# Properties

 Degree $16$ Conductor $6.993\times 10^{27}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·7-s + 8·19-s + 4·25-s − 8·31-s + 8·37-s + 12·49-s − 8·103-s − 56·109-s + 28·121-s + 127-s + 131-s − 64·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s − 32·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
 L(s)  = 1 − 3.02·7-s + 1.83·19-s + 4/5·25-s − 1.43·31-s + 1.31·37-s + 12/7·49-s − 0.788·103-s − 5.36·109-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.46·169-s + 0.0760·173-s − 2.41·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{32} \cdot 3^{24} \cdot 7^{8}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{3024} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{32} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.587239819$$ $$L(\frac12)$$ $$\approx$$ $$1.587239819$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$( 1 + 2 T + p T^{2} )^{4}$$
good5 $$( 1 - 2 T^{2} + 33 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
11 $$( 1 - 14 T^{2} + 129 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
13 $$( 1 - 16 T^{2} + 114 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
17 $$( 1 - 32 T^{2} + 762 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
19 $$( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4}$$
23 $$( 1 - 14 T^{2} - 351 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
29 $$( 1 + 8 T^{2} + 1050 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
31 $$( 1 + 2 T + 45 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4}$$
37 $$( 1 - T + p T^{2} )^{8}$$
41 $$( 1 - 50 T^{2} + 3825 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
43 $$( 1 - 64 T^{2} + 3570 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
47 $$( 1 + 8 T^{2} - 1398 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
53 $$( 1 - 4 T^{2} + 3030 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
59 $$( 1 + 128 T^{2} + 10410 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
61 $$( 1 - 110 T^{2} + p^{2} T^{4} )^{4}$$
67 $$( 1 - 136 T^{2} + 11010 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
71 $$( 1 - 266 T^{2} + 27753 T^{4} - 266 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
73 $$( 1 - 88 T^{2} + 2226 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
79 $$( 1 - 208 T^{2} + 22146 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
83 $$( 1 + 152 T^{2} + 13722 T^{4} + 152 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
89 $$( 1 - 146 T^{2} + 13233 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
97 $$( 1 - 172 T^{2} + 21606 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$