# Properties

 Label 16-42e16-1.1-c3e8-0-2 Degree $16$ Conductor $9.375\times 10^{25}$ Sign $1$ Analytic cond. $1.37694\times 10^{16}$ Root an. cond. $10.2019$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 276·25-s − 312·37-s + 1.18e3·43-s + 1.04e3·67-s − 2.65e3·79-s + 2.60e3·109-s − 780·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.03e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
 L(s)  = 1 + 2.20·25-s − 1.38·37-s + 4.19·43-s + 1.89·67-s − 3.78·79-s + 2.28·109-s − 0.586·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4.70·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{16} \cdot 3^{16} \cdot 7^{16}$$ Sign: $1$ Analytic conductor: $$1.37694\times 10^{16}$$ Root analytic conductor: $$10.2019$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1764} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$54.90911559$$ $$L(\frac12)$$ $$\approx$$ $$54.90911559$$ $$L(\frac{5}{2})$$ 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$$
good5 $$( 1 - 138 T^{2} + 3419 T^{4} - 138 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
11 $$( 1 + 390 T^{2} - 1619461 T^{4} + 390 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
13 $$( 1 - 2582 T^{2} + p^{6} T^{4} )^{4}$$
17 $$( 1 - 754 T^{2} - 23569053 T^{4} - 754 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
19 $$( 1 - 6742 T^{2} - 1591317 T^{4} - 6742 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
23 $$( 1 + 3134 T^{2} - 138213933 T^{4} + 3134 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
29 $$( 1 + 36570 T^{2} + p^{6} T^{4} )^{4}$$
31 $$( 1 - 52606 T^{2} + 1879887555 T^{4} - 52606 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
37 $$( 1 + 78 T - 44569 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
41 $$( 1 - 32510 T^{2} + p^{6} T^{4} )^{4}$$
43 $$( 1 - 148 T + p^{3} T^{2} )^{8}$$
47 $$( 1 + 9186 T^{2} - 10694832733 T^{4} + 9186 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
53 $$( 1 - 285546 T^{2} + 59372156987 T^{4} - 285546 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
59 $$( 1 - 107910 T^{2} - 30535965541 T^{4} - 107910 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
61 $$( 1 - 112138 T^{2} - 38945443317 T^{4} - 112138 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
67 $$( 1 - 260 T - 233163 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
71 $$( 1 + 200034 T^{2} + p^{6} T^{4} )^{4}$$
73 $$( 1 - 331570 T^{2} - 41395561389 T^{4} - 331570 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
79 $$( 1 + 664 T - 52143 T^{2} + 664 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
83 $$( 1 + 1127446 T^{2} + p^{6} T^{4} )^{4}$$
89 $$( 1 - 638370 T^{2} - 89465034061 T^{4} - 638370 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
97 $$( 1 + 458050 T^{2} + p^{6} T^{4} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$