# Properties

 Label 8-7800e4-1.1-c1e4-0-0 Degree $8$ Conductor $3.702\times 10^{15}$ Sign $1$ Analytic cond. $1.50482\times 10^{7}$ Root an. cond. $7.89197$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·3-s − 3·7-s + 10·9-s + 5·11-s − 4·13-s − 7·17-s + 3·19-s + 12·21-s − 8·23-s − 20·27-s + 2·29-s + 5·31-s − 20·33-s + 37-s + 16·39-s + 7·41-s − 6·43-s − 8·49-s + 28·51-s − 6·53-s − 12·57-s − 9·59-s + 19·61-s − 30·63-s + 4·67-s + 32·69-s + 19·71-s + ⋯
 L(s)  = 1 − 2.30·3-s − 1.13·7-s + 10/3·9-s + 1.50·11-s − 1.10·13-s − 1.69·17-s + 0.688·19-s + 2.61·21-s − 1.66·23-s − 3.84·27-s + 0.371·29-s + 0.898·31-s − 3.48·33-s + 0.164·37-s + 2.56·39-s + 1.09·41-s − 0.914·43-s − 8/7·49-s + 3.92·51-s − 0.824·53-s − 1.58·57-s − 1.17·59-s + 2.43·61-s − 3.77·63-s + 0.488·67-s + 3.85·69-s + 2.25·71-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$1.50482\times 10^{7}$$ Root analytic conductor: $$7.89197$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{7800} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.008627701$$ $$L(\frac12)$$ $$\approx$$ $$1.008627701$$ $$L(\frac{3}{2})$$ 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 $$1$$
3$C_1$ $$( 1 + T )^{4}$$
5 $$1$$
13$C_1$ $$( 1 + T )^{4}$$
good7$C_2 \wr S_4$ $$1 + 3 T + 17 T^{2} + 24 T^{3} + 120 T^{4} + 24 p T^{5} + 17 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr S_4$ $$1 - 5 T + 15 T^{2} - 6 T^{3} + 16 T^{4} - 6 p T^{5} + 15 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr S_4$ $$1 + 7 T + 39 T^{2} + 186 T^{3} + 964 T^{4} + 186 p T^{5} + 39 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr S_4$ $$1 - 3 T + 34 T^{2} - 27 T^{3} + 606 T^{4} - 27 p T^{5} + 34 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr S_4$ $$1 + 8 T + 84 T^{2} + 420 T^{3} + 2614 T^{4} + 420 p T^{5} + 84 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr S_4$ $$1 - 2 T + 60 T^{2} + 36 T^{3} + 1741 T^{4} + 36 p T^{5} + 60 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr S_4$ $$1 - 5 T + 79 T^{2} - 158 T^{3} + 2548 T^{4} - 158 p T^{5} + 79 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr S_4$ $$1 - T + 2 p T^{2} - 123 T^{3} + 3374 T^{4} - 123 p T^{5} + 2 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr S_4$ $$1 - 7 T + 72 T^{2} + 99 T^{3} + 298 T^{4} + 99 p T^{5} + 72 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr S_4$ $$1 + 6 T + 80 T^{2} + 714 T^{3} + 3174 T^{4} + 714 p T^{5} + 80 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
47$S_4\times C_2$ $$1 + 60 T^{2} - 18 T^{3} + 5171 T^{4} - 18 p T^{5} + 60 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2 \wr S_4$ $$1 + 6 T - 12 T^{2} - 24 T^{3} + 3257 T^{4} - 24 p T^{5} - 12 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr S_4$ $$1 + 9 T + 107 T^{2} + 576 T^{3} + 6690 T^{4} + 576 p T^{5} + 107 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr S_4$ $$1 - 19 T + 319 T^{2} - 3286 T^{3} + 30796 T^{4} - 3286 p T^{5} + 319 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr S_4$ $$1 - 4 T + 142 T^{2} - 382 T^{3} + 12817 T^{4} - 382 p T^{5} + 142 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr S_4$ $$1 - 19 T + 374 T^{2} - 3967 T^{3} + 42262 T^{4} - 3967 p T^{5} + 374 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr S_4$ $$1 - 6 T + 16 T^{2} - 486 T^{3} + 11694 T^{4} - 486 p T^{5} + 16 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr S_4$ $$1 + 9 T + 272 T^{2} + 1893 T^{3} + 30882 T^{4} + 1893 p T^{5} + 272 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr S_4$ $$1 + 21 T + 483 T^{2} + 5676 T^{3} + 66866 T^{4} + 5676 p T^{5} + 483 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr S_4$ $$1 - 14 T + 276 T^{2} - 3402 T^{3} + 34726 T^{4} - 3402 p T^{5} + 276 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr S_4$ $$1 - 6 T + 344 T^{2} - 1434 T^{3} + 47598 T^{4} - 1434 p T^{5} + 344 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$