# Properties

 Degree $16$ Conductor $2.180\times 10^{25}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·4-s + 4·5-s − 4·9-s + 10·16-s − 24·19-s − 16·20-s + 10·25-s + 16·29-s + 32·31-s + 16·36-s + 24·41-s − 16·45-s − 40·59-s + 24·61-s − 20·64-s − 40·71-s + 96·76-s + 16·79-s + 40·80-s + 10·81-s − 88·89-s − 96·95-s − 40·100-s + 48·101-s − 24·109-s − 64·116-s − 52·121-s + ⋯
 L(s)  = 1 − 2·4-s + 1.78·5-s − 4/3·9-s + 5/2·16-s − 5.50·19-s − 3.57·20-s + 2·25-s + 2.97·29-s + 5.74·31-s + 8/3·36-s + 3.74·41-s − 2.38·45-s − 5.20·59-s + 3.07·61-s − 5/2·64-s − 4.74·71-s + 11.0·76-s + 1.80·79-s + 4.47·80-s + 10/9·81-s − 9.32·89-s − 9.84·95-s − 4·100-s + 4.77·101-s − 2.29·109-s − 5.94·116-s − 4.72·121-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{16}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1470} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4177618022$$ $$L(\frac12)$$ $$\approx$$ $$0.4177618022$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 + T^{2} )^{4}$$
3 $$( 1 + T^{2} )^{4}$$
5 $$1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
7 $$1$$
good11 $$( 1 + 26 T^{2} - 16 T^{3} + 362 T^{4} - 16 p T^{5} + 26 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
13 $$1 - 4 p T^{2} + 1320 T^{4} - 23228 T^{6} + 329390 T^{8} - 23228 p^{2} T^{10} + 1320 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16}$$
17 $$1 - 56 T^{2} + 1268 T^{4} - 11176 T^{6} + 45670 T^{8} - 11176 p^{2} T^{10} + 1268 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16}$$
19 $$( 1 + 12 T + 88 T^{2} + 460 T^{3} + 2174 T^{4} + 460 p T^{5} + 88 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
23 $$1 - 96 T^{2} + 4220 T^{4} - 116640 T^{6} + 2693446 T^{8} - 116640 p^{2} T^{10} + 4220 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16}$$
29 $$( 1 - 8 T + 50 T^{2} - 360 T^{3} + 2786 T^{4} - 360 p T^{5} + 50 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
31 $$( 1 - 16 T + 202 T^{2} - 1616 T^{3} + 10666 T^{4} - 1616 p T^{5} + 202 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
37 $$1 - 4 p T^{2} + 9560 T^{4} - 370620 T^{6} + 12535566 T^{8} - 370620 p^{2} T^{10} + 9560 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16}$$
41 $$( 1 - 12 T + 190 T^{2} - 1356 T^{3} + 11842 T^{4} - 1356 p T^{5} + 190 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
43 $$1 - 108 T^{2} + 7400 T^{4} - 450340 T^{6} + 22241646 T^{8} - 450340 p^{2} T^{10} + 7400 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16}$$
47 $$1 - 172 T^{2} + 15464 T^{4} - 948164 T^{6} + 47811790 T^{8} - 948164 p^{2} T^{10} + 15464 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16}$$
53 $$1 - 240 T^{2} + 25532 T^{4} - 1715600 T^{6} + 94098918 T^{8} - 1715600 p^{2} T^{10} + 25532 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16}$$
59 $$( 1 + 20 T + 288 T^{2} + 2804 T^{3} + 24014 T^{4} + 2804 p T^{5} + 288 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
61 $$( 1 - 12 T + 118 T^{2} - 684 T^{3} + 6306 T^{4} - 684 p T^{5} + 118 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
67 $$1 - 492 T^{2} + 108392 T^{4} - 13982820 T^{6} + 1156204078 T^{8} - 13982820 p^{2} T^{10} + 108392 p^{4} T^{12} - 492 p^{6} T^{14} + p^{8} T^{16}$$
71 $$( 1 + 20 T + 248 T^{2} + 2788 T^{3} + 28078 T^{4} + 2788 p T^{5} + 248 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
73 $$1 - 268 T^{2} + 30568 T^{4} - 2051940 T^{6} + 125567630 T^{8} - 2051940 p^{2} T^{10} + 30568 p^{4} T^{12} - 268 p^{6} T^{14} + p^{8} T^{16}$$
79 $$( 1 - 8 T + 140 T^{2} - 104 T^{3} + 6054 T^{4} - 104 p T^{5} + 140 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
83 $$1 - 424 T^{2} + 80348 T^{4} - 9509080 T^{6} + 861447654 T^{8} - 9509080 p^{2} T^{10} + 80348 p^{4} T^{12} - 424 p^{6} T^{14} + p^{8} T^{16}$$
89 $$( 1 + 44 T + 1014 T^{2} + 15436 T^{3} + 169698 T^{4} + 15436 p T^{5} + 1014 p^{2} T^{6} + 44 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
97 $$1 - 348 T^{2} + 81160 T^{4} - 12144532 T^{6} + 1394593550 T^{8} - 12144532 p^{2} T^{10} + 81160 p^{4} T^{12} - 348 p^{6} T^{14} + p^{8} T^{16}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$