# Properties

 Label 8-363e4-1.1-c3e4-0-5 Degree $8$ Conductor $17363069361$ Sign $1$ Analytic cond. $210421.$ Root an. cond. $4.62792$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s + 12·3-s − 3·4-s + 14·5-s − 12·6-s + 20·7-s − 19·8-s + 90·9-s − 14·10-s − 36·12-s + 32·13-s − 20·14-s + 168·15-s + 49·16-s + 92·17-s − 90·18-s − 34·19-s − 42·20-s + 240·21-s − 26·23-s − 228·24-s + 15·25-s − 32·26-s + 540·27-s − 60·28-s + 174·29-s − 168·30-s + ⋯
 L(s)  = 1 − 0.353·2-s + 2.30·3-s − 3/8·4-s + 1.25·5-s − 0.816·6-s + 1.07·7-s − 0.839·8-s + 10/3·9-s − 0.442·10-s − 0.866·12-s + 0.682·13-s − 0.381·14-s + 2.89·15-s + 0.765·16-s + 1.31·17-s − 1.17·18-s − 0.410·19-s − 0.469·20-s + 2.49·21-s − 0.235·23-s − 1.93·24-s + 3/25·25-s − 0.241·26-s + 3.84·27-s − 0.404·28-s + 1.11·29-s − 1.02·30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{4} \cdot 11^{8}$$ Sign: $1$ Analytic conductor: $$210421.$$ Root analytic conductor: $$4.62792$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$25.58370602$$ $$L(\frac12)$$ $$\approx$$ $$25.58370602$$ $$L(\frac{5}{2})$$ 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$C_1$ $$( 1 - p T )^{4}$$
11 $$1$$
good2$C_2 \wr S_4$ $$1 + T + p^{2} T^{2} + 13 p T^{3} + p^{3} T^{4} + 13 p^{4} T^{5} + p^{8} T^{6} + p^{9} T^{7} + p^{12} T^{8}$$
5$C_2 \wr S_4$ $$1 - 14 T + 181 T^{2} - 34 T^{3} - 3712 T^{4} - 34 p^{3} T^{5} + 181 p^{6} T^{6} - 14 p^{9} T^{7} + p^{12} T^{8}$$
7$C_2 \wr S_4$ $$1 - 20 T + 745 T^{2} - 250 p T^{3} + 151460 T^{4} - 250 p^{4} T^{5} + 745 p^{6} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8}$$
13$C_2 \wr S_4$ $$1 - 32 T + 6283 T^{2} - 172324 T^{3} + 19678376 T^{4} - 172324 p^{3} T^{5} + 6283 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8}$$
17$C_2 \wr S_4$ $$1 - 92 T + 10015 T^{2} - 1708 p^{2} T^{3} + 48261836 T^{4} - 1708 p^{5} T^{5} + 10015 p^{6} T^{6} - 92 p^{9} T^{7} + p^{12} T^{8}$$
19$C_2 \wr S_4$ $$1 + 34 T + 7789 T^{2} + 491934 T^{3} + 100192812 T^{4} + 491934 p^{3} T^{5} + 7789 p^{6} T^{6} + 34 p^{9} T^{7} + p^{12} T^{8}$$
23$C_2 \wr S_4$ $$1 + 26 T + 5800 T^{2} + 628762 T^{3} + 272602574 T^{4} + 628762 p^{3} T^{5} + 5800 p^{6} T^{6} + 26 p^{9} T^{7} + p^{12} T^{8}$$
29$C_2 \wr S_4$ $$1 - 6 p T + 89405 T^{2} - 10079334 T^{3} + 3059530596 T^{4} - 10079334 p^{3} T^{5} + 89405 p^{6} T^{6} - 6 p^{10} T^{7} + p^{12} T^{8}$$
31$C_2 \wr S_4$ $$1 - 422 T + 121621 T^{2} - 813586 p T^{3} + 4715438036 T^{4} - 813586 p^{4} T^{5} + 121621 p^{6} T^{6} - 422 p^{9} T^{7} + p^{12} T^{8}$$
37$C_2 \wr S_4$ $$1 - 14 p T + 222196 T^{2} - 69969108 T^{3} + 17042241381 T^{4} - 69969108 p^{3} T^{5} + 222196 p^{6} T^{6} - 14 p^{10} T^{7} + p^{12} T^{8}$$
41$C_2 \wr S_4$ $$1 - 428 T + 281623 T^{2} - 72996448 T^{3} + 27843065288 T^{4} - 72996448 p^{3} T^{5} + 281623 p^{6} T^{6} - 428 p^{9} T^{7} + p^{12} T^{8}$$
43$C_2 \wr S_4$ $$1 + 550 T + 356860 T^{2} + 122234838 T^{3} + 43628494230 T^{4} + 122234838 p^{3} T^{5} + 356860 p^{6} T^{6} + 550 p^{9} T^{7} + p^{12} T^{8}$$
47$C_2 \wr S_4$ $$1 - 556 T + 464992 T^{2} - 159161852 T^{3} + 73095550526 T^{4} - 159161852 p^{3} T^{5} + 464992 p^{6} T^{6} - 556 p^{9} T^{7} + p^{12} T^{8}$$
53$C_2 \wr S_4$ $$1 - 882 T + 753641 T^{2} - 356933250 T^{3} + 170508810204 T^{4} - 356933250 p^{3} T^{5} + 753641 p^{6} T^{6} - 882 p^{9} T^{7} + p^{12} T^{8}$$
59$C_2 \wr S_4$ $$1 + 158 T + 455704 T^{2} - 10644746 T^{3} + 99834013694 T^{4} - 10644746 p^{3} T^{5} + 455704 p^{6} T^{6} + 158 p^{9} T^{7} + p^{12} T^{8}$$
61$C_2 \wr S_4$ $$1 - 290 T + 780649 T^{2} - 160512114 T^{3} + 249052940940 T^{4} - 160512114 p^{3} T^{5} + 780649 p^{6} T^{6} - 290 p^{9} T^{7} + p^{12} T^{8}$$
67$C_2 \wr S_4$ $$1 - 992 T + 1325281 T^{2} - 870459746 T^{3} + 613021090972 T^{4} - 870459746 p^{3} T^{5} + 1325281 p^{6} T^{6} - 992 p^{9} T^{7} + p^{12} T^{8}$$
71$C_2 \wr S_4$ $$1 + 42 T + 203372 T^{2} - 102129390 T^{3} - 39343991994 T^{4} - 102129390 p^{3} T^{5} + 203372 p^{6} T^{6} + 42 p^{9} T^{7} + p^{12} T^{8}$$
73$C_2 \wr S_4$ $$1 - 1274 T + 1771225 T^{2} - 1360918918 T^{3} + 1080945673244 T^{4} - 1360918918 p^{3} T^{5} + 1771225 p^{6} T^{6} - 1274 p^{9} T^{7} + p^{12} T^{8}$$
79$C_2 \wr S_4$ $$1 - 362 T + 1421725 T^{2} - 271468310 T^{3} + 902647752772 T^{4} - 271468310 p^{3} T^{5} + 1421725 p^{6} T^{6} - 362 p^{9} T^{7} + p^{12} T^{8}$$
83$C_2 \wr S_4$ $$1 - 1500 T + 2217920 T^{2} - 1968382332 T^{3} + 1823815616910 T^{4} - 1968382332 p^{3} T^{5} + 2217920 p^{6} T^{6} - 1500 p^{9} T^{7} + p^{12} T^{8}$$
89$C_2 \wr S_4$ $$1 - 1428 T + 2335535 T^{2} - 2459628540 T^{3} + 2471453298972 T^{4} - 2459628540 p^{3} T^{5} + 2335535 p^{6} T^{6} - 1428 p^{9} T^{7} + p^{12} T^{8}$$
97$C_2 \wr S_4$ $$1 - 1052 T + 2038366 T^{2} - 1884713816 T^{3} + 2895539158591 T^{4} - 1884713816 p^{3} T^{5} + 2038366 p^{6} T^{6} - 1052 p^{9} T^{7} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$