Properties

 Label 16-33e16-1.1-c2e8-0-0 Degree $16$ Conductor $1.978\times 10^{24}$ Sign $1$ Analytic cond. $6.01040\times 10^{11}$ Root an. cond. $5.44730$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 + 16·4-s + 126·16-s + 168·25-s + 96·31-s + 16·37-s − 360·49-s + 720·64-s + 216·67-s + 352·97-s + 2.68e3·100-s + 1.25e3·103-s + 1.53e3·124-s + 127-s + 131-s + 137-s + 139-s + 256·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 332·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
 L(s)  = 1 + 4·4-s + 63/8·16-s + 6.71·25-s + 3.09·31-s + 0.432·37-s − 7.34·49-s + 45/4·64-s + 3.22·67-s + 3.62·97-s + 26.8·100-s + 12.1·103-s + 12.3·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.72·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.96·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$16$$ Conductor: $$3^{16} \cdot 11^{16}$$ Sign: $1$ Analytic conductor: $$6.01040\times 10^{11}$$ Root analytic conductor: $$5.44730$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1089} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 3^{16} \cdot 11^{16} ,\ ( \ : [1]^{8} ),\ 1 )$$

Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.1084115308$$ $$L(\frac12)$$ $$\approx$$ $$0.1084115308$$ $$L(2)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
11 $$1$$
good2 $$( 1 - p^{3} T^{2} + 33 T^{4} - p^{7} T^{6} + p^{8} T^{8} )^{2}$$
5 $$( 1 - 84 T^{2} + 2999 T^{4} - 84 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
7 $$( 1 + 180 T^{2} + 12842 T^{4} + 180 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
13 $$( 1 + 166 T^{2} + 10011 T^{4} + 166 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
17 $$( 1 - 164 T^{2} + 49551 T^{4} - 164 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
19 $$( 1 + 1188 T^{2} + 604838 T^{4} + 1188 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
23 $$( 1 + 240 T^{2} + 150722 T^{4} + 240 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
29 $$( 1 - 2012 T^{2} + 1998183 T^{4} - 2012 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
31 $$( 1 - 24 T + 1826 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
37 $$( 1 - 4 T + 1527 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
41 $$( 1 - 1316 T^{2} + 2228751 T^{4} - 1316 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
43 $$( 1 + 3300 T^{2} + 7805642 T^{4} + 3300 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
47 $$( 1 - 6852 T^{2} + 20770838 T^{4} - 6852 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
53 $$( 1 - 1300 T^{2} + 14880327 T^{4} - 1300 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
59 $$( 1 - 1720 T^{2} + 18675762 T^{4} - 1720 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
61 $$( 1 + 10788 T^{2} + 56624678 T^{4} + 10788 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
67 $$( 1 - 54 T + 3092 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
71 $$( 1 - 9240 T^{2} + 64218002 T^{4} - 9240 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
73 $$( 1 - 5820 T^{2} + 62668742 T^{4} - 5820 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
79 $$( 1 + 4348 T^{2} - 21917562 T^{4} + 4348 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
83 $$( 1 - 17828 T^{2} + 165802998 T^{4} - 17828 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
89 $$( 1 - 27468 T^{2} + 309749423 T^{4} - 27468 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
97 $$( 1 - 88 T + 20019 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$