# Properties

 Degree $8$ Conductor $3.060\times 10^{13}$ Sign $1$ Motivic weight $3$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 12·3-s + 90·9-s − 48·17-s + 192·19-s − 192·23-s − 88·25-s + 540·27-s + 96·29-s + 48·31-s + 256·37-s − 1.00e3·41-s + 112·43-s + 864·47-s − 576·51-s − 648·53-s + 2.30e3·57-s + 336·59-s − 960·61-s − 720·67-s − 2.30e3·69-s + 1.34e3·71-s − 672·73-s − 1.05e3·75-s + 1.98e3·79-s + 2.83e3·81-s + 3.12e3·83-s + 1.15e3·87-s + ⋯
 L(s)  = 1 + 2.30·3-s + 10/3·9-s − 0.684·17-s + 2.31·19-s − 1.74·23-s − 0.703·25-s + 3.84·27-s + 0.614·29-s + 0.278·31-s + 1.13·37-s − 3.83·41-s + 0.397·43-s + 2.68·47-s − 1.58·51-s − 1.67·53-s + 5.35·57-s + 0.741·59-s − 2.01·61-s − 1.31·67-s − 4.01·69-s + 2.24·71-s − 1.07·73-s − 1.62·75-s + 2.82·79-s + 35/9·81-s + 4.12·83-s + 1.41·87-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \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(2^{16} \cdot 3^{4} \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: $$2^{16} \cdot 3^{4} \cdot 7^{8}$$ Sign: $1$ Motivic weight: $$3$$ Character: induced by $\chi_{2352} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$26.34873475$$ $$L(\frac12)$$ $$\approx$$ $$26.34873475$$ $$L(\frac{5}{2})$$ 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)$
bad2 $$1$$
3$C_1$ $$( 1 - p T )^{4}$$
7 $$1$$
good5$C_2 \wr C_2\wr C_2$ $$1 + 88 T^{2} + 576 T^{3} + 18498 T^{4} + 576 p^{3} T^{5} + 88 p^{6} T^{6} + p^{12} T^{8}$$
11$C_2 \wr C_2\wr C_2$ $$1 + 108 p T^{2} - 82944 T^{3} + 352406 T^{4} - 82944 p^{3} T^{5} + 108 p^{7} T^{6} + p^{12} T^{8}$$
13$C_2 \wr C_2\wr C_2$ $$1 + 1696 T^{2} + 31104 T^{3} + 7806642 T^{4} + 31104 p^{3} T^{5} + 1696 p^{6} T^{6} + p^{12} T^{8}$$
17$C_2 \wr C_2\wr C_2$ $$1 + 48 T + 9880 T^{2} + 116976 T^{3} + 43940658 T^{4} + 116976 p^{3} T^{5} + 9880 p^{6} T^{6} + 48 p^{9} T^{7} + p^{12} T^{8}$$
19$C_2 \wr C_2\wr C_2$ $$1 - 192 T + 24156 T^{2} - 2400960 T^{3} + 195309110 T^{4} - 2400960 p^{3} T^{5} + 24156 p^{6} T^{6} - 192 p^{9} T^{7} + p^{12} T^{8}$$
23$C_2 \wr C_2\wr C_2$ $$1 + 192 T + 52596 T^{2} + 6099648 T^{3} + 943244678 T^{4} + 6099648 p^{3} T^{5} + 52596 p^{6} T^{6} + 192 p^{9} T^{7} + p^{12} T^{8}$$
29$C_2 \wr C_2\wr C_2$ $$1 - 96 T + 85860 T^{2} - 6375072 T^{3} + 3037151990 T^{4} - 6375072 p^{3} T^{5} + 85860 p^{6} T^{6} - 96 p^{9} T^{7} + p^{12} T^{8}$$
31$C_2 \wr C_2\wr C_2$ $$1 - 48 T + 75532 T^{2} + 1415952 T^{3} + 2535445158 T^{4} + 1415952 p^{3} T^{5} + 75532 p^{6} T^{6} - 48 p^{9} T^{7} + p^{12} T^{8}$$
37$C_2 \wr C_2\wr C_2$ $$1 - 256 T + 66772 T^{2} + 2259200 T^{3} + 153679222 T^{4} + 2259200 p^{3} T^{5} + 66772 p^{6} T^{6} - 256 p^{9} T^{7} + p^{12} T^{8}$$
41$C_2 \wr C_2\wr C_2$ $$1 + 1008 T + 609784 T^{2} + 250033968 T^{3} + 76165467474 T^{4} + 250033968 p^{3} T^{5} + 609784 p^{6} T^{6} + 1008 p^{9} T^{7} + p^{12} T^{8}$$
43$C_2 \wr C_2\wr C_2$ $$1 - 112 T + 246988 T^{2} - 11722480 T^{3} + 25842449782 T^{4} - 11722480 p^{3} T^{5} + 246988 p^{6} T^{6} - 112 p^{9} T^{7} + p^{12} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 - 864 T + 642220 T^{2} - 282465504 T^{3} + 110627505318 T^{4} - 282465504 p^{3} T^{5} + 642220 p^{6} T^{6} - 864 p^{9} T^{7} + p^{12} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 + 648 T + 386892 T^{2} + 114656472 T^{3} + 50365144694 T^{4} + 114656472 p^{3} T^{5} + 386892 p^{6} T^{6} + 648 p^{9} T^{7} + p^{12} T^{8}$$
59$C_2 \wr C_2\wr C_2$ $$1 - 336 T + 534364 T^{2} - 159356496 T^{3} + 154119406422 T^{4} - 159356496 p^{3} T^{5} + 534364 p^{6} T^{6} - 336 p^{9} T^{7} + p^{12} T^{8}$$
61$C_2 \wr C_2\wr C_2$ $$1 + 960 T + 577728 T^{2} + 153988800 T^{3} + 52569538418 T^{4} + 153988800 p^{3} T^{5} + 577728 p^{6} T^{6} + 960 p^{9} T^{7} + p^{12} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 + 720 T + 533740 T^{2} + 318964176 T^{3} + 154478344470 T^{4} + 318964176 p^{3} T^{5} + 533740 p^{6} T^{6} + 720 p^{9} T^{7} + p^{12} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 - 1344 T + 1892084 T^{2} - 1427151168 T^{3} + 1080205217862 T^{4} - 1427151168 p^{3} T^{5} + 1892084 p^{6} T^{6} - 1344 p^{9} T^{7} + p^{12} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 + 672 T + 1088640 T^{2} + 325175712 T^{3} + 453343664738 T^{4} + 325175712 p^{3} T^{5} + 1088640 p^{6} T^{6} + 672 p^{9} T^{7} + p^{12} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 - 1984 T + 1725436 T^{2} - 666213568 T^{3} + 202189693510 T^{4} - 666213568 p^{3} T^{5} + 1725436 p^{6} T^{6} - 1984 p^{9} T^{7} + p^{12} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 - 3120 T + 5531020 T^{2} - 6555475248 T^{3} + 5745505983510 T^{4} - 6555475248 p^{3} T^{5} + 5531020 p^{6} T^{6} - 3120 p^{9} T^{7} + p^{12} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 + 2160 T + 3343000 T^{2} + 3573514800 T^{3} + 3407992760850 T^{4} + 3573514800 p^{3} T^{5} + 3343000 p^{6} T^{6} + 2160 p^{9} T^{7} + p^{12} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 + 2016 T + 2898432 T^{2} + 2840274144 T^{3} + 3044116636418 T^{4} + 2840274144 p^{3} T^{5} + 2898432 p^{6} T^{6} + 2016 p^{9} T^{7} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$