# Properties

 Label 8-2352e4-1.1-c3e4-0-1 Degree $8$ Conductor $3.060\times 10^{13}$ Sign $1$ Analytic cond. $3.70863\times 10^{8}$ Root an. cond. $11.7801$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 12·3-s − 8·5-s + 90·9-s + 40·11-s − 48·13-s + 96·15-s + 152·17-s + 224·19-s + 8·23-s − 232·25-s − 540·27-s − 144·29-s + 400·31-s − 480·33-s − 304·37-s + 576·39-s + 152·41-s − 160·43-s − 720·45-s + 544·47-s − 1.82e3·51-s − 1.32e3·53-s − 320·55-s − 2.68e3·57-s + 1.04e3·59-s − 896·61-s + 384·65-s + ⋯
 L(s)  = 1 − 2.30·3-s − 0.715·5-s + 10/3·9-s + 1.09·11-s − 1.02·13-s + 1.65·15-s + 2.16·17-s + 2.70·19-s + 0.0725·23-s − 1.85·25-s − 3.84·27-s − 0.922·29-s + 2.31·31-s − 2.53·33-s − 1.35·37-s + 2.36·39-s + 0.578·41-s − 0.567·43-s − 2.38·45-s + 1.68·47-s − 5.00·51-s − 3.42·53-s − 0.784·55-s − 6.24·57-s + 2.29·59-s − 1.88·61-s + 0.732·65-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: $1$ Analytic conductor: $$3.70863\times 10^{8}$$ Root analytic conductor: $$11.7801$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes 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

 $$L(2)$$ $$\approx$$ $$0.3792958004$$ $$L(\frac12)$$ $$\approx$$ $$0.3792958004$$ $$L(\frac{5}{2})$$ 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$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 296 T^{2} + 2312 T^{3} + 48194 T^{4} + 2312 p^{3} T^{5} + 296 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8}$$
11$C_2 \wr C_2\wr C_2$ $$1 - 40 T + 2404 T^{2} - 79752 T^{3} + 4895510 T^{4} - 79752 p^{3} T^{5} + 2404 p^{6} T^{6} - 40 p^{9} T^{7} + p^{12} T^{8}$$
13$C_2 \wr C_2\wr C_2$ $$1 + 48 T + 4080 T^{2} + 163056 T^{3} + 7581746 T^{4} + 163056 p^{3} T^{5} + 4080 p^{6} T^{6} + 48 p^{9} T^{7} + p^{12} T^{8}$$
17$C_2 \wr C_2\wr C_2$ $$1 - 152 T + 15832 T^{2} - 1323320 T^{3} + 98297586 T^{4} - 1323320 p^{3} T^{5} + 15832 p^{6} T^{6} - 152 p^{9} T^{7} + p^{12} T^{8}$$
19$C_2 \wr C_2\wr C_2$ $$1 - 224 T + 42396 T^{2} - 4864224 T^{3} + 486114742 T^{4} - 4864224 p^{3} T^{5} + 42396 p^{6} T^{6} - 224 p^{9} T^{7} + p^{12} T^{8}$$
23$C_2 \wr C_2\wr C_2$ $$1 - 8 T + 9780 T^{2} - 1509576 T^{3} + 54064134 T^{4} - 1509576 p^{3} T^{5} + 9780 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8}$$
29$C_2 \wr C_2\wr C_2$ $$1 + 144 T + 70628 T^{2} + 7042032 T^{3} + 2292651510 T^{4} + 7042032 p^{3} T^{5} + 70628 p^{6} T^{6} + 144 p^{9} T^{7} + p^{12} T^{8}$$
31$C_2 \wr C_2\wr C_2$ $$1 - 400 T + 171340 T^{2} - 38254288 T^{3} + 8472479270 T^{4} - 38254288 p^{3} T^{5} + 171340 p^{6} T^{6} - 400 p^{9} T^{7} + p^{12} T^{8}$$
37$C_2 \wr C_2\wr C_2$ $$1 + 304 T + 169172 T^{2} + 40635984 T^{3} + 11843462518 T^{4} + 40635984 p^{3} T^{5} + 169172 p^{6} T^{6} + 304 p^{9} T^{7} + p^{12} T^{8}$$
41$C_2 \wr C_2\wr C_2$ $$1 - 152 T + 250232 T^{2} - 28847096 T^{3} + 25157924114 T^{4} - 28847096 p^{3} T^{5} + 250232 p^{6} T^{6} - 152 p^{9} T^{7} + p^{12} T^{8}$$
43$C_2 \wr C_2\wr C_2$ $$1 + 160 T + 147340 T^{2} + 16282528 T^{3} + 15975380470 T^{4} + 16282528 p^{3} T^{5} + 147340 p^{6} T^{6} + 160 p^{9} T^{7} + p^{12} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 - 544 T + 378668 T^{2} - 154041760 T^{3} + 58194085286 T^{4} - 154041760 p^{3} T^{5} + 378668 p^{6} T^{6} - 544 p^{9} T^{7} + p^{12} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 + 1320 T + 959116 T^{2} + 462674104 T^{3} + 189646882294 T^{4} + 462674104 p^{3} T^{5} + 959116 p^{6} T^{6} + 1320 p^{9} T^{7} + p^{12} T^{8}$$
59$C_2 \wr C_2\wr C_2$ $$1 - 1040 T + 962396 T^{2} - 555546128 T^{3} + 297086579030 T^{4} - 555546128 p^{3} T^{5} + 962396 p^{6} T^{6} - 1040 p^{9} T^{7} + p^{12} T^{8}$$
61$C_2 \wr C_2\wr C_2$ $$1 + 896 T + 1188912 T^{2} + 644590848 T^{3} + 437823850738 T^{4} + 644590848 p^{3} T^{5} + 1188912 p^{6} T^{6} + 896 p^{9} T^{7} + p^{12} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 - 416 T + 995500 T^{2} - 335375264 T^{3} + 430142403478 T^{4} - 335375264 p^{3} T^{5} + 995500 p^{6} T^{6} - 416 p^{9} T^{7} + p^{12} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 + 248 T + 996916 T^{2} + 246307512 T^{3} + 457673544134 T^{4} + 246307512 p^{3} T^{5} + 996916 p^{6} T^{6} + 248 p^{9} T^{7} + p^{12} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 - 752 T + 1272832 T^{2} - 641462384 T^{3} + 655763261282 T^{4} - 641462384 p^{3} T^{5} + 1272832 p^{6} T^{6} - 752 p^{9} T^{7} + p^{12} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 + 864 T + 1864956 T^{2} + 1235423200 T^{3} + 1357772054598 T^{4} + 1235423200 p^{3} T^{5} + 1864956 p^{6} T^{6} + 864 p^{9} T^{7} + p^{12} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 - 1456 T + 2297228 T^{2} - 2130405040 T^{3} + 2028397026326 T^{4} - 2130405040 p^{3} T^{5} + 2297228 p^{6} T^{6} - 1456 p^{9} T^{7} + p^{12} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 + 2936 T + 5403960 T^{2} + 6890733528 T^{3} + 6651967591762 T^{4} + 6890733528 p^{3} T^{5} + 5403960 p^{6} T^{6} + 2936 p^{9} T^{7} + p^{12} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 + 144 T + 2591968 T^{2} + 824150160 T^{3} + 3033197727234 T^{4} + 824150160 p^{3} T^{5} + 2591968 p^{6} T^{6} + 144 p^{9} T^{7} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$