# Properties

 Label 16-42e16-1.1-c2e8-0-0 Degree $16$ Conductor $9.375\times 10^{25}$ Sign $1$ Analytic cond. $2.84884\times 10^{13}$ Root an. cond. $6.93293$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 16·13-s − 40·19-s − 68·25-s + 40·31-s − 32·37-s − 32·43-s − 272·61-s − 168·67-s + 144·73-s − 232·79-s − 192·97-s − 328·103-s − 240·109-s − 368·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 760·169-s + 173-s + 179-s + 181-s + ⋯
 L(s)  = 1 + 1.23·13-s − 2.10·19-s − 2.71·25-s + 1.29·31-s − 0.864·37-s − 0.744·43-s − 4.45·61-s − 2.50·67-s + 1.97·73-s − 2.93·79-s − 1.97·97-s − 3.18·103-s − 2.20·109-s − 3.04·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4.49·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-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(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{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: $$2.84884\times 10^{13}$$ Root analytic conductor: $$6.93293$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1764} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.007738254147$$ $$L(\frac12)$$ $$\approx$$ $$0.007738254147$$ $$L(2)$$ 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 + 68 T^{2} + 466 p T^{4} + 70992 T^{6} + 1981811 T^{8} + 70992 p^{4} T^{10} + 466 p^{9} T^{12} + 68 p^{12} T^{14} + p^{16} T^{16}$$
11 $$1 + 368 T^{2} + 72734 T^{4} + 12294144 T^{6} + 1732934435 T^{8} + 12294144 p^{4} T^{10} + 72734 p^{8} T^{12} + 368 p^{12} T^{14} + p^{16} T^{16}$$
13 $$( 1 - 4 T + 230 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
17 $$1 + 452 T^{2} - 11014 T^{4} + 21820752 T^{6} + 22322409299 T^{8} + 21820752 p^{4} T^{10} - 11014 p^{8} T^{12} + 452 p^{12} T^{14} + p^{16} T^{16}$$
19 $$( 1 + 20 T - 170 T^{2} - 160 p T^{3} + 139 p^{2} T^{4} - 160 p^{3} T^{5} - 170 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
23 $$1 + 1808 T^{2} + 1913918 T^{4} + 1437837312 T^{6} + 842805507011 T^{8} + 1437837312 p^{4} T^{10} + 1913918 p^{8} T^{12} + 1808 p^{12} T^{14} + p^{16} T^{16}$$
29 $$( 1 + 112 T^{2} - 475102 T^{4} + 112 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
31 $$( 1 - 20 T - 1594 T^{2} - 1440 T^{3} + 2668115 T^{4} - 1440 p^{2} T^{5} - 1594 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
37 $$( 1 + 16 T - 1538 T^{2} - 15104 T^{3} + 993811 T^{4} - 15104 p^{2} T^{5} - 1538 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
41 $$( 1 - 6436 T^{2} + 15997974 T^{4} - 6436 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
43 $$( 1 + 8 T + 2706 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
47 $$1 + 2644 T^{2} + 709162 T^{4} - 9195271472 T^{6} - 18465676379501 T^{8} - 9195271472 p^{4} T^{10} + 709162 p^{8} T^{12} + 2644 p^{12} T^{14} + p^{16} T^{16}$$
53 $$( 1 + 3440 T^{2} + 3943119 T^{4} + 3440 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
59 $$1 + 6836 T^{2} + 19594250 T^{4} + 19837552464 T^{6} + 23311981839539 T^{8} + 19837552464 p^{4} T^{10} + 19594250 p^{8} T^{12} + 6836 p^{12} T^{14} + p^{16} T^{16}$$
61 $$( 1 + 136 T + 6458 T^{2} + 625056 T^{3} + 62243987 T^{4} + 625056 p^{2} T^{5} + 6458 p^{4} T^{6} + 136 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
67 $$( 1 + 84 T + 1802 T^{2} - 312816 T^{3} - 24220989 T^{4} - 312816 p^{2} T^{5} + 1802 p^{4} T^{6} + 84 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
71 $$( 1 - 4496 T^{2} + 27200834 T^{4} - 4496 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
73 $$( 1 - 72 T - 6742 T^{2} - 91296 T^{3} + 86205699 T^{4} - 91296 p^{2} T^{5} - 6742 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
79 $$( 1 + 116 T + 4778 T^{2} - 441264 T^{3} - 47621293 T^{4} - 441264 p^{2} T^{5} + 4778 p^{4} T^{6} + 116 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
83 $$( 1 - 8228 T^{2} + 33565286 T^{4} - 8228 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
89 $$1 - 17852 T^{2} + 125955514 T^{4} - 1200616765616 T^{6} + 13215928444091155 T^{8} - 1200616765616 p^{4} T^{10} + 125955514 p^{8} T^{12} - 17852 p^{12} T^{14} + p^{16} T^{16}$$
97 $$( 1 + 48 T + 11302 T^{2} + 48 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}$$