# Properties

 Label 10-230e5-1.1-c5e5-0-1 Degree $10$ Conductor $643634300000$ Sign $1$ Analytic cond. $6.83033\times 10^{7}$ Root an. cond. $6.07357$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 20·2-s + 3-s + 240·4-s + 125·5-s − 20·6-s + 102·7-s − 2.24e3·8-s − 440·9-s − 2.50e3·10-s + 251·11-s + 240·12-s + 1.74e3·13-s − 2.04e3·14-s + 125·15-s + 1.79e4·16-s + 1.94e3·17-s + 8.80e3·18-s − 845·19-s + 3.00e4·20-s + 102·21-s − 5.02e3·22-s − 2.64e3·23-s − 2.24e3·24-s + 9.37e3·25-s − 3.48e4·26-s + 288·27-s + 2.44e4·28-s + ⋯
 L(s)  = 1 − 3.53·2-s + 0.0641·3-s + 15/2·4-s + 2.23·5-s − 0.226·6-s + 0.786·7-s − 12.3·8-s − 1.81·9-s − 7.90·10-s + 0.625·11-s + 0.481·12-s + 2.86·13-s − 2.78·14-s + 0.143·15-s + 35/2·16-s + 1.63·17-s + 6.40·18-s − 0.536·19-s + 16.7·20-s + 0.0504·21-s − 2.21·22-s − 1.04·23-s − 0.793·24-s + 3·25-s − 10.1·26-s + 0.0760·27-s + 5.90·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$10$$ Conductor: $$2^{5} \cdot 5^{5} \cdot 23^{5}$$ Sign: $1$ Analytic conductor: $$6.83033\times 10^{7}$$ Root analytic conductor: $$6.07357$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{230} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(10,\ 2^{5} \cdot 5^{5} \cdot 23^{5} ,\ ( \ : 5/2, 5/2, 5/2, 5/2, 5/2 ),\ 1 )$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$4.298736081$$ $$L(\frac12)$$ $$\approx$$ $$4.298736081$$ $$L(\frac{7}{2})$$ 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$C_1$ $$( 1 + p^{2} T )^{5}$$
5$C_1$ $$( 1 - p^{2} T )^{5}$$
23$C_1$ $$( 1 + p^{2} T )^{5}$$
good3$C_2 \wr S_5$ $$1 - T + 49 p^{2} T^{2} - 1169 T^{3} + 92531 T^{4} - 98636 p T^{5} + 92531 p^{5} T^{6} - 1169 p^{10} T^{7} + 49 p^{17} T^{8} - p^{20} T^{9} + p^{25} T^{10}$$
7$C_2 \wr S_5$ $$1 - 102 T + 1699 T^{2} + 441095 T^{3} - 173075738 T^{4} + 12005933570 T^{5} - 173075738 p^{5} T^{6} + 441095 p^{10} T^{7} + 1699 p^{15} T^{8} - 102 p^{20} T^{9} + p^{25} T^{10}$$
11$C_2 \wr S_5$ $$1 - 251 T + 396934 T^{2} + 42201726 T^{3} + 35916833921 T^{4} + 26830642335754 T^{5} + 35916833921 p^{5} T^{6} + 42201726 p^{10} T^{7} + 396934 p^{15} T^{8} - 251 p^{20} T^{9} + p^{25} T^{10}$$
13$C_2 \wr S_5$ $$1 - 1743 T + 2320333 T^{2} - 2149376173 T^{3} + 1719321763603 T^{4} - 84590325518470 p T^{5} + 1719321763603 p^{5} T^{6} - 2149376173 p^{10} T^{7} + 2320333 p^{15} T^{8} - 1743 p^{20} T^{9} + p^{25} T^{10}$$
17$C_2 \wr S_5$ $$1 - 1944 T + 5592385 T^{2} - 7962664989 T^{3} + 12463122646414 T^{4} - 14622158363680074 T^{5} + 12463122646414 p^{5} T^{6} - 7962664989 p^{10} T^{7} + 5592385 p^{15} T^{8} - 1944 p^{20} T^{9} + p^{25} T^{10}$$
19$C_2 \wr S_5$ $$1 + 845 T + 3332472 T^{2} + 6308082522 T^{3} + 14730843199923 T^{4} + 15110077512695874 T^{5} + 14730843199923 p^{5} T^{6} + 6308082522 p^{10} T^{7} + 3332472 p^{15} T^{8} + 845 p^{20} T^{9} + p^{25} T^{10}$$
29$C_2 \wr S_5$ $$1 + 4021 T + 2438630 p T^{2} + 321579765099 T^{3} + 2393071055288039 T^{4} + 9733424656331106760 T^{5} + 2393071055288039 p^{5} T^{6} + 321579765099 p^{10} T^{7} + 2438630 p^{16} T^{8} + 4021 p^{20} T^{9} + p^{25} T^{10}$$
31$C_2 \wr S_5$ $$1 + 15752 T + 147580386 T^{2} + 977642232087 T^{3} + 5502243646667061 T^{4} + 28632909996326311755 T^{5} + 5502243646667061 p^{5} T^{6} + 977642232087 p^{10} T^{7} + 147580386 p^{15} T^{8} + 15752 p^{20} T^{9} + p^{25} T^{10}$$
37$C_2 \wr S_5$ $$1 + 3455 T + 239666121 T^{2} + 1102201441332 T^{3} + 27331653143107866 T^{4} +$$$$11\!\cdots\!50$$$$T^{5} + 27331653143107866 p^{5} T^{6} + 1102201441332 p^{10} T^{7} + 239666121 p^{15} T^{8} + 3455 p^{20} T^{9} + p^{25} T^{10}$$
41$C_2 \wr S_5$ $$1 + 11898 T + 487603420 T^{2} + 5343653490621 T^{3} + 102374034000426175 T^{4} +$$$$91\!\cdots\!99$$$$T^{5} + 102374034000426175 p^{5} T^{6} + 5343653490621 p^{10} T^{7} + 487603420 p^{15} T^{8} + 11898 p^{20} T^{9} + p^{25} T^{10}$$
43$C_2 \wr S_5$ $$1 - 6968 T + 689663315 T^{2} - 3917775258416 T^{3} + 196764408262203430 T^{4} -$$$$85\!\cdots\!60$$$$T^{5} + 196764408262203430 p^{5} T^{6} - 3917775258416 p^{10} T^{7} + 689663315 p^{15} T^{8} - 6968 p^{20} T^{9} + p^{25} T^{10}$$
47$C_2 \wr S_5$ $$1 - 13412 T + 664650340 T^{2} - 5989528533990 T^{3} + 210505665900463931 T^{4} -$$$$13\!\cdots\!60$$$$T^{5} + 210505665900463931 p^{5} T^{6} - 5989528533990 p^{10} T^{7} + 664650340 p^{15} T^{8} - 13412 p^{20} T^{9} + p^{25} T^{10}$$
53$C_2 \wr S_5$ $$1 - 53029 T + 1673284285 T^{2} - 21538482907524 T^{3} - 70750143334792234 T^{4} +$$$$82\!\cdots\!26$$$$T^{5} - 70750143334792234 p^{5} T^{6} - 21538482907524 p^{10} T^{7} + 1673284285 p^{15} T^{8} - 53029 p^{20} T^{9} + p^{25} T^{10}$$
59$C_2 \wr S_5$ $$1 + 31223 T + 51749977 T^{2} - 15962857373976 T^{3} + 507833043660274652 T^{4} +$$$$32\!\cdots\!38$$$$T^{5} + 507833043660274652 p^{5} T^{6} - 15962857373976 p^{10} T^{7} + 51749977 p^{15} T^{8} + 31223 p^{20} T^{9} + p^{25} T^{10}$$
61$C_2 \wr S_5$ $$1 - 71477 T + 5596916510 T^{2} - 232699931306900 T^{3} + 10103225672512544845 T^{4} -$$$$28\!\cdots\!46$$$$T^{5} + 10103225672512544845 p^{5} T^{6} - 232699931306900 p^{10} T^{7} + 5596916510 p^{15} T^{8} - 71477 p^{20} T^{9} + p^{25} T^{10}$$
67$C_2 \wr S_5$ $$1 - 76003 T + 7653403455 T^{2} - 380755045384332 T^{3} + 21549101354298004122 T^{4} -$$$$75\!\cdots\!70$$$$T^{5} + 21549101354298004122 p^{5} T^{6} - 380755045384332 p^{10} T^{7} + 7653403455 p^{15} T^{8} - 76003 p^{20} T^{9} + p^{25} T^{10}$$
71$C_2 \wr S_5$ $$1 - 54418 T + 7989039538 T^{2} - 347459045595051 T^{3} + 27083179725751514873 T^{4} -$$$$90\!\cdots\!53$$$$T^{5} + 27083179725751514873 p^{5} T^{6} - 347459045595051 p^{10} T^{7} + 7989039538 p^{15} T^{8} - 54418 p^{20} T^{9} + p^{25} T^{10}$$
73$C_2 \wr S_5$ $$1 - 69418 T + 8297287422 T^{2} - 411024721190724 T^{3} + 28905032376530491725 T^{4} -$$$$11\!\cdots\!52$$$$T^{5} + 28905032376530491725 p^{5} T^{6} - 411024721190724 p^{10} T^{7} + 8297287422 p^{15} T^{8} - 69418 p^{20} T^{9} + p^{25} T^{10}$$
79$C_2 \wr S_5$ $$1 - 105024 T + 12983382259 T^{2} - 849845049806656 T^{3} + 68345856269270634322 T^{4} -$$$$34\!\cdots\!28$$$$T^{5} + 68345856269270634322 p^{5} T^{6} - 849845049806656 p^{10} T^{7} + 12983382259 p^{15} T^{8} - 105024 p^{20} T^{9} + p^{25} T^{10}$$
83$C_2 \wr S_5$ $$1 - 89399 T + 13489049947 T^{2} - 785663657597640 T^{3} + 82884005072260939046 T^{4} -$$$$40\!\cdots\!62$$$$T^{5} + 82884005072260939046 p^{5} T^{6} - 785663657597640 p^{10} T^{7} + 13489049947 p^{15} T^{8} - 89399 p^{20} T^{9} + p^{25} T^{10}$$
89$C_2 \wr S_5$ $$1 - 96240 T + 25021582105 T^{2} - 1571226370781136 T^{3} +$$$$24\!\cdots\!30$$$$T^{4} -$$$$11\!\cdots\!48$$$$T^{5} +$$$$24\!\cdots\!30$$$$p^{5} T^{6} - 1571226370781136 p^{10} T^{7} + 25021582105 p^{15} T^{8} - 96240 p^{20} T^{9} + p^{25} T^{10}$$
97$C_2 \wr S_5$ $$1 - 216087 T + 51336597460 T^{2} - 6535822162578682 T^{3} +$$$$88\!\cdots\!11$$$$T^{4} -$$$$79\!\cdots\!66$$$$T^{5} +$$$$88\!\cdots\!11$$$$p^{5} T^{6} - 6535822162578682 p^{10} T^{7} + 51336597460 p^{15} T^{8} - 216087 p^{20} T^{9} + p^{25} T^{10}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$