# Properties

 Label 10-1856e5-1.1-c3e5-0-0 Degree $10$ Conductor $2.202\times 10^{16}$ Sign $1$ Analytic cond. $1.57478\times 10^{10}$ Root an. cond. $10.4645$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·3-s − 10·5-s + 32·7-s − 45·9-s − 36·11-s − 26·13-s + 40·15-s + 82·17-s − 156·19-s − 128·21-s + 336·23-s − 187·25-s + 120·27-s − 145·29-s + 432·31-s + 144·33-s − 320·35-s + 18·37-s + 104·39-s + 82·41-s − 340·43-s + 450·45-s + 680·47-s − 403·49-s − 328·51-s + 102·53-s + 360·55-s + ⋯
 L(s)  = 1 − 0.769·3-s − 0.894·5-s + 1.72·7-s − 5/3·9-s − 0.986·11-s − 0.554·13-s + 0.688·15-s + 1.16·17-s − 1.88·19-s − 1.33·21-s + 3.04·23-s − 1.49·25-s + 0.855·27-s − 0.928·29-s + 2.50·31-s + 0.759·33-s − 1.54·35-s + 0.0799·37-s + 0.427·39-s + 0.312·41-s − 1.20·43-s + 1.49·45-s + 2.11·47-s − 1.17·49-s − 0.900·51-s + 0.264·53-s + 0.882·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$10$$ Conductor: $$2^{30} \cdot 29^{5}$$ Sign: $1$ Analytic conductor: $$1.57478\times 10^{10}$$ Root analytic conductor: $$10.4645$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1856} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(10,\ 2^{30} \cdot 29^{5} ,\ ( \ : 3/2, 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.8480202717$$ $$L(\frac12)$$ $$\approx$$ $$0.8480202717$$ $$L(\frac{5}{2})$$ 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 $$1$$
29$C_1$ $$( 1 + p T )^{5}$$
good3$C_2 \wr S_5$ $$1 + 4 T + 61 T^{2} + 304 T^{3} + 2821 T^{4} + 9244 T^{5} + 2821 p^{3} T^{6} + 304 p^{6} T^{7} + 61 p^{9} T^{8} + 4 p^{12} T^{9} + p^{15} T^{10}$$
5$C_2 \wr S_5$ $$1 + 2 p T + 287 T^{2} + 2696 T^{3} + 11009 p T^{4} + 371994 T^{5} + 11009 p^{4} T^{6} + 2696 p^{6} T^{7} + 287 p^{9} T^{8} + 2 p^{13} T^{9} + p^{15} T^{10}$$
7$C_2 \wr S_5$ $$1 - 32 T + 1427 T^{2} - 36032 T^{3} + 18282 p^{2} T^{4} - 17455168 T^{5} + 18282 p^{5} T^{6} - 36032 p^{6} T^{7} + 1427 p^{9} T^{8} - 32 p^{12} T^{9} + p^{15} T^{10}$$
11$C_2 \wr S_5$ $$1 + 36 T + 5309 T^{2} + 177032 T^{3} + 12820837 T^{4} + 341964932 T^{5} + 12820837 p^{3} T^{6} + 177032 p^{6} T^{7} + 5309 p^{9} T^{8} + 36 p^{12} T^{9} + p^{15} T^{10}$$
13$C_2 \wr S_5$ $$1 + 2 p T + 5839 T^{2} + 200496 T^{3} + 22116069 T^{4} + 524580946 T^{5} + 22116069 p^{3} T^{6} + 200496 p^{6} T^{7} + 5839 p^{9} T^{8} + 2 p^{13} T^{9} + p^{15} T^{10}$$
17$C_2 \wr S_5$ $$1 - 82 T + 21789 T^{2} - 1579928 T^{3} + 201344338 T^{4} - 11565860396 T^{5} + 201344338 p^{3} T^{6} - 1579928 p^{6} T^{7} + 21789 p^{9} T^{8} - 82 p^{12} T^{9} + p^{15} T^{10}$$
19$C_2 \wr S_5$ $$1 + 156 T + 38511 T^{2} + 4147088 T^{3} + 557756866 T^{4} + 42213259048 T^{5} + 557756866 p^{3} T^{6} + 4147088 p^{6} T^{7} + 38511 p^{9} T^{8} + 156 p^{12} T^{9} + p^{15} T^{10}$$
23$C_2 \wr S_5$ $$1 - 336 T + 68587 T^{2} - 9632544 T^{3} + 1169621794 T^{4} - 5540381280 p T^{5} + 1169621794 p^{3} T^{6} - 9632544 p^{6} T^{7} + 68587 p^{9} T^{8} - 336 p^{12} T^{9} + p^{15} T^{10}$$
31$C_2 \wr S_5$ $$1 - 432 T + 201929 T^{2} - 52526032 T^{3} + 13543164109 T^{4} - 2362357787192 T^{5} + 13543164109 p^{3} T^{6} - 52526032 p^{6} T^{7} + 201929 p^{9} T^{8} - 432 p^{12} T^{9} + p^{15} T^{10}$$
37$C_2 \wr S_5$ $$1 - 18 T + 164945 T^{2} + 1995160 T^{3} + 13003010010 T^{4} + 309782516404 T^{5} + 13003010010 p^{3} T^{6} + 1995160 p^{6} T^{7} + 164945 p^{9} T^{8} - 18 p^{12} T^{9} + p^{15} T^{10}$$
41$C_2 \wr S_5$ $$1 - 2 p T + 134797 T^{2} + 4859064 T^{3} + 11312527994 T^{4} + 717309134036 T^{5} + 11312527994 p^{3} T^{6} + 4859064 p^{6} T^{7} + 134797 p^{9} T^{8} - 2 p^{13} T^{9} + p^{15} T^{10}$$
43$C_2 \wr S_5$ $$1 + 340 T + 78557 T^{2} + 13668248 T^{3} + 883246085 T^{4} - 1553774172796 T^{5} + 883246085 p^{3} T^{6} + 13668248 p^{6} T^{7} + 78557 p^{9} T^{8} + 340 p^{12} T^{9} + p^{15} T^{10}$$
47$C_2 \wr S_5$ $$1 - 680 T + 384593 T^{2} - 98072184 T^{3} + 24706007645 T^{4} - 3145150453528 T^{5} + 24706007645 p^{3} T^{6} - 98072184 p^{6} T^{7} + 384593 p^{9} T^{8} - 680 p^{12} T^{9} + p^{15} T^{10}$$
53$C_2 \wr S_5$ $$1 - 102 T + 654439 T^{2} - 51999000 T^{3} + 183063886741 T^{4} - 11065291160166 T^{5} + 183063886741 p^{3} T^{6} - 51999000 p^{6} T^{7} + 654439 p^{9} T^{8} - 102 p^{12} T^{9} + p^{15} T^{10}$$
59$C_2 \wr S_5$ $$1 + 924 T + 866103 T^{2} + 337060304 T^{3} + 171323060562 T^{4} + 43701048197672 T^{5} + 171323060562 p^{3} T^{6} + 337060304 p^{6} T^{7} + 866103 p^{9} T^{8} + 924 p^{12} T^{9} + p^{15} T^{10}$$
61$C_2 \wr S_5$ $$1 - 618 T + 750561 T^{2} - 351831128 T^{3} + 243423855586 T^{4} - 98404847105884 T^{5} + 243423855586 p^{3} T^{6} - 351831128 p^{6} T^{7} + 750561 p^{9} T^{8} - 618 p^{12} T^{9} + p^{15} T^{10}$$
67$C_2 \wr S_5$ $$1 + 44 T + 597703 T^{2} + 81163152 T^{3} + 249970117114 T^{4} + 11056591462856 T^{5} + 249970117114 p^{3} T^{6} + 81163152 p^{6} T^{7} + 597703 p^{9} T^{8} + 44 p^{12} T^{9} + p^{15} T^{10}$$
71$C_2 \wr S_5$ $$1 - 1032 T + 1465835 T^{2} - 983439264 T^{3} + 931340637650 T^{4} - 489572777520432 T^{5} + 931340637650 p^{3} T^{6} - 983439264 p^{6} T^{7} + 1465835 p^{9} T^{8} - 1032 p^{12} T^{9} + p^{15} T^{10}$$
73$C_2 \wr S_5$ $$1 + 1078 T + 1544061 T^{2} + 895316056 T^{3} + 816827199978 T^{4} + 365553950129028 T^{5} + 816827199978 p^{3} T^{6} + 895316056 p^{6} T^{7} + 1544061 p^{9} T^{8} + 1078 p^{12} T^{9} + p^{15} T^{10}$$
79$C_2 \wr S_5$ $$1 - 200 T + 2450913 T^{2} - 390614776 T^{3} + 2409756860989 T^{4} - 287955059167144 T^{5} + 2409756860989 p^{3} T^{6} - 390614776 p^{6} T^{7} + 2450913 p^{9} T^{8} - 200 p^{12} T^{9} + p^{15} T^{10}$$
83$C_2 \wr S_5$ $$1 + 452 T + 1144543 T^{2} + 431114608 T^{3} + 1056135128754 T^{4} + 379282762835224 T^{5} + 1056135128754 p^{3} T^{6} + 431114608 p^{6} T^{7} + 1144543 p^{9} T^{8} + 452 p^{12} T^{9} + p^{15} T^{10}$$
89$C_2 \wr S_5$ $$1 + 1790 T + 3080861 T^{2} + 2722201848 T^{3} + 2858190637178 T^{4} + 1978995927638068 T^{5} + 2858190637178 p^{3} T^{6} + 2722201848 p^{6} T^{7} + 3080861 p^{9} T^{8} + 1790 p^{12} T^{9} + p^{15} T^{10}$$
97$C_2 \wr S_5$ $$1 + 2518 T + 6513717 T^{2} + 9477719064 T^{3} + 13480346210298 T^{4} + 13064195578094340 T^{5} + 13480346210298 p^{3} T^{6} + 9477719064 p^{6} T^{7} + 6513717 p^{9} T^{8} + 2518 p^{12} T^{9} + p^{15} T^{10}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$