# Properties

 Label 10-2960e5-1.1-c1e5-0-2 Degree $10$ Conductor $2.272\times 10^{17}$ Sign $1$ Analytic cond. $7.37639\times 10^{6}$ Root an. cond. $4.86165$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 5·3-s + 5·5-s + 5·7-s + 8·9-s + 7·11-s − 6·13-s + 25·15-s + 12·19-s + 25·21-s + 6·23-s + 15·25-s − 27-s + 6·29-s + 10·31-s + 35·33-s + 25·35-s − 5·37-s − 30·39-s + 7·41-s + 22·43-s + 40·45-s + 13·47-s + 2·49-s − 11·53-s + 35·55-s + 60·57-s + 10·59-s + ⋯
 L(s)  = 1 + 2.88·3-s + 2.23·5-s + 1.88·7-s + 8/3·9-s + 2.11·11-s − 1.66·13-s + 6.45·15-s + 2.75·19-s + 5.45·21-s + 1.25·23-s + 3·25-s − 0.192·27-s + 1.11·29-s + 1.79·31-s + 6.09·33-s + 4.22·35-s − 0.821·37-s − 4.80·39-s + 1.09·41-s + 3.35·43-s + 5.96·45-s + 1.89·47-s + 2/7·49-s − 1.51·53-s + 4.71·55-s + 7.94·57-s + 1.30·59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$10$$ Conductor: $$2^{20} \cdot 5^{5} \cdot 37^{5}$$ Sign: $1$ Analytic conductor: $$7.37639\times 10^{6}$$ Root analytic conductor: $$4.86165$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(10,\ 2^{20} \cdot 5^{5} \cdot 37^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$114.3272780$$ $$L(\frac12)$$ $$\approx$$ $$114.3272780$$ $$L(\frac{3}{2})$$ 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 $$1$$
5$C_1$ $$( 1 - T )^{5}$$
37$C_1$ $$( 1 + T )^{5}$$
good3$C_2 \wr S_5$ $$1 - 5 T + 17 T^{2} - 44 T^{3} + 100 T^{4} - 188 T^{5} + 100 p T^{6} - 44 p^{2} T^{7} + 17 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10}$$
7$C_2 \wr S_5$ $$1 - 5 T + 23 T^{2} - 40 T^{3} + 78 T^{4} - 8 T^{5} + 78 p T^{6} - 40 p^{2} T^{7} + 23 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10}$$
11$C_2 \wr S_5$ $$1 - 7 T + 5 p T^{2} - 260 T^{3} + 1226 T^{4} - 4058 T^{5} + 1226 p T^{6} - 260 p^{2} T^{7} + 5 p^{4} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10}$$
13$C_2 \wr S_5$ $$1 + 6 T + 37 T^{2} + 140 T^{3} + 402 T^{4} + 1604 T^{5} + 402 p T^{6} + 140 p^{2} T^{7} + 37 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10}$$
17$C_2 \wr S_5$ $$1 + 9 T^{2} - 100 T^{3} + 282 T^{4} - 504 T^{5} + 282 p T^{6} - 100 p^{2} T^{7} + 9 p^{3} T^{8} + p^{5} T^{10}$$
19$C_2 \wr S_5$ $$1 - 12 T + 121 T^{2} - 850 T^{3} + 5046 T^{4} - 23668 T^{5} + 5046 p T^{6} - 850 p^{2} T^{7} + 121 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10}$$
23$C_2 \wr S_5$ $$1 - 6 T + 67 T^{2} - 120 T^{3} + 1098 T^{4} + 1052 T^{5} + 1098 p T^{6} - 120 p^{2} T^{7} + 67 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10}$$
29$C_2 \wr S_5$ $$1 - 6 T + 97 T^{2} - 264 T^{3} + 3354 T^{4} - 4996 T^{5} + 3354 p T^{6} - 264 p^{2} T^{7} + 97 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10}$$
31$C_2 \wr S_5$ $$1 - 10 T + 145 T^{2} - 870 T^{3} + 7474 T^{4} - 33608 T^{5} + 7474 p T^{6} - 870 p^{2} T^{7} + 145 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10}$$
41$C_2 \wr S_5$ $$1 - 7 T + 5 p T^{2} - 1100 T^{3} + 16826 T^{4} - 66698 T^{5} + 16826 p T^{6} - 1100 p^{2} T^{7} + 5 p^{4} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10}$$
43$C_2 \wr S_5$ $$1 - 22 T + 367 T^{2} - 4168 T^{3} + 38338 T^{4} - 277060 T^{5} + 38338 p T^{6} - 4168 p^{2} T^{7} + 367 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10}$$
47$C_2 \wr S_5$ $$1 - 13 T + 259 T^{2} - 2168 T^{3} + 24246 T^{4} - 145120 T^{5} + 24246 p T^{6} - 2168 p^{2} T^{7} + 259 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10}$$
53$C_2 \wr S_5$ $$1 + 11 T + 73 T^{2} - 188 T^{3} - 30 p T^{4} - 23182 T^{5} - 30 p^{2} T^{6} - 188 p^{2} T^{7} + 73 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10}$$
59$C_2 \wr S_5$ $$1 - 10 T + 253 T^{2} - 1874 T^{3} + 27242 T^{4} - 152960 T^{5} + 27242 p T^{6} - 1874 p^{2} T^{7} + 253 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10}$$
61$C_2 \wr S_5$ $$1 + 177 T^{2} + 144 T^{3} + 17738 T^{4} + 6752 T^{5} + 17738 p T^{6} + 144 p^{2} T^{7} + 177 p^{3} T^{8} + p^{5} T^{10}$$
67$C_2 \wr S_5$ $$1 - 24 T + 385 T^{2} - 4850 T^{3} + 50386 T^{4} - 435420 T^{5} + 50386 p T^{6} - 4850 p^{2} T^{7} + 385 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10}$$
71$C_2 \wr S_5$ $$1 - 13 T + 143 T^{2} - 1068 T^{3} + 14598 T^{4} - 119726 T^{5} + 14598 p T^{6} - 1068 p^{2} T^{7} + 143 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10}$$
73$C_2 \wr S_5$ $$1 + 13 T + 257 T^{2} + 2884 T^{3} + 32406 T^{4} + 294142 T^{5} + 32406 p T^{6} + 2884 p^{2} T^{7} + 257 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10}$$
79$C_2 \wr S_5$ $$1 + 2 T + 285 T^{2} + 286 T^{3} + 39270 T^{4} + 31680 T^{5} + 39270 p T^{6} + 286 p^{2} T^{7} + 285 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10}$$
83$C_2 \wr S_5$ $$1 - 13 T + 161 T^{2} - 2704 T^{3} + 28628 T^{4} - 234316 T^{5} + 28628 p T^{6} - 2704 p^{2} T^{7} + 161 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10}$$
89$C_2 \wr S_5$ $$1 - 8 T + 301 T^{2} - 3120 T^{3} + 41770 T^{4} - 426640 T^{5} + 41770 p T^{6} - 3120 p^{2} T^{7} + 301 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10}$$
97$C_2 \wr S_5$ $$1 + 20 T + 405 T^{2} + 4224 T^{3} + 51002 T^{4} + 411352 T^{5} + 51002 p T^{6} + 4224 p^{2} T^{7} + 405 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$