# Properties

 Label 12-234e6-1.1-c5e6-0-3 Degree $12$ Conductor $1.642\times 10^{14}$ Sign $1$ Analytic cond. $2.79420\times 10^{9}$ Root an. cond. $6.12615$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 48·4-s + 530·13-s + 1.53e3·16-s + 836·17-s + 416·23-s + 9.73e3·25-s − 1.87e4·29-s − 2.42e4·43-s + 4.89e4·49-s − 2.54e4·52-s + 4.23e4·53-s − 3.19e3·61-s − 4.09e4·64-s − 4.01e4·68-s − 1.69e5·79-s − 1.99e4·92-s − 4.67e5·100-s − 1.38e5·101-s − 3.64e5·103-s + 5.22e4·107-s + 3.13e5·113-s + 9.01e5·116-s + 4.88e5·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
 L(s)  = 1 − 3/2·4-s + 0.869·13-s + 3/2·16-s + 0.701·17-s + 0.163·23-s + 3.11·25-s − 4.14·29-s − 1.99·43-s + 2.90·49-s − 1.30·52-s + 2.07·53-s − 0.109·61-s − 5/4·64-s − 1.05·68-s − 3.05·79-s − 0.245·92-s − 4.67·100-s − 1.35·101-s − 3.38·103-s + 0.440·107-s + 2.31·113-s + 6.22·116-s + 3.03·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$2^{6} \cdot 3^{12} \cdot 13^{6}$$ Sign: $1$ Analytic conductor: $$2.79420\times 10^{9}$$ Root analytic conductor: $$6.12615$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 2^{6} \cdot 3^{12} \cdot 13^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$6.117908104$$ $$L(\frac12)$$ $$\approx$$ $$6.117908104$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 + p^{4} T^{2} )^{3}$$
3 $$1$$
13 $$1 - 530 T - 12785 p T^{2} + 92996 p^{3} T^{3} - 12785 p^{6} T^{4} - 530 p^{10} T^{5} + p^{15} T^{6}$$
good5 $$1 - 9734 T^{2} + 45022279 T^{4} - 151321754836 T^{6} + 45022279 p^{10} T^{8} - 9734 p^{20} T^{10} + p^{30} T^{12}$$
7 $$1 - 998 p^{2} T^{2} + 1501179695 T^{4} - 28969294894196 T^{6} + 1501179695 p^{10} T^{8} - 998 p^{22} T^{10} + p^{30} T^{12}$$
11 $$1 - 488550 T^{2} + 122692550487 T^{4} - 21659528044972276 T^{6} + 122692550487 p^{10} T^{8} - 488550 p^{20} T^{10} + p^{30} T^{12}$$
17 $$( 1 - 418 T + 1867887 T^{2} + 596217860 T^{3} + 1867887 p^{5} T^{4} - 418 p^{10} T^{5} + p^{15} T^{6} )^{2}$$
19 $$1 - 4259886 T^{2} + 15059505195735 T^{4} - 52704394717503498500 T^{6} + 15059505195735 p^{10} T^{8} - 4259886 p^{20} T^{10} + p^{30} T^{12}$$
23 $$( 1 - 208 T + 13024485 T^{2} - 5279785312 T^{3} + 13024485 p^{5} T^{4} - 208 p^{10} T^{5} + p^{15} T^{6} )^{2}$$
29 $$( 1 + 9394 T + 75091011 T^{2} + 387910039372 T^{3} + 75091011 p^{5} T^{4} + 9394 p^{10} T^{5} + p^{15} T^{6} )^{2}$$
31 $$1 - 116335670 T^{2} + 6009258573309407 T^{4} -$$$$20\!\cdots\!76$$$$T^{6} + 6009258573309407 p^{10} T^{8} - 116335670 p^{20} T^{10} + p^{30} T^{12}$$
37 $$1 - 289018862 T^{2} + 38604779572315415 T^{4} -$$$$32\!\cdots\!76$$$$T^{6} + 38604779572315415 p^{10} T^{8} - 289018862 p^{20} T^{10} + p^{30} T^{12}$$
41 $$1 - 461990702 T^{2} + 96326950481430703 T^{4} -$$$$13\!\cdots\!04$$$$T^{6} + 96326950481430703 p^{10} T^{8} - 461990702 p^{20} T^{10} + p^{30} T^{12}$$
43 $$( 1 + 12100 T + 192229169 T^{2} + 42503254792 p T^{3} + 192229169 p^{5} T^{4} + 12100 p^{10} T^{5} + p^{15} T^{6} )^{2}$$
47 $$1 - 5196642 p T^{2} + 98471047533657423 T^{4} -$$$$23\!\cdots\!32$$$$T^{6} + 98471047533657423 p^{10} T^{8} - 5196642 p^{21} T^{10} + p^{30} T^{12}$$
53 $$( 1 - 21198 T + 82289499 T^{2} + 9230236358604 T^{3} + 82289499 p^{5} T^{4} - 21198 p^{10} T^{5} + p^{15} T^{6} )^{2}$$
59 $$1 - 2269058726 T^{2} + 2942781419203692343 T^{4} -$$$$25\!\cdots\!56$$$$T^{6} + 2942781419203692343 p^{10} T^{8} - 2269058726 p^{20} T^{10} + p^{30} T^{12}$$
61 $$( 1 + 1598 T + 927987683 T^{2} + 20079713107796 T^{3} + 927987683 p^{5} T^{4} + 1598 p^{10} T^{5} + p^{15} T^{6} )^{2}$$
67 $$1 - 3626333582 T^{2} + 8616065023269477815 T^{4} -$$$$14\!\cdots\!76$$$$T^{6} + 8616065023269477815 p^{10} T^{8} - 3626333582 p^{20} T^{10} + p^{30} T^{12}$$
71 $$1 - 5654364942 T^{2} + 254047452086597385 p T^{4} -$$$$37\!\cdots\!88$$$$T^{6} + 254047452086597385 p^{11} T^{8} - 5654364942 p^{20} T^{10} + p^{30} T^{12}$$
73 $$1 - 10335109526 T^{2} + 47996850402258820223 T^{4} -$$$$12\!\cdots\!28$$$$T^{6} + 47996850402258820223 p^{10} T^{8} - 10335109526 p^{20} T^{10} + p^{30} T^{12}$$
79 $$( 1 + 84664 T + 9958851181 T^{2} + 485196394773392 T^{3} + 9958851181 p^{5} T^{4} + 84664 p^{10} T^{5} + p^{15} T^{6} )^{2}$$
83 $$1 - 22485370838 T^{2} +$$$$21\!\cdots\!95$$$$T^{4} -$$$$11\!\cdots\!04$$$$T^{6} +$$$$21\!\cdots\!95$$$$p^{10} T^{8} - 22485370838 p^{20} T^{10} + p^{30} T^{12}$$
89 $$1 - 25940313102 T^{2} +$$$$31\!\cdots\!55$$$$T^{4} -$$$$22\!\cdots\!68$$$$T^{6} +$$$$31\!\cdots\!55$$$$p^{10} T^{8} - 25940313102 p^{20} T^{10} + p^{30} T^{12}$$
97 $$1 - 34764784422 T^{2} +$$$$54\!\cdots\!75$$$$T^{4} -$$$$54\!\cdots\!16$$$$T^{6} +$$$$54\!\cdots\!75$$$$p^{10} T^{8} - 34764784422 p^{20} T^{10} + p^{30} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$