# Properties

 Label 8-570e4-1.1-c5e4-0-4 Degree $8$ Conductor $105560010000$ Sign $1$ Analytic cond. $6.98460\times 10^{7}$ Root an. cond. $9.56131$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·2-s − 36·3-s + 160·4-s − 100·5-s + 576·6-s + 108·7-s − 1.28e3·8-s + 810·9-s + 1.60e3·10-s − 460·11-s − 5.76e3·12-s + 296·13-s − 1.72e3·14-s + 3.60e3·15-s + 8.96e3·16-s − 412·17-s − 1.29e4·18-s − 1.44e3·19-s − 1.60e4·20-s − 3.88e3·21-s + 7.36e3·22-s − 768·23-s + 4.60e4·24-s + 6.25e3·25-s − 4.73e3·26-s − 1.45e4·27-s + 1.72e4·28-s + ⋯
 L(s)  = 1 − 2.82·2-s − 2.30·3-s + 5·4-s − 1.78·5-s + 6.53·6-s + 0.833·7-s − 7.07·8-s + 10/3·9-s + 5.05·10-s − 1.14·11-s − 11.5·12-s + 0.485·13-s − 2.35·14-s + 4.13·15-s + 35/4·16-s − 0.345·17-s − 9.42·18-s − 0.917·19-s − 8.94·20-s − 1.92·21-s + 3.24·22-s − 0.302·23-s + 16.3·24-s + 2·25-s − 1.37·26-s − 3.84·27-s + 4.16·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}$$ Sign: $1$ Analytic conductor: $$6.98460\times 10^{7}$$ Root analytic conductor: $$9.56131$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{570} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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$C_1$ $$( 1 + p^{2} T )^{4}$$
3$C_1$ $$( 1 + p^{2} T )^{4}$$
5$C_1$ $$( 1 + p^{2} T )^{4}$$
19$C_1$ $$( 1 + p^{2} T )^{4}$$
good7$C_2 \wr S_4$ $$1 - 108 T + 22592 T^{2} - 335172 p T^{3} + 23650626 p T^{4} - 335172 p^{6} T^{5} + 22592 p^{10} T^{6} - 108 p^{15} T^{7} + p^{20} T^{8}$$
11$C_2 \wr S_4$ $$1 + 460 T + 634272 T^{2} + 207171372 T^{3} + 153127224190 T^{4} + 207171372 p^{5} T^{5} + 634272 p^{10} T^{6} + 460 p^{15} T^{7} + p^{20} T^{8}$$
13$C_2 \wr S_4$ $$1 - 296 T + 1026368 T^{2} - 369139656 T^{3} + 498364489310 T^{4} - 369139656 p^{5} T^{5} + 1026368 p^{10} T^{6} - 296 p^{15} T^{7} + p^{20} T^{8}$$
17$C_2 \wr S_4$ $$1 + 412 T + 1895348 T^{2} + 217134820 T^{3} + 3423837223510 T^{4} + 217134820 p^{5} T^{5} + 1895348 p^{10} T^{6} + 412 p^{15} T^{7} + p^{20} T^{8}$$
23$C_2 \wr S_4$ $$1 + 768 T + 6342144 T^{2} + 20807267184 T^{3} + 55000877068958 T^{4} + 20807267184 p^{5} T^{5} + 6342144 p^{10} T^{6} + 768 p^{15} T^{7} + p^{20} T^{8}$$
29$C_2 \wr S_4$ $$1 + 1828 T + 45521568 T^{2} + 199682159052 T^{3} + 965967761755582 T^{4} + 199682159052 p^{5} T^{5} + 45521568 p^{10} T^{6} + 1828 p^{15} T^{7} + p^{20} T^{8}$$
31$C_2 \wr S_4$ $$1 - 3856 T + 93218140 T^{2} - 258961384912 T^{3} + 3740109452966854 T^{4} - 258961384912 p^{5} T^{5} + 93218140 p^{10} T^{6} - 3856 p^{15} T^{7} + p^{20} T^{8}$$
37$C_2 \wr S_4$ $$1 - 11456 T + 153795592 T^{2} - 1859026238912 T^{3} + 14827439369510638 T^{4} - 1859026238912 p^{5} T^{5} + 153795592 p^{10} T^{6} - 11456 p^{15} T^{7} + p^{20} T^{8}$$
41$C_2 \wr S_4$ $$1 + 12904 T + 372993128 T^{2} + 3896517353272 T^{3} + 62214038619055870 T^{4} + 3896517353272 p^{5} T^{5} + 372993128 p^{10} T^{6} + 12904 p^{15} T^{7} + p^{20} T^{8}$$
43$C_2 \wr S_4$ $$1 - 15096 T + 358977688 T^{2} - 3664713039048 T^{3} + 59284047791232078 T^{4} - 3664713039048 p^{5} T^{5} + 358977688 p^{10} T^{6} - 15096 p^{15} T^{7} + p^{20} T^{8}$$
47$C_2 \wr S_4$ $$1 - 17040 T + 234338432 T^{2} - 130304429280 T^{3} - 29945625228689922 T^{4} - 130304429280 p^{5} T^{5} + 234338432 p^{10} T^{6} - 17040 p^{15} T^{7} + p^{20} T^{8}$$
53$C_2 \wr S_4$ $$1 - 16728 T + 1363547600 T^{2} - 18614385658728 T^{3} + 808606586283795054 T^{4} - 18614385658728 p^{5} T^{5} + 1363547600 p^{10} T^{6} - 16728 p^{15} T^{7} + p^{20} T^{8}$$
59$C_2 \wr S_4$ $$1 + 19760 T + 2421223932 T^{2} + 35339003907888 T^{3} + 2463150953500873654 T^{4} + 35339003907888 p^{5} T^{5} + 2421223932 p^{10} T^{6} + 19760 p^{15} T^{7} + p^{20} T^{8}$$
61$C_2 \wr S_4$ $$1 - 47168 T + 3649900588 T^{2} - 105880714264256 T^{3} + 4575221539431375670 T^{4} - 105880714264256 p^{5} T^{5} + 3649900588 p^{10} T^{6} - 47168 p^{15} T^{7} + p^{20} T^{8}$$
67$C_2 \wr S_4$ $$1 - 104580 T + 8164162892 T^{2} - 415397968014372 T^{3} + 17856114887408360694 T^{4} - 415397968014372 p^{5} T^{5} + 8164162892 p^{10} T^{6} - 104580 p^{15} T^{7} + p^{20} T^{8}$$
71$C_2 \wr S_4$ $$1 + 36764 T + 6591663084 T^{2} + 185108107455948 T^{3} + 17317985101394794246 T^{4} + 185108107455948 p^{5} T^{5} + 6591663084 p^{10} T^{6} + 36764 p^{15} T^{7} + p^{20} T^{8}$$
73$C_2 \wr S_4$ $$1 - 74356 T + 9790499396 T^{2} - 473710574281644 T^{3} + 32025142509597093782 T^{4} - 473710574281644 p^{5} T^{5} + 9790499396 p^{10} T^{6} - 74356 p^{15} T^{7} + p^{20} T^{8}$$
79$C_2 \wr S_4$ $$1 - 80920 T + 7826986556 T^{2} - 539026104326712 T^{3} + 36863551059304082246 T^{4} - 539026104326712 p^{5} T^{5} + 7826986556 p^{10} T^{6} - 80920 p^{15} T^{7} + p^{20} T^{8}$$
83$C_2 \wr S_4$ $$1 - 19416 T + 13103987592 T^{2} - 177655097333208 T^{3} + 72489990968666978078 T^{4} - 177655097333208 p^{5} T^{5} + 13103987592 p^{10} T^{6} - 19416 p^{15} T^{7} + p^{20} T^{8}$$
89$C_2 \wr S_4$ $$1 + 4760 T + 19300981752 T^{2} + 89242926342888 T^{3} +$$$$15\!\cdots\!14$$$$T^{4} + 89242926342888 p^{5} T^{5} + 19300981752 p^{10} T^{6} + 4760 p^{15} T^{7} + p^{20} T^{8}$$
97$C_2 \wr S_4$ $$1 - 139572 T + 26521339240 T^{2} - 2482193854344348 T^{3} +$$$$32\!\cdots\!50$$$$T^{4} - 2482193854344348 p^{5} T^{5} + 26521339240 p^{10} T^{6} - 139572 p^{15} T^{7} + p^{20} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$