# Properties

 Degree 10 Conductor $2^{10} \cdot 3^{20}$ Sign $-1$ Motivic weight 5 Primitive no Self-dual yes Analytic rank 5

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 21·5-s − 29·7-s + 177·11-s + 181·13-s − 1.14e3·17-s − 416·19-s + 399·23-s − 5.20e3·25-s − 6.03e3·29-s − 2.75e3·31-s + 609·35-s − 7.58e3·37-s − 1.84e4·41-s − 1.46e3·43-s − 2.51e4·47-s − 3.95e4·49-s − 5.84e4·53-s − 3.71e3·55-s − 9.05e4·59-s − 1.40e3·61-s − 3.80e3·65-s − 1.39e4·67-s − 1.14e5·71-s + 7.60e3·73-s − 5.13e3·77-s − 2.99e4·79-s − 2.28e5·83-s + ⋯
 L(s)  = 1 − 0.375·5-s − 0.223·7-s + 0.441·11-s + 0.297·13-s − 0.956·17-s − 0.264·19-s + 0.157·23-s − 1.66·25-s − 1.33·29-s − 0.515·31-s + 0.0840·35-s − 0.910·37-s − 1.71·41-s − 0.121·43-s − 1.66·47-s − 2.35·49-s − 2.85·53-s − 0.165·55-s − 3.38·59-s − 0.0482·61-s − 0.111·65-s − 0.378·67-s − 2.69·71-s + 0.166·73-s − 0.0986·77-s − 0.540·79-s − 3.64·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$10$$ $$N$$ = $$2^{10} \cdot 3^{20}$$ $$\varepsilon$$ = $-1$ motivic weight = $$5$$ character : induced by $\chi_{324} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$5$$ Selberg data = $$(10,\ 2^{10} \cdot 3^{20} ,\ ( \ : 5/2, 5/2, 5/2, 5/2, 5/2 ),\ -1 )$$ $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 10. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 9.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5$C_2 \wr S_5$ $$1 + 21 T + 5644 T^{2} + 319227 T^{3} + 444607 p^{2} T^{4} + 1519942932 T^{5} + 444607 p^{7} T^{6} + 319227 p^{10} T^{7} + 5644 p^{15} T^{8} + 21 p^{20} T^{9} + p^{25} T^{10}$$
7$C_2 \wr S_5$ $$1 + 29 T + 40410 T^{2} + 2467167 T^{3} + 1121022921 T^{4} + 42673614444 T^{5} + 1121022921 p^{5} T^{6} + 2467167 p^{10} T^{7} + 40410 p^{15} T^{8} + 29 p^{20} T^{9} + p^{25} T^{10}$$
11$C_2 \wr S_5$ $$1 - 177 T + 427561 T^{2} - 2122014 T^{3} + 77616186361 T^{4} + 7157256911361 T^{5} + 77616186361 p^{5} T^{6} - 2122014 p^{10} T^{7} + 427561 p^{15} T^{8} - 177 p^{20} T^{9} + p^{25} T^{10}$$
13$C_2 \wr S_5$ $$1 - 181 T + 1045092 T^{2} - 87339735 T^{3} + 629954553255 T^{4} - 52722114809928 T^{5} + 629954553255 p^{5} T^{6} - 87339735 p^{10} T^{7} + 1045092 p^{15} T^{8} - 181 p^{20} T^{9} + p^{25} T^{10}$$
17$C_2 \wr S_5$ $$1 + 1140 T + 4980550 T^{2} + 3443850354 T^{3} + 10068870522169 T^{4} + 5069379208548852 T^{5} + 10068870522169 p^{5} T^{6} + 3443850354 p^{10} T^{7} + 4980550 p^{15} T^{8} + 1140 p^{20} T^{9} + p^{25} T^{10}$$
19$C_2 \wr S_5$ $$1 + 416 T + 5046258 T^{2} + 6215761044 T^{3} + 20272296121125 T^{4} + 15898268281316088 T^{5} + 20272296121125 p^{5} T^{6} + 6215761044 p^{10} T^{7} + 5046258 p^{15} T^{8} + 416 p^{20} T^{9} + p^{25} T^{10}$$
23$C_2 \wr S_5$ $$1 - 399 T + 16236442 T^{2} + 15815242731 T^{3} + 4968768180959 p T^{4} + 214964375471995932 T^{5} + 4968768180959 p^{6} T^{6} + 15815242731 p^{10} T^{7} + 16236442 p^{15} T^{8} - 399 p^{20} T^{9} + p^{25} T^{10}$$
29$C_2 \wr S_5$ $$1 + 6033 T + 32744932 T^{2} + 82954489011 T^{3} + 288237740873407 T^{4} + 1430006647746298968 T^{5} + 288237740873407 p^{5} T^{6} + 82954489011 p^{10} T^{7} + 32744932 p^{15} T^{8} + 6033 p^{20} T^{9} + p^{25} T^{10}$$
31$C_2 \wr S_5$ $$1 + 89 p T + 62514558 T^{2} - 8483492775 T^{3} + 1879317626697489 T^{4} - 2373615527060533968 T^{5} + 1879317626697489 p^{5} T^{6} - 8483492775 p^{10} T^{7} + 62514558 p^{15} T^{8} + 89 p^{21} T^{9} + p^{25} T^{10}$$
37$C_2 \wr S_5$ $$1 + 7586 T + 201201093 T^{2} + 803146672896 T^{3} + 19241810738464926 T^{4} + 60351714230064941916 T^{5} + 19241810738464926 p^{5} T^{6} + 803146672896 p^{10} T^{7} + 201201093 p^{15} T^{8} + 7586 p^{20} T^{9} + p^{25} T^{10}$$
41$C_2 \wr S_5$ $$1 + 18435 T + 457528267 T^{2} + 6389512835910 T^{3} + 92046695080587061 T^{4} +$$$$99\!\cdots\!97$$$$T^{5} + 92046695080587061 p^{5} T^{6} + 6389512835910 p^{10} T^{7} + 457528267 p^{15} T^{8} + 18435 p^{20} T^{9} + p^{25} T^{10}$$
43$C_2 \wr S_5$ $$1 + 1469 T + 274021497 T^{2} + 2209484086710 T^{3} + 59712901022608185 T^{4} +$$$$32\!\cdots\!87$$$$T^{5} + 59712901022608185 p^{5} T^{6} + 2209484086710 p^{10} T^{7} + 274021497 p^{15} T^{8} + 1469 p^{20} T^{9} + p^{25} T^{10}$$
47$C_2 \wr S_5$ $$1 + 25155 T + 1034020258 T^{2} + 20179979725617 T^{3} + 464012894081647969 T^{4} +$$$$66\!\cdots\!16$$$$T^{5} + 464012894081647969 p^{5} T^{6} + 20179979725617 p^{10} T^{7} + 1034020258 p^{15} T^{8} + 25155 p^{20} T^{9} + p^{25} T^{10}$$
53$C_2 \wr S_5$ $$1 + 58422 T + 3354568213 T^{2} + 110313236959296 T^{3} + 3390725554692289246 T^{4} +$$$$71\!\cdots\!28$$$$T^{5} + 3390725554692289246 p^{5} T^{6} + 110313236959296 p^{10} T^{7} + 3354568213 p^{15} T^{8} + 58422 p^{20} T^{9} + p^{25} T^{10}$$
59$C_2 \wr S_5$ $$1 + 90537 T + 5365830529 T^{2} + 236000523803862 T^{3} + 8392018437003638425 T^{4} +$$$$24\!\cdots\!43$$$$T^{5} + 8392018437003638425 p^{5} T^{6} + 236000523803862 p^{10} T^{7} + 5365830529 p^{15} T^{8} + 90537 p^{20} T^{9} + p^{25} T^{10}$$
61$C_2 \wr S_5$ $$1 + 23 p T + 3538874292 T^{2} + 2256100311765 T^{3} + 5454602684103950271 T^{4} +$$$$18\!\cdots\!56$$$$T^{5} + 5454602684103950271 p^{5} T^{6} + 2256100311765 p^{10} T^{7} + 3538874292 p^{15} T^{8} + 23 p^{21} T^{9} + p^{25} T^{10}$$
67$C_2 \wr S_5$ $$1 + 13907 T + 4070090193 T^{2} - 10411373671926 T^{3} + 6695736735425021001 T^{4} -$$$$80\!\cdots\!07$$$$T^{5} + 6695736735425021001 p^{5} T^{6} - 10411373671926 p^{10} T^{7} + 4070090193 p^{15} T^{8} + 13907 p^{20} T^{9} + p^{25} T^{10}$$
71$C_2 \wr S_5$ $$1 + 114684 T + 7758380659 T^{2} + 426246123888336 T^{3} + 19260501229393543450 T^{4} +$$$$77\!\cdots\!40$$$$T^{5} + 19260501229393543450 p^{5} T^{6} + 426246123888336 p^{10} T^{7} + 7758380659 p^{15} T^{8} + 114684 p^{20} T^{9} + p^{25} T^{10}$$
73$C_2 \wr S_5$ $$1 - 7600 T + 3606834246 T^{2} - 31056473559714 T^{3} + 12288417972789256281 T^{4} -$$$$80\!\cdots\!84$$$$T^{5} + 12288417972789256281 p^{5} T^{6} - 31056473559714 p^{10} T^{7} + 3606834246 p^{15} T^{8} - 7600 p^{20} T^{9} + p^{25} T^{10}$$
79$C_2 \wr S_5$ $$1 + 29993 T + 6251931678 T^{2} - 85699438105257 T^{3} + 15422207141619743649 T^{4} -$$$$73\!\cdots\!92$$$$T^{5} + 15422207141619743649 p^{5} T^{6} - 85699438105257 p^{10} T^{7} + 6251931678 p^{15} T^{8} + 29993 p^{20} T^{9} + p^{25} T^{10}$$
83$C_2 \wr S_5$ $$1 + 228951 T + 373676246 p T^{2} + 2949331181889681 T^{3} +$$$$22\!\cdots\!25$$$$T^{4} +$$$$14\!\cdots\!60$$$$T^{5} +$$$$22\!\cdots\!25$$$$p^{5} T^{6} + 2949331181889681 p^{10} T^{7} + 373676246 p^{16} T^{8} + 228951 p^{20} T^{9} + p^{25} T^{10}$$
89$C_2 \wr S_5$ $$1 + 299166 T + 52616244181 T^{2} + 6660261403977288 T^{3} +$$$$67\!\cdots\!10$$$$T^{4} +$$$$55\!\cdots\!64$$$$T^{5} +$$$$67\!\cdots\!10$$$$p^{5} T^{6} + 6660261403977288 p^{10} T^{7} + 52616244181 p^{15} T^{8} + 299166 p^{20} T^{9} + p^{25} T^{10}$$
97$C_2 \wr S_5$ $$1 + 40541 T + 19537068819 T^{2} + 1527692999826186 T^{3} +$$$$22\!\cdots\!25$$$$T^{4} +$$$$18\!\cdots\!11$$$$T^{5} +$$$$22\!\cdots\!25$$$$p^{5} T^{6} + 1527692999826186 p^{10} T^{7} + 19537068819 p^{15} T^{8} + 40541 p^{20} T^{9} + p^{25} T^{10}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}