# Properties

 Degree 8 Conductor $2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s + 10·4-s + 4·5-s + 7-s + 20·8-s + 16·10-s + 3·11-s + 2·13-s + 4·14-s + 35·16-s + 2·17-s + 2·19-s + 40·20-s + 12·22-s + 12·23-s + 10·25-s + 8·26-s + 10·28-s − 2·29-s + 10·31-s + 56·32-s + 8·34-s + 4·35-s + 3·37-s + 8·38-s + 80·40-s − 6·43-s + ⋯
 L(s)  = 1 + 2.82·2-s + 5·4-s + 1.78·5-s + 0.377·7-s + 7.07·8-s + 5.05·10-s + 0.904·11-s + 0.554·13-s + 1.06·14-s + 35/4·16-s + 0.485·17-s + 0.458·19-s + 8.94·20-s + 2.55·22-s + 2.50·23-s + 2·25-s + 1.56·26-s + 1.88·28-s − 0.371·29-s + 1.79·31-s + 9.89·32-s + 1.37·34-s + 0.676·35-s + 0.493·37-s + 1.29·38-s + 12.6·40-s − 0.914·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{6030} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$ $$L(1)$$ $$\approx$$ $$200.6831135$$ $$L(\frac12)$$ $$\approx$$ $$200.6831135$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;67\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;3,\;5,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ $$( 1 - T )^{4}$$
3 $$1$$
5$C_1$ $$( 1 - T )^{4}$$
67$C_1$ $$( 1 + T )^{4}$$
good7$C_2 \wr S_4$ $$1 - T + 8 T^{2} - 9 T^{3} + 66 T^{4} - 9 p T^{5} + 8 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr S_4$ $$1 - 3 T + 24 T^{2} - 67 T^{3} + 398 T^{4} - 67 p T^{5} + 24 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr S_4$ $$1 - 2 T + 32 T^{2} - 50 T^{3} + 486 T^{4} - 50 p T^{5} + 32 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr S_4$ $$1 - 2 T + 36 T^{2} - 22 T^{3} + 678 T^{4} - 22 p T^{5} + 36 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr S_4$ $$1 - 2 T + 44 T^{2} - 34 T^{3} + 982 T^{4} - 34 p T^{5} + 44 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr S_4$ $$1 - 12 T + 100 T^{2} - 624 T^{3} + 3094 T^{4} - 624 p T^{5} + 100 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr S_4$ $$1 + 2 T + 64 T^{2} + 322 T^{3} + 1918 T^{4} + 322 p T^{5} + 64 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr S_4$ $$1 - 10 T + 140 T^{2} - 878 T^{3} + 6646 T^{4} - 878 p T^{5} + 140 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr S_4$ $$1 - 3 T + 122 T^{2} - 273 T^{3} + 6346 T^{4} - 273 p T^{5} + 122 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr S_4$ $$1 + 92 T^{2} - 16 T^{3} + 4358 T^{4} - 16 p T^{5} + 92 p^{2} T^{6} + p^{4} T^{8}$$
43$C_2 \wr S_4$ $$1 + 6 T + 92 T^{2} + 518 T^{3} + 6006 T^{4} + 518 p T^{5} + 92 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr S_4$ $$1 - 14 T + 116 T^{2} - 762 T^{3} + 4862 T^{4} - 762 p T^{5} + 116 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr S_4$ $$1 - 10 T + 168 T^{2} - 1230 T^{3} + 12542 T^{4} - 1230 p T^{5} + 168 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr S_4$ $$1 + 164 T^{2} - 16 T^{3} + 12566 T^{4} - 16 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2 \wr S_4$ $$1 - 9 T + 254 T^{2} - 1599 T^{3} + 23518 T^{4} - 1599 p T^{5} + 254 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr S_4$ $$1 - 9 T + 242 T^{2} - 1313 T^{3} + 22814 T^{4} - 1313 p T^{5} + 242 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr S_4$ $$1 + 14 T + 248 T^{2} + 2274 T^{3} + 25134 T^{4} + 2274 p T^{5} + 248 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr S_4$ $$1 - 12 T + 200 T^{2} - 992 T^{3} + 13686 T^{4} - 992 p T^{5} + 200 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr S_4$ $$1 - 25 T + 324 T^{2} - 3793 T^{3} + 40262 T^{4} - 3793 p T^{5} + 324 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr S_4$ $$1 - 9 T + 122 T^{2} - 783 T^{3} + 10410 T^{4} - 783 p T^{5} + 122 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr S_4$ $$1 + 3 T + 368 T^{2} + 841 T^{3} + 52686 T^{4} + 841 p T^{5} + 368 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}