.51251147950086715389097290488, 2.75750791098450057495496417973, 3.50787378425920602948612573599, 4.14678549143656247135259834678, 4.27696240740388187614970845924, 4.66350039491162786629787560024, 5.58621097731803228636301926928, 5.79321802723758171688565278358, 6.00844911252045306958664154283, 6.46357810473545174548790919213, 6.71958737707093313517450815223, 7.60494472825424610602861201375, 7.69201545213449970226854442205, 8.045360654252855019094626248255, 8.805942642734660003602883950522, 8.833735353102634924544331122145, 8.886191116102974503470666933340, 8.895482721571723595682031093014, 9.642911815724621114655167451281, 9.787207065563630190217342159878, 10.25601475548394904271188041891, 10.62674435111164435369475841147, 10.67756232988677772205142636585

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