# Properties

 Label 8-1840e4-1.1-c3e4-0-1 Degree $8$ Conductor $1.146\times 10^{13}$ Sign $1$ Analytic cond. $1.38910\times 10^{8}$ Root an. cond. $10.4193$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·3-s − 20·5-s − 26·7-s − 14·9-s − 93·11-s + 32·13-s + 80·15-s + 108·17-s − 185·19-s + 104·21-s − 92·23-s + 250·25-s + 27·27-s + 294·29-s + 211·31-s + 372·33-s + 520·35-s + 5·37-s − 128·39-s − 369·41-s + 100·43-s + 280·45-s + 363·47-s − 48·49-s − 432·51-s + 21·53-s + 1.86e3·55-s + ⋯
 L(s)  = 1 − 0.769·3-s − 1.78·5-s − 1.40·7-s − 0.518·9-s − 2.54·11-s + 0.682·13-s + 1.37·15-s + 1.54·17-s − 2.23·19-s + 1.08·21-s − 0.834·23-s + 2·25-s + 0.192·27-s + 1.88·29-s + 1.22·31-s + 1.96·33-s + 2.51·35-s + 0.0222·37-s − 0.525·39-s − 1.40·41-s + 0.354·43-s + 0.927·45-s + 1.12·47-s − 0.139·49-s − 1.18·51-s + 0.0544·53-s + 4.56·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 5^{4} \cdot 23^{4}$$ Sign: $1$ Analytic conductor: $$1.38910\times 10^{8}$$ Root analytic conductor: $$10.4193$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{16} \cdot 5^{4} \cdot 23^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## 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)$
bad2 $$1$$
5$C_1$ $$( 1 + p T )^{4}$$
23$C_1$ $$( 1 + p T )^{4}$$
good3$C_2 \wr S_4$ $$1 + 4 T + 10 p T^{2} + 149 T^{3} + 1310 T^{4} + 149 p^{3} T^{5} + 10 p^{7} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8}$$
7$C_2 \wr S_4$ $$1 + 26 T + 724 T^{2} + 10099 T^{3} + 260534 T^{4} + 10099 p^{3} T^{5} + 724 p^{6} T^{6} + 26 p^{9} T^{7} + p^{12} T^{8}$$
11$C_2 \wr S_4$ $$1 + 93 T + 6725 T^{2} + 31627 p T^{3} + 14317128 T^{4} + 31627 p^{4} T^{5} + 6725 p^{6} T^{6} + 93 p^{9} T^{7} + p^{12} T^{8}$$
13$C_2 \wr S_4$ $$1 - 32 T + 976 T^{2} + 52631 T^{3} - 4682512 T^{4} + 52631 p^{3} T^{5} + 976 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8}$$
17$C_2 \wr S_4$ $$1 - 108 T + 18260 T^{2} - 93099 p T^{3} + 131243742 T^{4} - 93099 p^{4} T^{5} + 18260 p^{6} T^{6} - 108 p^{9} T^{7} + p^{12} T^{8}$$
19$C_2 \wr S_4$ $$1 + 185 T + 33223 T^{2} + 186623 p T^{3} + 363066172 T^{4} + 186623 p^{4} T^{5} + 33223 p^{6} T^{6} + 185 p^{9} T^{7} + p^{12} T^{8}$$
29$C_2 \wr S_4$ $$1 - 294 T + 42335 T^{2} + 2109052 T^{3} - 859672512 T^{4} + 2109052 p^{3} T^{5} + 42335 p^{6} T^{6} - 294 p^{9} T^{7} + p^{12} T^{8}$$
31$C_2 \wr S_4$ $$1 - 211 T + 103384 T^{2} - 13064236 T^{3} + 4073558761 T^{4} - 13064236 p^{3} T^{5} + 103384 p^{6} T^{6} - 211 p^{9} T^{7} + p^{12} T^{8}$$
37$C_2 \wr S_4$ $$1 - 5 T + 134830 T^{2} + 4948045 T^{3} + 8513448058 T^{4} + 4948045 p^{3} T^{5} + 134830 p^{6} T^{6} - 5 p^{9} T^{7} + p^{12} T^{8}$$
41$C_2 \wr S_4$ $$1 + 9 p T + 291554 T^{2} + 74201826 T^{3} + 30573978075 T^{4} + 74201826 p^{3} T^{5} + 291554 p^{6} T^{6} + 9 p^{10} T^{7} + p^{12} T^{8}$$
43$C_2 \wr S_4$ $$1 - 100 T + 82204 T^{2} - 9571140 T^{3} + 522019158 T^{4} - 9571140 p^{3} T^{5} + 82204 p^{6} T^{6} - 100 p^{9} T^{7} + p^{12} T^{8}$$
47$C_2 \wr S_4$ $$1 - 363 T + 163502 T^{2} + 6031301 T^{3} + 2659565058 T^{4} + 6031301 p^{3} T^{5} + 163502 p^{6} T^{6} - 363 p^{9} T^{7} + p^{12} T^{8}$$
53$C_2 \wr S_4$ $$1 - 21 T + 269018 T^{2} + 89970593 T^{3} + 27668065578 T^{4} + 89970593 p^{3} T^{5} + 269018 p^{6} T^{6} - 21 p^{9} T^{7} + p^{12} T^{8}$$
59$C_2 \wr S_4$ $$1 - 33 T + 343586 T^{2} - 84316061 T^{3} + 56145878106 T^{4} - 84316061 p^{3} T^{5} + 343586 p^{6} T^{6} - 33 p^{9} T^{7} + p^{12} T^{8}$$
61$C_2 \wr S_4$ $$1 + 307 T + 487855 T^{2} + 298124379 T^{3} + 113148939756 T^{4} + 298124379 p^{3} T^{5} + 487855 p^{6} T^{6} + 307 p^{9} T^{7} + p^{12} T^{8}$$
67$C_2 \wr S_4$ $$1 + 725 T + 912826 T^{2} + 351937325 T^{3} + 318440704618 T^{4} + 351937325 p^{3} T^{5} + 912826 p^{6} T^{6} + 725 p^{9} T^{7} + p^{12} T^{8}$$
71$C_2 \wr S_4$ $$1 - 1257 T + 1758566 T^{2} - 1300005098 T^{3} + 998915687661 T^{4} - 1300005098 p^{3} T^{5} + 1758566 p^{6} T^{6} - 1257 p^{9} T^{7} + p^{12} T^{8}$$
73$C_2 \wr S_4$ $$1 - 509 T + 697972 T^{2} - 203398343 T^{3} + 321017582158 T^{4} - 203398343 p^{3} T^{5} + 697972 p^{6} T^{6} - 509 p^{9} T^{7} + p^{12} T^{8}$$
79$C_2 \wr S_4$ $$1 + 1202 T + 1003684 T^{2} + 146546698 T^{3} + 4581734390 T^{4} + 146546698 p^{3} T^{5} + 1003684 p^{6} T^{6} + 1202 p^{9} T^{7} + p^{12} T^{8}$$
83$C_2 \wr S_4$ $$1 - 1377 T + 1324064 T^{2} - 677164645 T^{3} + 435520077246 T^{4} - 677164645 p^{3} T^{5} + 1324064 p^{6} T^{6} - 1377 p^{9} T^{7} + p^{12} T^{8}$$
89$C_2 \wr S_4$ $$1 + 984 T + 2444900 T^{2} + 1775501064 T^{3} + 2545577861478 T^{4} + 1775501064 p^{3} T^{5} + 2444900 p^{6} T^{6} + 984 p^{9} T^{7} + p^{12} T^{8}$$
97$C_2 \wr S_4$ $$1 - 137 T + 867295 T^{2} - 1257176833 T^{3} + 651849046340 T^{4} - 1257176833 p^{3} T^{5} + 867295 p^{6} T^{6} - 137 p^{9} T^{7} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$