# Properties

 Label 6-768e3-1.1-c3e3-0-2 Degree $6$ Conductor $452984832$ Sign $-1$ Analytic cond. $93042.6$ Root an. cond. $6.73152$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $3$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 9·3-s − 10·5-s + 14·7-s + 54·9-s − 52·13-s + 90·15-s + 26·17-s − 28·19-s − 126·21-s + 164·23-s − 111·25-s − 270·27-s − 174·29-s + 318·31-s − 140·35-s − 296·37-s + 468·39-s − 118·41-s + 260·43-s − 540·45-s + 204·47-s − 253·49-s − 234·51-s − 1.08e3·53-s + 252·57-s − 196·59-s − 1.53e3·61-s + ⋯
 L(s)  = 1 − 1.73·3-s − 0.894·5-s + 0.755·7-s + 2·9-s − 1.10·13-s + 1.54·15-s + 0.370·17-s − 0.338·19-s − 1.30·21-s + 1.48·23-s − 0.887·25-s − 1.92·27-s − 1.11·29-s + 1.84·31-s − 0.676·35-s − 1.31·37-s + 1.92·39-s − 0.449·41-s + 0.922·43-s − 1.78·45-s + 0.633·47-s − 0.737·49-s − 0.642·51-s − 2.81·53-s + 0.585·57-s − 0.432·59-s − 3.22·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$2^{24} \cdot 3^{3}$$ Sign: $-1$ Analytic conductor: $$93042.6$$ Root analytic conductor: $$6.73152$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(6,\ 2^{24} \cdot 3^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 $$1$$
3$C_1$ $$( 1 + p T )^{3}$$
good5$S_4\times C_2$ $$1 + 2 p T + 211 T^{2} + 2396 T^{3} + 211 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6}$$
7$S_4\times C_2$ $$1 - 2 p T + 449 T^{2} - 3788 T^{3} + 449 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6}$$
11$S_4\times C_2$ $$1 + 107 p T^{2} - 49152 T^{3} + 107 p^{4} T^{4} + p^{9} T^{6}$$
13$S_4\times C_2$ $$1 + 4 p T + 5487 T^{2} + 172616 T^{3} + 5487 p^{3} T^{4} + 4 p^{7} T^{5} + p^{9} T^{6}$$
17$S_4\times C_2$ $$1 - 26 T + 3615 T^{2} + 222100 T^{3} + 3615 p^{3} T^{4} - 26 p^{6} T^{5} + p^{9} T^{6}$$
19$S_4\times C_2$ $$1 + 28 T + 9489 T^{2} + 658856 T^{3} + 9489 p^{3} T^{4} + 28 p^{6} T^{5} + p^{9} T^{6}$$
23$S_4\times C_2$ $$1 - 164 T + 42885 T^{2} - 4036280 T^{3} + 42885 p^{3} T^{4} - 164 p^{6} T^{5} + p^{9} T^{6}$$
29$S_4\times C_2$ $$1 + 6 p T + 77131 T^{2} + 8426004 T^{3} + 77131 p^{3} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6}$$
31$S_4\times C_2$ $$1 - 318 T + 93849 T^{2} - 15197452 T^{3} + 93849 p^{3} T^{4} - 318 p^{6} T^{5} + p^{9} T^{6}$$
37$S_4\times C_2$ $$1 + 8 p T + 105879 T^{2} + 29903632 T^{3} + 105879 p^{3} T^{4} + 8 p^{7} T^{5} + p^{9} T^{6}$$
41$S_4\times C_2$ $$1 + 118 T + 89463 T^{2} - 3720620 T^{3} + 89463 p^{3} T^{4} + 118 p^{6} T^{5} + p^{9} T^{6}$$
43$S_4\times C_2$ $$1 - 260 T + 157545 T^{2} - 32742232 T^{3} + 157545 p^{3} T^{4} - 260 p^{6} T^{5} + p^{9} T^{6}$$
47$S_4\times C_2$ $$1 - 204 T + 283677 T^{2} - 40395048 T^{3} + 283677 p^{3} T^{4} - 204 p^{6} T^{5} + p^{9} T^{6}$$
53$S_4\times C_2$ $$1 + 1086 T + 736099 T^{2} + 344026068 T^{3} + 736099 p^{3} T^{4} + 1086 p^{6} T^{5} + p^{9} T^{6}$$
59$S_4\times C_2$ $$1 + 196 T + 566137 T^{2} + 71984984 T^{3} + 566137 p^{3} T^{4} + 196 p^{6} T^{5} + p^{9} T^{6}$$
61$S_4\times C_2$ $$1 + 1536 T + 1409583 T^{2} + 802126848 T^{3} + 1409583 p^{3} T^{4} + 1536 p^{6} T^{5} + p^{9} T^{6}$$
67$S_4\times C_2$ $$1 - 660 T + 592833 T^{2} - 364778232 T^{3} + 592833 p^{3} T^{4} - 660 p^{6} T^{5} + p^{9} T^{6}$$
71$S_4\times C_2$ $$1 + 12 p T + 1006773 T^{2} + 524795352 T^{3} + 1006773 p^{3} T^{4} + 12 p^{7} T^{5} + p^{9} T^{6}$$
73$S_4\times C_2$ $$1 + 478 T + 911095 T^{2} + 251066948 T^{3} + 911095 p^{3} T^{4} + 478 p^{6} T^{5} + p^{9} T^{6}$$
79$S_4\times C_2$ $$1 - 22 T + 1407593 T^{2} - 13791100 T^{3} + 1407593 p^{3} T^{4} - 22 p^{6} T^{5} + p^{9} T^{6}$$
83$S_4\times C_2$ $$1 + 1136 T + 2100385 T^{2} + 1337053600 T^{3} + 2100385 p^{3} T^{4} + 1136 p^{6} T^{5} + p^{9} T^{6}$$
89$S_4\times C_2$ $$1 - 110 T + 2073543 T^{2} - 156516836 T^{3} + 2073543 p^{3} T^{4} - 110 p^{6} T^{5} + p^{9} T^{6}$$
97$S_4\times C_2$ $$1 + 1222 T + 2989679 T^{2} + 2155770388 T^{3} + 2989679 p^{3} T^{4} + 1222 p^{6} T^{5} + p^{9} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$