# Properties

 Degree 20 Conductor $2^{10} \cdot 3^{10} \cdot 59^{10}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 10·2-s − 3-s + 55·4-s + 10·6-s − 2·7-s − 220·8-s + 2·9-s + 4·11-s − 55·12-s + 20·14-s + 715·16-s − 20·18-s − 6·19-s + 2·21-s − 40·22-s − 8·23-s + 220·24-s + 27·25-s − 5·27-s − 110·28-s − 2.00e3·32-s − 4·33-s + 110·36-s + 60·38-s − 20·42-s + 220·44-s + 80·46-s + ⋯
 L(s)  = 1 − 7.07·2-s − 0.577·3-s + 55/2·4-s + 4.08·6-s − 0.755·7-s − 77.7·8-s + 2/3·9-s + 1.20·11-s − 15.8·12-s + 5.34·14-s + 178.·16-s − 4.71·18-s − 1.37·19-s + 0.436·21-s − 8.52·22-s − 1.66·23-s + 44.9·24-s + 27/5·25-s − 0.962·27-s − 20.7·28-s − 353.·32-s − 0.696·33-s + 55/3·36-s + 9.73·38-s − 3.08·42-s + 33.1·44-s + 11.7·46-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 59^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 59^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$20$$ $$N$$ = $$2^{10} \cdot 3^{10} \cdot 59^{10}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{354} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(20,\ 2^{10} \cdot 3^{10} \cdot 59^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )$ $L(1)$ $\approx$ $0.0181330$ $L(\frac12)$ $\approx$ $0.0181330$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;59\}$, $$F_p(T)$$ is a polynomial of degree 20. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 19.
$p$$F_p(T)$
bad2 $$( 1 + T )^{10}$$
3 $$1 + T - T^{2} + 2 T^{3} + 2 T^{4} - 14 T^{5} + 2 p T^{6} + 2 p^{2} T^{7} - p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10}$$
59 $$1 - 20 T + 71 T^{2} + 896 T^{3} - 6134 T^{4} + 9960 T^{5} - 6134 p T^{6} + 896 p^{2} T^{7} + 71 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10}$$
good5 $$1 - 27 T^{2} + 79 p T^{4} - 3938 T^{6} + 29068 T^{8} - 165022 T^{10} + 29068 p^{2} T^{12} - 3938 p^{4} T^{14} + 79 p^{7} T^{16} - 27 p^{8} T^{18} + p^{10} T^{20}$$
7 $$( 1 + T + 16 T^{2} + 17 T^{3} + 179 T^{4} + 148 T^{5} + 179 p T^{6} + 17 p^{2} T^{7} + 16 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2}$$
11 $$( 1 - 2 T + 30 T^{2} - 34 T^{3} + 439 T^{4} - 372 T^{5} + 439 p T^{6} - 34 p^{2} T^{7} + 30 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2}$$
13 $$1 - 4 p T^{2} + 88 p T^{4} - 10254 T^{6} - 57459 T^{8} + 2166640 T^{10} - 57459 p^{2} T^{12} - 10254 p^{4} T^{14} + 88 p^{7} T^{16} - 4 p^{9} T^{18} + p^{10} T^{20}$$
17 $$1 - 108 T^{2} + 5426 T^{4} - 171278 T^{6} + 3938173 T^{8} - 73013740 T^{10} + 3938173 p^{2} T^{12} - 171278 p^{4} T^{14} + 5426 p^{6} T^{16} - 108 p^{8} T^{18} + p^{10} T^{20}$$
19 $$( 1 + 3 T + 77 T^{2} + 178 T^{3} + 2608 T^{4} + 4670 T^{5} + 2608 p T^{6} + 178 p^{2} T^{7} + 77 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2}$$
23 $$( 1 + 4 T + 29 T^{2} + 188 T^{3} + 1084 T^{4} + 2688 T^{5} + 1084 p T^{6} + 188 p^{2} T^{7} + 29 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2}$$
29 $$1 - 155 T^{2} + 12907 T^{4} - 729682 T^{6} + 30825980 T^{8} - 1008918398 T^{10} + 30825980 p^{2} T^{12} - 729682 p^{4} T^{14} + 12907 p^{6} T^{16} - 155 p^{8} T^{18} + p^{10} T^{20}$$
31 $$1 - 66 T^{2} + 2027 T^{4} - 68420 T^{6} + 2683060 T^{8} - 85000756 T^{10} + 2683060 p^{2} T^{12} - 68420 p^{4} T^{14} + 2027 p^{6} T^{16} - 66 p^{8} T^{18} + p^{10} T^{20}$$
37 $$1 - 188 T^{2} + 14592 T^{4} - 563614 T^{6} + 9281381 T^{8} - 55855104 T^{10} + 9281381 p^{2} T^{12} - 563614 p^{4} T^{14} + 14592 p^{6} T^{16} - 188 p^{8} T^{18} + p^{10} T^{20}$$
41 $$1 - 209 T^{2} + 23428 T^{4} - 1817599 T^{6} + 106670111 T^{8} - 4914350696 T^{10} + 106670111 p^{2} T^{12} - 1817599 p^{4} T^{14} + 23428 p^{6} T^{16} - 209 p^{8} T^{18} + p^{10} T^{20}$$
43 $$1 - 124 T^{2} + 8986 T^{4} - 558234 T^{6} + 29482965 T^{8} - 1340267732 T^{10} + 29482965 p^{2} T^{12} - 558234 p^{4} T^{14} + 8986 p^{6} T^{16} - 124 p^{8} T^{18} + p^{10} T^{20}$$
47 $$( 1 + 149 T^{2} + 36 T^{3} + 11260 T^{4} + 2520 T^{5} + 11260 p T^{6} + 36 p^{2} T^{7} + 149 p^{3} T^{8} + p^{5} T^{10} )^{2}$$
53 $$1 - 355 T^{2} + 60783 T^{4} - 6738486 T^{6} + 540355392 T^{8} - 32753382654 T^{10} + 540355392 p^{2} T^{12} - 6738486 p^{4} T^{14} + 60783 p^{6} T^{16} - 355 p^{8} T^{18} + p^{10} T^{20}$$
61 $$1 - 226 T^{2} + 28531 T^{4} - 2771220 T^{6} + 216253284 T^{8} - 14141970452 T^{10} + 216253284 p^{2} T^{12} - 2771220 p^{4} T^{14} + 28531 p^{6} T^{16} - 226 p^{8} T^{18} + p^{10} T^{20}$$
67 $$1 - 226 T^{2} + 24469 T^{4} - 2099544 T^{6} + 180779250 T^{8} - 13661508812 T^{10} + 180779250 p^{2} T^{12} - 2099544 p^{4} T^{14} + 24469 p^{6} T^{16} - 226 p^{8} T^{18} + p^{10} T^{20}$$
71 $$1 - 404 T^{2} + 85792 T^{4} - 12232858 T^{6} + 1288692677 T^{8} - 103955631344 T^{10} + 1288692677 p^{2} T^{12} - 12232858 p^{4} T^{14} + 85792 p^{6} T^{16} - 404 p^{8} T^{18} + p^{10} T^{20}$$
73 $$1 - 234 T^{2} + 40625 T^{4} - 4874480 T^{6} + 488701726 T^{8} - 38421788236 T^{10} + 488701726 p^{2} T^{12} - 4874480 p^{4} T^{14} + 40625 p^{6} T^{16} - 234 p^{8} T^{18} + p^{10} T^{20}$$
79 $$( 1 - 3 T + 192 T^{2} - 299 T^{3} + 24163 T^{4} - 44052 T^{5} + 24163 p T^{6} - 299 p^{2} T^{7} + 192 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2}$$
83 $$( 1 - 6 T + 260 T^{2} - 1794 T^{3} + 33211 T^{4} - 211464 T^{5} + 33211 p T^{6} - 1794 p^{2} T^{7} + 260 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{2}$$
89 $$( 1 + 8 T + 201 T^{2} + 220 T^{3} + 15682 T^{4} - 31200 T^{5} + 15682 p T^{6} + 220 p^{2} T^{7} + 201 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2}$$
97 $$1 - 682 T^{2} + 227521 T^{4} - 48709872 T^{6} + 7388315838 T^{8} - 828434696780 T^{10} + 7388315838 p^{2} T^{12} - 48709872 p^{4} T^{14} + 227521 p^{6} T^{16} - 682 p^{8} T^{18} + p^{10} T^{20}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}