# Properties

 Label 8-21e8-1.1-c3e4-0-9 Degree $8$ Conductor $37822859361$ Sign $1$ Analytic cond. $458372.$ Root an. cond. $5.10096$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 13·4-s − 102·13-s + 47·16-s − 222·19-s − 67·25-s − 220·31-s − 374·37-s − 838·43-s + 1.32e3·52-s − 1.33e3·61-s + 143·64-s + 1.89e3·67-s − 1.75e3·73-s + 2.88e3·76-s + 8·79-s − 3.01e3·97-s + 871·100-s − 2.04e3·103-s − 1.01e3·109-s − 1.38e3·121-s + 2.86e3·124-s + 127-s + 131-s + 137-s + 139-s + 4.86e3·148-s + 149-s + ⋯
 L(s)  = 1 − 1.62·4-s − 2.17·13-s + 0.734·16-s − 2.68·19-s − 0.535·25-s − 1.27·31-s − 1.66·37-s − 2.97·43-s + 3.53·52-s − 2.79·61-s + 0.279·64-s + 3.44·67-s − 2.80·73-s + 4.35·76-s + 0.0113·79-s − 3.15·97-s + 0.870·100-s − 1.95·103-s − 0.887·109-s − 1.04·121-s + 2.07·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 2.70·148-s + 0.000549·149-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$458372.$$ Root analytic conductor: $$5.10096$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{441} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## 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)$
bad3 $$1$$
7 $$1$$
good2$C_2^2 \wr C_2$ $$1 + 13 T^{2} + 61 p T^{4} + 13 p^{6} T^{6} + p^{12} T^{8}$$
5$C_2^2 \wr C_2$ $$1 + 67 T^{2} + 18428 T^{4} + 67 p^{6} T^{6} + p^{12} T^{8}$$
11$C_2^2 \wr C_2$ $$1 + 1387 T^{2} + 837200 T^{4} + 1387 p^{6} T^{6} + p^{12} T^{8}$$
13$D_{4}$ $$( 1 + 51 T + 4996 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
17$C_2^2 \wr C_2$ $$1 + 4256 T^{2} + 51629310 T^{4} + 4256 p^{6} T^{6} + p^{12} T^{8}$$
19$D_{4}$ $$( 1 + 111 T + 10960 T^{2} + 111 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
23$C_2^2 \wr C_2$ $$1 + 8216 T^{2} - 51412626 T^{4} + 8216 p^{6} T^{6} + p^{12} T^{8}$$
29$C_2^2 \wr C_2$ $$1 + 59371 T^{2} + 1802370044 T^{4} + 59371 p^{6} T^{6} + p^{12} T^{8}$$
31$D_{4}$ $$( 1 + 110 T + 1397 p T^{2} + 110 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
37$D_{4}$ $$( 1 + 187 T + 79892 T^{2} + 187 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
41$C_2^2 \wr C_2$ $$1 + 85540 T^{2} + 8435951270 T^{4} + 85540 p^{6} T^{6} + p^{12} T^{8}$$
43$D_{4}$ $$( 1 + 419 T + 167730 T^{2} + 419 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
47$C_2^2 \wr C_2$ $$1 + 279692 T^{2} + 41075202726 T^{4} + 279692 p^{6} T^{6} + p^{12} T^{8}$$
53$C_2^2 \wr C_2$ $$1 + 574307 T^{2} + 126700633692 T^{4} + 574307 p^{6} T^{6} + p^{12} T^{8}$$
59$C_2^2 \wr C_2$ $$1 + 615851 T^{2} + 176286397968 T^{4} + 615851 p^{6} T^{6} + p^{12} T^{8}$$
61$D_{4}$ $$( 1 + 666 T + 462754 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
67$D_{4}$ $$( 1 - 945 T + 807364 T^{2} - 945 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
71$C_2^2 \wr C_2$ $$1 + 874364 T^{2} + 392544633894 T^{4} + 874364 p^{6} T^{6} + p^{12} T^{8}$$
73$D_{4}$ $$( 1 + 875 T + 967076 T^{2} + 875 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
79$D_{4}$ $$( 1 - 4 T + 900969 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
83$C_2^2 \wr C_2$ $$1 + 1482323 T^{2} + 1051851285456 T^{4} + 1482323 p^{6} T^{6} + p^{12} T^{8}$$
89$C_2^2 \wr C_2$ $$1 + 627904 T^{2} + 1046910984158 T^{4} + 627904 p^{6} T^{6} + p^{12} T^{8}$$
97$D_{4}$ $$( 1 + 1505 T + 1992044 T^{2} + 1505 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$