# Properties

 Degree 16 Conductor $2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 12·3-s − 4·4-s + 78·9-s − 4·11-s − 48·12-s + 4·16-s + 84·17-s + 108·19-s + 12·23-s + 10·25-s + 360·27-s + 72·29-s − 132·31-s − 48·33-s − 312·36-s − 96·37-s − 112·43-s + 16·44-s − 24·47-s + 48·48-s + 78·49-s + 1.00e3·51-s + 32·53-s + 1.29e3·57-s + 132·59-s + 96·61-s + 16·64-s + ⋯
 L(s)  = 1 + 4·3-s − 4-s + 26/3·9-s − 0.363·11-s − 4·12-s + 1/4·16-s + 4.94·17-s + 5.68·19-s + 0.521·23-s + 2/5·25-s + 40/3·27-s + 2.48·29-s − 4.25·31-s − 1.45·33-s − 8.66·36-s − 2.59·37-s − 2.60·43-s + 4/11·44-s − 0.510·47-s + 48-s + 1.59·49-s + 19.7·51-s + 0.603·53-s + 22.7·57-s + 2.23·59-s + 1.57·61-s + 1/4·64-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{210} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )$$ $$L(\frac{3}{2})$$ $$\approx$$ $$51.7240$$ $$L(\frac12)$$ $$\approx$$ $$51.7240$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 16. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
3 $$( 1 - p T + p T^{2} )^{4}$$
5 $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
7 $$1 - 78 T^{2} + 107 p^{2} T^{4} - 78 p^{4} T^{6} + p^{8} T^{8}$$
good11 $$1 + 4 T - 168 T^{2} + 1928 T^{3} + 16250 T^{4} - 341700 T^{5} + 2353408 T^{6} + 32486668 T^{7} - 296735661 T^{8} + 32486668 p^{2} T^{9} + 2353408 p^{4} T^{10} - 341700 p^{6} T^{11} + 16250 p^{8} T^{12} + 1928 p^{10} T^{13} - 168 p^{12} T^{14} + 4 p^{14} T^{15} + p^{16} T^{16}$$
13 $$1 - 860 T^{2} + 357498 T^{4} - 95438800 T^{6} + 18541152803 T^{8} - 95438800 p^{4} T^{10} + 357498 p^{8} T^{12} - 860 p^{12} T^{14} + p^{16} T^{16}$$
17 $$1 - 84 T + 4208 T^{2} - 155904 T^{3} + 4746250 T^{4} - 123749124 T^{5} + 2818194464 T^{6} - 56792172564 T^{7} + 1020121281523 T^{8} - 56792172564 p^{2} T^{9} + 2818194464 p^{4} T^{10} - 123749124 p^{6} T^{11} + 4746250 p^{8} T^{12} - 155904 p^{10} T^{13} + 4208 p^{12} T^{14} - 84 p^{14} T^{15} + p^{16} T^{16}$$
19 $$1 - 108 T + 6142 T^{2} - 243432 T^{3} + 7548345 T^{4} - 198492408 T^{5} + 4660679966 T^{6} - 100240172052 T^{7} + 1986343374068 T^{8} - 100240172052 p^{2} T^{9} + 4660679966 p^{4} T^{10} - 198492408 p^{6} T^{11} + 7548345 p^{8} T^{12} - 243432 p^{10} T^{13} + 6142 p^{12} T^{14} - 108 p^{14} T^{15} + p^{16} T^{16}$$
23 $$1 - 12 T - 1512 T^{2} + 15144 T^{3} + 1340890 T^{4} - 9662580 T^{5} - 850652928 T^{6} + 2479269564 T^{7} + 454065720819 T^{8} + 2479269564 p^{2} T^{9} - 850652928 p^{4} T^{10} - 9662580 p^{6} T^{11} + 1340890 p^{8} T^{12} + 15144 p^{10} T^{13} - 1512 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16}$$
29 $$( 1 - 36 T + 2904 T^{2} - 82956 T^{3} + 3490070 T^{4} - 82956 p^{2} T^{5} + 2904 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
31 $$1 + 132 T + 11278 T^{2} + 722040 T^{3} + 38602473 T^{4} + 1778529960 T^{5} + 72524765390 T^{6} + 2637318955548 T^{7} + 86278260959732 T^{8} + 2637318955548 p^{2} T^{9} + 72524765390 p^{4} T^{10} + 1778529960 p^{6} T^{11} + 38602473 p^{8} T^{12} + 722040 p^{10} T^{13} + 11278 p^{12} T^{14} + 132 p^{14} T^{15} + p^{16} T^{16}$$
37 $$1 + 96 T + 30 p T^{2} - 46464 T^{3} + 7652401 T^{4} + 360085920 T^{5} - 3532027338 T^{6} + 170035213440 T^{7} + 29111429055396 T^{8} + 170035213440 p^{2} T^{9} - 3532027338 p^{4} T^{10} + 360085920 p^{6} T^{11} + 7652401 p^{8} T^{12} - 46464 p^{10} T^{13} + 30 p^{13} T^{14} + 96 p^{14} T^{15} + p^{16} T^{16}$$
41 $$1 - 5456 T^{2} + 486924 p T^{4} - 49821420976 T^{6} + 96897341176550 T^{8} - 49821420976 p^{4} T^{10} + 486924 p^{9} T^{12} - 5456 p^{12} T^{14} + p^{16} T^{16}$$
43 $$( 1 + 56 T + 5082 T^{2} + 180688 T^{3} + 11078435 T^{4} + 180688 p^{2} T^{5} + 5082 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
47 $$1 + 24 T + 3580 T^{2} + 81312 T^{3} + 3430986 T^{4} + 200307384 T^{5} + 115413136 p T^{6} + 942687331128 T^{7} + 25778360177363 T^{8} + 942687331128 p^{2} T^{9} + 115413136 p^{5} T^{10} + 200307384 p^{6} T^{11} + 3430986 p^{8} T^{12} + 81312 p^{10} T^{13} + 3580 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16}$$
53 $$1 - 32 T - 3572 T^{2} + 570944 T^{3} - 366070 T^{4} - 2018388320 T^{5} + 123180800432 T^{6} + 3809526158624 T^{7} - 445561676340941 T^{8} + 3809526158624 p^{2} T^{9} + 123180800432 p^{4} T^{10} - 2018388320 p^{6} T^{11} - 366070 p^{8} T^{12} + 570944 p^{10} T^{13} - 3572 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16}$$
59 $$1 - 132 T + 15632 T^{2} - 1296768 T^{3} + 88851370 T^{4} - 4629146772 T^{5} + 218386477856 T^{6} - 7743785317668 T^{7} + 393482793133843 T^{8} - 7743785317668 p^{2} T^{9} + 218386477856 p^{4} T^{10} - 4629146772 p^{6} T^{11} + 88851370 p^{8} T^{12} - 1296768 p^{10} T^{13} + 15632 p^{12} T^{14} - 132 p^{14} T^{15} + p^{16} T^{16}$$
61 $$1 - 96 T + 11380 T^{2} - 797568 T^{3} + 60840138 T^{4} - 3862327392 T^{5} + 197844384848 T^{6} - 13588740778464 T^{7} + 649379857320947 T^{8} - 13588740778464 p^{2} T^{9} + 197844384848 p^{4} T^{10} - 3862327392 p^{6} T^{11} + 60840138 p^{8} T^{12} - 797568 p^{10} T^{13} + 11380 p^{12} T^{14} - 96 p^{14} T^{15} + p^{16} T^{16}$$
67 $$1 + 120 T - 4522 T^{2} - 749040 T^{3} + 46360561 T^{4} + 4026690720 T^{5} - 240439443082 T^{6} - 2848051223400 T^{7} + 1781423144538724 T^{8} - 2848051223400 p^{2} T^{9} - 240439443082 p^{4} T^{10} + 4026690720 p^{6} T^{11} + 46360561 p^{8} T^{12} - 749040 p^{10} T^{13} - 4522 p^{12} T^{14} + 120 p^{14} T^{15} + p^{16} T^{16}$$
71 $$( 1 - 4 T + 13176 T^{2} - 338828 T^{3} + 79144886 T^{4} - 338828 p^{2} T^{5} + 13176 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
73 $$1 - 24 T + 9630 T^{2} - 226512 T^{3} + 47274361 T^{4} + 2709573216 T^{5} - 18252715698 T^{6} + 34150744827912 T^{7} - 1287593588654412 T^{8} + 34150744827912 p^{2} T^{9} - 18252715698 p^{4} T^{10} + 2709573216 p^{6} T^{11} + 47274361 p^{8} T^{12} - 226512 p^{10} T^{13} + 9630 p^{12} T^{14} - 24 p^{14} T^{15} + p^{16} T^{16}$$
79 $$1 - 12 T - 2418 T^{2} + 386472 T^{3} - 58917335 T^{4} - 324907032 T^{5} + 66125106126 T^{6} - 10581169939908 T^{7} + 1991860414252788 T^{8} - 10581169939908 p^{2} T^{9} + 66125106126 p^{4} T^{10} - 324907032 p^{6} T^{11} - 58917335 p^{8} T^{12} + 386472 p^{10} T^{13} - 2418 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16}$$
83 $$1 - 2384 T^{2} + 54393900 T^{4} - 170213772976 T^{6} + 3204332319622694 T^{8} - 170213772976 p^{4} T^{10} + 54393900 p^{8} T^{12} - 2384 p^{12} T^{14} + p^{16} T^{16}$$
89 $$1 - 492 T + 137248 T^{2} - 27827520 T^{3} + 4512312618 T^{4} - 615183153660 T^{5} + 72837035721440 T^{6} - 7634393359602348 T^{7} + 716618506839255347 T^{8} - 7634393359602348 p^{2} T^{9} + 72837035721440 p^{4} T^{10} - 615183153660 p^{6} T^{11} + 4512312618 p^{8} T^{12} - 27827520 p^{10} T^{13} + 137248 p^{12} T^{14} - 492 p^{14} T^{15} + p^{16} T^{16}$$
97 $$1 - 35880 T^{2} + 724155484 T^{4} - 9811041724440 T^{6} + 103976225413471686 T^{8} - 9811041724440 p^{4} T^{10} + 724155484 p^{8} T^{12} - 35880 p^{12} T^{14} + p^{16} T^{16}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}