# Properties

 Label 10-546e5-1.1-c7e5-0-5 Degree $10$ Conductor $4.852\times 10^{13}$ Sign $-1$ Analytic cond. $1.44349\times 10^{11}$ Root an. cond. $13.0599$ Motivic weight $7$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $5$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 40·2-s + 135·3-s + 960·4-s − 340·5-s − 5.40e3·6-s + 1.71e3·7-s − 1.79e4·8-s + 1.09e4·9-s + 1.36e4·10-s − 1.30e3·11-s + 1.29e5·12-s + 1.09e4·13-s − 6.86e4·14-s − 4.59e4·15-s + 2.86e5·16-s − 4.24e3·17-s − 4.37e5·18-s − 1.69e4·19-s − 3.26e5·20-s + 2.31e5·21-s + 5.21e4·22-s − 7.80e4·23-s − 2.41e6·24-s − 1.77e5·25-s − 4.39e5·26-s + 6.88e5·27-s + 1.64e6·28-s + ⋯
 L(s)  = 1 − 3.53·2-s + 2.88·3-s + 15/2·4-s − 1.21·5-s − 10.2·6-s + 1.88·7-s − 12.3·8-s + 5·9-s + 4.30·10-s − 0.295·11-s + 21.6·12-s + 1.38·13-s − 6.68·14-s − 3.51·15-s + 35/2·16-s − 0.209·17-s − 17.6·18-s − 0.568·19-s − 9.12·20-s + 5.45·21-s + 1.04·22-s − 1.33·23-s − 35.7·24-s − 2.26·25-s − 4.90·26-s + 6.73·27-s + 14.1·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 7^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(8-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 7^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$10$$ Conductor: $$2^{5} \cdot 3^{5} \cdot 7^{5} \cdot 13^{5}$$ Sign: $-1$ Analytic conductor: $$1.44349\times 10^{11}$$ Root analytic conductor: $$13.0599$$ Motivic weight: $$7$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{546} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$5$$ Selberg data: $$(10,\ 2^{5} \cdot 3^{5} \cdot 7^{5} \cdot 13^{5} ,\ ( \ : 7/2, 7/2, 7/2, 7/2, 7/2 ),\ -1 )$$

## Particular Values

 $$L(4)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ $$( 1 + p^{3} T )^{5}$$
3$C_1$ $$( 1 - p^{3} T )^{5}$$
7$C_1$ $$( 1 - p^{3} T )^{5}$$
13$C_1$ $$( 1 - p^{3} T )^{5}$$
good5$C_2 \wr S_5$ $$1 + 68 p T + 58578 p T^{2} + 608986 p^{3} T^{3} + 1444916821 p^{2} T^{4} + 61790108436 p^{3} T^{5} + 1444916821 p^{9} T^{6} + 608986 p^{17} T^{7} + 58578 p^{22} T^{8} + 68 p^{29} T^{9} + p^{35} T^{10}$$
11$C_2 \wr S_5$ $$1 + 1303 T + 42108237 T^{2} - 24597842260 T^{3} + 916878414481288 T^{4} - 1475553724993041462 T^{5} + 916878414481288 p^{7} T^{6} - 24597842260 p^{14} T^{7} + 42108237 p^{21} T^{8} + 1303 p^{28} T^{9} + p^{35} T^{10}$$
17$C_2 \wr S_5$ $$1 + 4247 T + 1613690547 T^{2} + 8149276487656 T^{3} + 1156582243463969020 T^{4} +$$$$52\!\cdots\!66$$$$T^{5} + 1156582243463969020 p^{7} T^{6} + 8149276487656 p^{14} T^{7} + 1613690547 p^{21} T^{8} + 4247 p^{28} T^{9} + p^{35} T^{10}$$
19$C_2 \wr S_5$ $$1 + 16984 T + 83138398 p T^{2} + 24210102187460 T^{3} + 2225069258269514477 T^{4} +$$$$31\!\cdots\!08$$$$T^{5} + 2225069258269514477 p^{7} T^{6} + 24210102187460 p^{14} T^{7} + 83138398 p^{22} T^{8} + 16984 p^{28} T^{9} + p^{35} T^{10}$$
23$C_2 \wr S_5$ $$1 + 78072 T + 13572347822 T^{2} + 650463352125228 T^{3} + 71145272160697806769 T^{4} +$$$$25\!\cdots\!72$$$$T^{5} + 71145272160697806769 p^{7} T^{6} + 650463352125228 p^{14} T^{7} + 13572347822 p^{21} T^{8} + 78072 p^{28} T^{9} + p^{35} T^{10}$$
29$C_2 \wr S_5$ $$1 + 213142 T + 64225137588 T^{2} + 8938676973052754 T^{3} + 59267307802432797791 p T^{4} +$$$$18\!\cdots\!96$$$$T^{5} + 59267307802432797791 p^{8} T^{6} + 8938676973052754 p^{14} T^{7} + 64225137588 p^{21} T^{8} + 213142 p^{28} T^{9} + p^{35} T^{10}$$
31$C_2 \wr S_5$ $$1 + 186027 T + 103628303579 T^{2} + 15060090909010996 T^{3} +$$$$49\!\cdots\!38$$$$T^{4} +$$$$57\!\cdots\!18$$$$T^{5} +$$$$49\!\cdots\!38$$$$p^{7} T^{6} + 15060090909010996 p^{14} T^{7} + 103628303579 p^{21} T^{8} + 186027 p^{28} T^{9} + p^{35} T^{10}$$
37$C_2 \wr S_5$ $$1 - 101025 T + 252470877075 T^{2} - 9467060729277200 T^{3} +$$$$34\!\cdots\!80$$$$T^{4} -$$$$90\!\cdots\!50$$$$T^{5} +$$$$34\!\cdots\!80$$$$p^{7} T^{6} - 9467060729277200 p^{14} T^{7} + 252470877075 p^{21} T^{8} - 101025 p^{28} T^{9} + p^{35} T^{10}$$
41$C_2 \wr S_5$ $$1 + 23976 T + 545534485325 T^{2} + 514334524827024 p T^{3} +$$$$15\!\cdots\!70$$$$T^{4} +$$$$54\!\cdots\!36$$$$T^{5} +$$$$15\!\cdots\!70$$$$p^{7} T^{6} + 514334524827024 p^{15} T^{7} + 545534485325 p^{21} T^{8} + 23976 p^{28} T^{9} + p^{35} T^{10}$$
43$C_2 \wr S_5$ $$1 + 55528 T + 673914196698 T^{2} + 141772770098277572 T^{3} +$$$$26\!\cdots\!81$$$$T^{4} +$$$$54\!\cdots\!16$$$$T^{5} +$$$$26\!\cdots\!81$$$$p^{7} T^{6} + 141772770098277572 p^{14} T^{7} + 673914196698 p^{21} T^{8} + 55528 p^{28} T^{9} + p^{35} T^{10}$$
47$C_2 \wr S_5$ $$1 + 985981 T + 1276446419051 T^{2} + 624419720944947068 T^{3} +$$$$52\!\cdots\!70$$$$T^{4} +$$$$20\!\cdots\!90$$$$T^{5} +$$$$52\!\cdots\!70$$$$p^{7} T^{6} + 624419720944947068 p^{14} T^{7} + 1276446419051 p^{21} T^{8} + 985981 p^{28} T^{9} + p^{35} T^{10}$$
53$C_2 \wr S_5$ $$1 + 1891657 T + 3098769350825 T^{2} + 3032598338798101916 T^{3} +$$$$52\!\cdots\!30$$$$T^{4} +$$$$55\!\cdots\!18$$$$T^{5} +$$$$52\!\cdots\!30$$$$p^{7} T^{6} + 3032598338798101916 p^{14} T^{7} + 3098769350825 p^{21} T^{8} + 1891657 p^{28} T^{9} + p^{35} T^{10}$$
59$C_2 \wr S_5$ $$1 + 2802208 T + 8769236776287 T^{2} + 18662528794625869616 T^{3} +$$$$38\!\cdots\!46$$$$T^{4} +$$$$60\!\cdots\!76$$$$T^{5} +$$$$38\!\cdots\!46$$$$p^{7} T^{6} + 18662528794625869616 p^{14} T^{7} + 8769236776287 p^{21} T^{8} + 2802208 p^{28} T^{9} + p^{35} T^{10}$$
61$C_2 \wr S_5$ $$1 - 1140591 T + 4932699775605 T^{2} - 11864523329163884 T^{3} +$$$$25\!\cdots\!10$$$$T^{4} -$$$$30\!\cdots\!06$$$$p T^{5} +$$$$25\!\cdots\!10$$$$p^{7} T^{6} - 11864523329163884 p^{14} T^{7} + 4932699775605 p^{21} T^{8} - 1140591 p^{28} T^{9} + p^{35} T^{10}$$
67$C_2 \wr S_5$ $$1 - 265168 T + 24241180772395 T^{2} - 11578680895063746576 T^{3} +$$$$25\!\cdots\!10$$$$T^{4} -$$$$12\!\cdots\!52$$$$T^{5} +$$$$25\!\cdots\!10$$$$p^{7} T^{6} - 11578680895063746576 p^{14} T^{7} + 24241180772395 p^{21} T^{8} - 265168 p^{28} T^{9} + p^{35} T^{10}$$
71$C_2 \wr S_5$ $$1 + 4483276 T + 39774202712459 T^{2} +$$$$12\!\cdots\!80$$$$T^{3} +$$$$64\!\cdots\!46$$$$T^{4} +$$$$16\!\cdots\!80$$$$T^{5} +$$$$64\!\cdots\!46$$$$p^{7} T^{6} +$$$$12\!\cdots\!80$$$$p^{14} T^{7} + 39774202712459 p^{21} T^{8} + 4483276 p^{28} T^{9} + p^{35} T^{10}$$
73$C_2 \wr S_5$ $$1 + 2350578 T + 31153469599224 T^{2} + 91122815818795101526 T^{3} +$$$$50\!\cdots\!35$$$$T^{4} +$$$$14\!\cdots\!80$$$$T^{5} +$$$$50\!\cdots\!35$$$$p^{7} T^{6} + 91122815818795101526 p^{14} T^{7} + 31153469599224 p^{21} T^{8} + 2350578 p^{28} T^{9} + p^{35} T^{10}$$
79$C_2 \wr S_5$ $$1 + 4079889 T + 631965451299 p T^{2} +$$$$12\!\cdots\!68$$$$T^{3} +$$$$15\!\cdots\!88$$$$T^{4} +$$$$38\!\cdots\!06$$$$T^{5} +$$$$15\!\cdots\!88$$$$p^{7} T^{6} +$$$$12\!\cdots\!68$$$$p^{14} T^{7} + 631965451299 p^{22} T^{8} + 4079889 p^{28} T^{9} + p^{35} T^{10}$$
83$C_2 \wr S_5$ $$1 + 8731571 T + 105915930047289 T^{2} +$$$$58\!\cdots\!24$$$$T^{3} +$$$$42\!\cdots\!68$$$$T^{4} +$$$$19\!\cdots\!10$$$$T^{5} +$$$$42\!\cdots\!68$$$$p^{7} T^{6} +$$$$58\!\cdots\!24$$$$p^{14} T^{7} + 105915930047289 p^{21} T^{8} + 8731571 p^{28} T^{9} + p^{35} T^{10}$$
89$C_2 \wr S_5$ $$1 + 20077879 T + 366698441511765 T^{2} +$$$$40\!\cdots\!04$$$$T^{3} +$$$$39\!\cdots\!50$$$$T^{4} +$$$$27\!\cdots\!54$$$$T^{5} +$$$$39\!\cdots\!50$$$$p^{7} T^{6} +$$$$40\!\cdots\!04$$$$p^{14} T^{7} + 366698441511765 p^{21} T^{8} + 20077879 p^{28} T^{9} + p^{35} T^{10}$$
97$C_2 \wr S_5$ $$1 - 3780209 T + 1536217199893 T^{2} +$$$$63\!\cdots\!36$$$$T^{3} +$$$$96\!\cdots\!94$$$$T^{4} -$$$$41\!\cdots\!14$$$$T^{5} +$$$$96\!\cdots\!94$$$$p^{7} T^{6} +$$$$63\!\cdots\!36$$$$p^{14} T^{7} + 1536217199893 p^{21} T^{8} - 3780209 p^{28} T^{9} + p^{35} T^{10}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$