# Properties

 Degree 32 Conductor $2^{32} \cdot 3^{32} \cdot 7^{32}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·13-s − 16-s + 28·25-s − 8·37-s − 16·61-s + 10·64-s − 8·73-s + 88·97-s + 24·109-s − 60·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 68·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
 L(s)  = 1 − 2.21·13-s − 1/4·16-s + 28/5·25-s − 1.31·37-s − 2.04·61-s + 5/4·64-s − 0.936·73-s + 8.93·97-s + 2.29·109-s − 5.45·121-s + 0.0887·127-s + 0.0873·131-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 − 5.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$32$$ $$N$$ = $$2^{32} \cdot 3^{32} \cdot 7^{32}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{1764} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(32,\ 2^{32} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )$$ $$L(1)$$ $$\approx$$ $$10.49628234$$ $$L(\frac12)$$ $$\approx$$ $$10.49628234$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 32. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 $$1 + T^{4} - 5 p T^{6} + p^{2} T^{8} - 5 p^{3} T^{10} + p^{4} T^{12} + p^{8} T^{16}$$
3 $$1$$
7 $$1$$
good5 $$( 1 - 14 T^{2} + 117 T^{4} - 698 T^{6} + 3576 T^{8} - 698 p^{2} T^{10} + 117 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
11 $$( 1 + 30 T^{2} + 757 T^{4} + 11666 T^{6} + 154360 T^{8} + 11666 p^{2} T^{10} + 757 p^{4} T^{12} + 30 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
13 $$( 1 + 2 T + 27 T^{2} + 38 T^{3} + 378 T^{4} + 38 p T^{5} + 27 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4}$$
17 $$( 1 - 76 T^{2} + 3096 T^{4} - 83940 T^{6} + 1656750 T^{8} - 83940 p^{2} T^{10} + 3096 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
19 $$( 1 - 34 T^{2} + 1041 T^{4} - 25110 T^{6} + 610032 T^{8} - 25110 p^{2} T^{10} + 1041 p^{4} T^{12} - 34 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
23 $$( 1 + 116 T^{2} + 6936 T^{4} + 269340 T^{6} + 7319790 T^{8} + 269340 p^{2} T^{10} + 6936 p^{4} T^{12} + 116 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
29 $$( 1 - 146 T^{2} + 10849 T^{4} - 18010 p T^{6} + 17802004 T^{8} - 18010 p^{3} T^{10} + 10849 p^{4} T^{12} - 146 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
31 $$( 1 - 132 T^{2} + 6430 T^{4} - 130264 T^{6} + 1736719 T^{8} - 130264 p^{2} T^{10} + 6430 p^{4} T^{12} - 132 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
37 $$( 1 + 2 T + 115 T^{2} + 282 T^{3} + 5758 T^{4} + 282 p T^{5} + 115 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4}$$
41 $$( 1 - 188 T^{2} + 15912 T^{4} - 846644 T^{6} + 36194574 T^{8} - 846644 p^{2} T^{10} + 15912 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
43 $$( 1 - 202 T^{2} + 17289 T^{4} - 881742 T^{6} + 37179072 T^{8} - 881742 p^{2} T^{10} + 17289 p^{4} T^{12} - 202 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
47 $$( 1 + 200 T^{2} + 22212 T^{4} + 1666392 T^{6} + 91084806 T^{8} + 1666392 p^{2} T^{10} + 22212 p^{4} T^{12} + 200 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
53 $$( 1 - 282 T^{2} + 34177 T^{4} - 2497906 T^{6} + 140957380 T^{8} - 2497906 p^{2} T^{10} + 34177 p^{4} T^{12} - 282 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
59 $$( 1 + 186 T^{2} + 18161 T^{4} + 1259962 T^{6} + 77631844 T^{8} + 1259962 p^{2} T^{10} + 18161 p^{4} T^{12} + 186 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
61 $$( 1 + 4 T + 162 T^{2} + 588 T^{3} + 12546 T^{4} + 588 p T^{5} + 162 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4}$$
67 $$( 1 - 254 T^{2} + 32925 T^{4} - 3201102 T^{6} + 246095804 T^{8} - 3201102 p^{2} T^{10} + 32925 p^{4} T^{12} - 254 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
71 $$( 1 + 360 T^{2} + 62596 T^{4} + 7074488 T^{6} + 580384198 T^{8} + 7074488 p^{2} T^{10} + 62596 p^{4} T^{12} + 360 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
73 $$( 1 + 2 T + 141 T^{2} - 58 T^{3} + 12336 T^{4} - 58 p T^{5} + 141 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4}$$
79 $$( 1 - 332 T^{2} + 56998 T^{4} - 6691240 T^{6} + 595616087 T^{8} - 6691240 p^{2} T^{10} + 56998 p^{4} T^{12} - 332 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
83 $$( 1 + 110 T^{2} + 7989 T^{4} + 466194 T^{6} + 66686616 T^{8} + 466194 p^{2} T^{10} + 7989 p^{4} T^{12} + 110 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
89 $$( 1 - 352 T^{2} + 72316 T^{4} - 10162848 T^{6} + 1047015622 T^{8} - 10162848 p^{2} T^{10} + 72316 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
97 $$( 1 - 22 T + 373 T^{2} - 3806 T^{3} + 41336 T^{4} - 3806 p T^{5} + 373 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}