# Properties

 Label 10-207e5-1.1-c5e5-0-0 Degree $10$ Conductor $380059617807$ Sign $-1$ Analytic cond. $4.03324\times 10^{7}$ Root an. cond. $5.76189$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $5$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·2-s + 11·4-s − 94·5-s + 272·7-s + 82·8-s + 752·10-s − 1.10e3·11-s − 978·13-s − 2.17e3·14-s − 915·16-s − 2.52e3·17-s + 2.06e3·19-s − 1.03e3·20-s + 8.80e3·22-s + 2.64e3·23-s + 2.62e3·25-s + 7.82e3·26-s + 2.99e3·28-s − 1.52e3·29-s − 7.39e3·31-s + 7.53e3·32-s + 2.01e4·34-s − 2.55e4·35-s − 8.21e3·37-s − 1.64e4·38-s − 7.70e3·40-s − 2.12e4·41-s + ⋯
 L(s)  = 1 − 1.41·2-s + 0.343·4-s − 1.68·5-s + 2.09·7-s + 0.452·8-s + 2.37·10-s − 2.74·11-s − 1.60·13-s − 2.96·14-s − 0.893·16-s − 2.11·17-s + 1.30·19-s − 0.578·20-s + 3.87·22-s + 1.04·23-s + 0.839·25-s + 2.26·26-s + 0.721·28-s − 0.336·29-s − 1.38·31-s + 1.30·32-s + 2.99·34-s − 3.52·35-s − 0.985·37-s − 1.85·38-s − 0.761·40-s − 1.97·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \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(3^{10} \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: $$3^{10} \cdot 23^{5}$$ Sign: $-1$ Analytic conductor: $$4.03324\times 10^{7}$$ Root analytic conductor: $$5.76189$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{207} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$5$$ Selberg data: $$(10,\ 3^{10} \cdot 23^{5} ,\ ( \ : 5/2, 5/2, 5/2, 5/2, 5/2 ),\ -1 )$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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)$
bad3 $$1$$
23$C_1$ $$( 1 - p^{2} T )^{5}$$
good2$C_2 \wr S_5$ $$1 + p^{3} T + 53 T^{2} + 127 p T^{3} + 427 p^{2} T^{4} + 789 p^{3} T^{5} + 427 p^{7} T^{6} + 127 p^{11} T^{7} + 53 p^{15} T^{8} + p^{23} T^{9} + p^{25} T^{10}$$
5$C_2 \wr S_5$ $$1 + 94 T + 6213 T^{2} + 233272 T^{3} + 16438526 T^{4} + 918832724 T^{5} + 16438526 p^{5} T^{6} + 233272 p^{10} T^{7} + 6213 p^{15} T^{8} + 94 p^{20} T^{9} + p^{25} T^{10}$$
7$C_2 \wr S_5$ $$1 - 272 T + 92999 T^{2} - 15921576 T^{3} + 3139766110 T^{4} - 382935152112 T^{5} + 3139766110 p^{5} T^{6} - 15921576 p^{10} T^{7} + 92999 p^{15} T^{8} - 272 p^{20} T^{9} + p^{25} T^{10}$$
11$C_2 \wr S_5$ $$1 + 100 p T + 950167 T^{2} + 47832944 p T^{3} + 266170957962 T^{4} + 106836876092680 T^{5} + 266170957962 p^{5} T^{6} + 47832944 p^{11} T^{7} + 950167 p^{15} T^{8} + 100 p^{21} T^{9} + p^{25} T^{10}$$
13$C_2 \wr S_5$ $$1 + 978 T + 1491369 T^{2} + 962007576 T^{3} + 67101754522 p T^{4} + 442247737686732 T^{5} + 67101754522 p^{6} T^{6} + 962007576 p^{10} T^{7} + 1491369 p^{15} T^{8} + 978 p^{20} T^{9} + p^{25} T^{10}$$
17$C_2 \wr S_5$ $$1 + 2522 T + 8504033 T^{2} + 13985869736 T^{3} + 25829300837998 T^{4} + 29533869359617308 T^{5} + 25829300837998 p^{5} T^{6} + 13985869736 p^{10} T^{7} + 8504033 p^{15} T^{8} + 2522 p^{20} T^{9} + p^{25} T^{10}$$
19$C_2 \wr S_5$ $$1 - 2060 T + 429073 p T^{2} - 6053371160 T^{3} + 17747857284814 T^{4} - 1707014975296600 T^{5} + 17747857284814 p^{5} T^{6} - 6053371160 p^{10} T^{7} + 429073 p^{16} T^{8} - 2060 p^{20} T^{9} + p^{25} T^{10}$$
29$C_2 \wr S_5$ $$1 + 1526 T + 53450201 T^{2} + 160216816776 T^{3} + 1463826216452722 T^{4} + 4967385842939808356 T^{5} + 1463826216452722 p^{5} T^{6} + 160216816776 p^{10} T^{7} + 53450201 p^{15} T^{8} + 1526 p^{20} T^{9} + p^{25} T^{10}$$
31$C_2 \wr S_5$ $$1 + 7392 T + 143582859 T^{2} + 814586108224 T^{3} + 8222478851768122 T^{4} + 34540989938511230912 T^{5} + 8222478851768122 p^{5} T^{6} + 814586108224 p^{10} T^{7} + 143582859 p^{15} T^{8} + 7392 p^{20} T^{9} + p^{25} T^{10}$$
37$C_2 \wr S_5$ $$1 + 8210 T + 333147057 T^{2} + 2158433364600 T^{3} + 45276559134316162 T^{4} +$$$$22\!\cdots\!00$$$$T^{5} + 45276559134316162 p^{5} T^{6} + 2158433364600 p^{10} T^{7} + 333147057 p^{15} T^{8} + 8210 p^{20} T^{9} + p^{25} T^{10}$$
41$C_2 \wr S_5$ $$1 + 21250 T + 285983317 T^{2} + 1967560460248 T^{3} + 17015437600068850 T^{4} +$$$$11\!\cdots\!08$$$$T^{5} + 17015437600068850 p^{5} T^{6} + 1967560460248 p^{10} T^{7} + 285983317 p^{15} T^{8} + 21250 p^{20} T^{9} + p^{25} T^{10}$$
43$C_2 \wr S_5$ $$1 + 4548 T + 287763707 T^{2} + 3207650046632 T^{3} + 68028185295500398 T^{4} +$$$$50\!\cdots\!76$$$$T^{5} + 68028185295500398 p^{5} T^{6} + 3207650046632 p^{10} T^{7} + 287763707 p^{15} T^{8} + 4548 p^{20} T^{9} + p^{25} T^{10}$$
47$C_2 \wr S_5$ $$1 + 536 T + 264020283 T^{2} + 2501526488736 T^{3} + 112580424453867674 T^{4} +$$$$14\!\cdots\!12$$$$T^{5} + 112580424453867674 p^{5} T^{6} + 2501526488736 p^{10} T^{7} + 264020283 p^{15} T^{8} + 536 p^{20} T^{9} + p^{25} T^{10}$$
53$C_2 \wr S_5$ $$1 - 11482 T + 1116302917 T^{2} + 941657798856 T^{3} + 483029536217021550 T^{4} +$$$$40\!\cdots\!16$$$$T^{5} + 483029536217021550 p^{5} T^{6} + 941657798856 p^{10} T^{7} + 1116302917 p^{15} T^{8} - 11482 p^{20} T^{9} + p^{25} T^{10}$$
59$C_2 \wr S_5$ $$1 + 74676 T + 4089190183 T^{2} + 163708505958576 T^{3} + 5812674984278443178 T^{4} +$$$$16\!\cdots\!84$$$$T^{5} + 5812674984278443178 p^{5} T^{6} + 163708505958576 p^{10} T^{7} + 4089190183 p^{15} T^{8} + 74676 p^{20} T^{9} + p^{25} T^{10}$$
61$C_2 \wr S_5$ $$1 + 44618 T + 4437282569 T^{2} + 138965692205272 T^{3} + 7551174855833493282 T^{4} +$$$$17\!\cdots\!68$$$$T^{5} + 7551174855833493282 p^{5} T^{6} + 138965692205272 p^{10} T^{7} + 4437282569 p^{15} T^{8} + 44618 p^{20} T^{9} + p^{25} T^{10}$$
67$C_2 \wr S_5$ $$1 + 1412 T + 4040959155 T^{2} - 20057652231432 T^{3} + 7903786473880913422 T^{4} -$$$$60\!\cdots\!48$$$$T^{5} + 7903786473880913422 p^{5} T^{6} - 20057652231432 p^{10} T^{7} + 4040959155 p^{15} T^{8} + 1412 p^{20} T^{9} + p^{25} T^{10}$$
71$C_2 \wr S_5$ $$1 + 37912 T + 1466467779 T^{2} - 7879531293536 T^{3} + 2981493544848827562 T^{4} +$$$$14\!\cdots\!32$$$$T^{5} + 2981493544848827562 p^{5} T^{6} - 7879531293536 p^{10} T^{7} + 1466467779 p^{15} T^{8} + 37912 p^{20} T^{9} + p^{25} T^{10}$$
73$C_2 \wr S_5$ $$1 - 46546 T + 8346436661 T^{2} - 319480387049752 T^{3} + 31398063765510760370 T^{4} -$$$$92\!\cdots\!96$$$$T^{5} + 31398063765510760370 p^{5} T^{6} - 319480387049752 p^{10} T^{7} + 8346436661 p^{15} T^{8} - 46546 p^{20} T^{9} + p^{25} T^{10}$$
79$C_2 \wr S_5$ $$1 - 50544 T + 6561571119 T^{2} - 53943736845944 T^{3} + 15591070525869009022 T^{4} +$$$$22\!\cdots\!12$$$$T^{5} + 15591070525869009022 p^{5} T^{6} - 53943736845944 p^{10} T^{7} + 6561571119 p^{15} T^{8} - 50544 p^{20} T^{9} + p^{25} T^{10}$$
83$C_2 \wr S_5$ $$1 + 89588 T + 10451924351 T^{2} + 775319273971120 T^{3} + 61260146630661454026 T^{4} +$$$$36\!\cdots\!20$$$$T^{5} + 61260146630661454026 p^{5} T^{6} + 775319273971120 p^{10} T^{7} + 10451924351 p^{15} T^{8} + 89588 p^{20} T^{9} + p^{25} T^{10}$$
89$C_2 \wr S_5$ $$1 + 280410 T + 47711380473 T^{2} + 5653018849483864 T^{3} +$$$$53\!\cdots\!70$$$$T^{4} +$$$$42\!\cdots\!76$$$$T^{5} +$$$$53\!\cdots\!70$$$$p^{5} T^{6} + 5653018849483864 p^{10} T^{7} + 47711380473 p^{15} T^{8} + 280410 p^{20} T^{9} + p^{25} T^{10}$$
97$C_2 \wr S_5$ $$1 - 90074 T + 36682650013 T^{2} - 2072067856619224 T^{3} +$$$$53\!\cdots\!94$$$$T^{4} -$$$$21\!\cdots\!28$$$$T^{5} +$$$$53\!\cdots\!94$$$$p^{5} T^{6} - 2072067856619224 p^{10} T^{7} + 36682650013 p^{15} T^{8} - 90074 p^{20} T^{9} + p^{25} T^{10}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$