# Properties

 Label 12-9680e6-1.1-c1e6-0-4 Degree $12$ Conductor $8.227\times 10^{23}$ Sign $1$ Analytic cond. $2.13262\times 10^{11}$ Root an. cond. $8.79176$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·3-s + 6·5-s − 4·7-s − 5·9-s + 12·15-s + 8·17-s − 12·19-s − 8·21-s + 8·23-s + 21·25-s − 10·27-s + 16·29-s + 4·31-s − 24·35-s + 8·37-s + 32·41-s + 4·43-s − 30·45-s + 6·47-s − 5·49-s + 16·51-s + 8·53-s − 24·57-s − 4·59-s + 16·61-s + 20·63-s + 2·67-s + ⋯
 L(s)  = 1 + 1.15·3-s + 2.68·5-s − 1.51·7-s − 5/3·9-s + 3.09·15-s + 1.94·17-s − 2.75·19-s − 1.74·21-s + 1.66·23-s + 21/5·25-s − 1.92·27-s + 2.97·29-s + 0.718·31-s − 4.05·35-s + 1.31·37-s + 4.99·41-s + 0.609·43-s − 4.47·45-s + 0.875·47-s − 5/7·49-s + 2.24·51-s + 1.09·53-s − 3.17·57-s − 0.520·59-s + 2.04·61-s + 2.51·63-s + 0.244·67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$2^{24} \cdot 5^{6} \cdot 11^{12}$$ Sign: $1$ Analytic conductor: $$2.13262\times 10^{11}$$ Root analytic conductor: $$8.79176$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 2^{24} \cdot 5^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$50.56711796$$ $$L(\frac12)$$ $$\approx$$ $$50.56711796$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$( 1 - T )^{6}$$
11 $$1$$
good3 $$1 - 2 T + p^{2} T^{2} - 2 p^{2} T^{3} + 50 T^{4} - 82 T^{5} + 181 T^{6} - 82 p T^{7} + 50 p^{2} T^{8} - 2 p^{5} T^{9} + p^{6} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}$$
7 $$1 + 4 T + 3 p T^{2} + 8 p T^{3} + 218 T^{4} + 636 T^{5} + 1993 T^{6} + 636 p T^{7} + 218 p^{2} T^{8} + 8 p^{4} T^{9} + 3 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}$$
13 $$1 + 46 T^{2} - 16 T^{3} + 935 T^{4} - 560 T^{5} + 13172 T^{6} - 560 p T^{7} + 935 p^{2} T^{8} - 16 p^{3} T^{9} + 46 p^{4} T^{10} + p^{6} T^{12}$$
17 $$1 - 8 T + 86 T^{2} - 488 T^{3} + 191 p T^{4} - 14352 T^{5} + 70772 T^{6} - 14352 p T^{7} + 191 p^{3} T^{8} - 488 p^{3} T^{9} + 86 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}$$
19 $$1 + 12 T + 142 T^{2} + 1028 T^{3} + 6983 T^{4} + 36232 T^{5} + 176372 T^{6} + 36232 p T^{7} + 6983 p^{2} T^{8} + 1028 p^{3} T^{9} + 142 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12}$$
23 $$1 - 8 T + 78 T^{2} - 464 T^{3} + 3647 T^{4} - 17448 T^{5} + 99796 T^{6} - 17448 p T^{7} + 3647 p^{2} T^{8} - 464 p^{3} T^{9} + 78 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}$$
29 $$1 - 16 T + 158 T^{2} - 976 T^{3} + 3655 T^{4} - 5856 T^{5} - 10876 T^{6} - 5856 p T^{7} + 3655 p^{2} T^{8} - 976 p^{3} T^{9} + 158 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12}$$
31 $$1 - 4 T + 106 T^{2} - 580 T^{3} + 5471 T^{4} - 35008 T^{5} + 194684 T^{6} - 35008 p T^{7} + 5471 p^{2} T^{8} - 580 p^{3} T^{9} + 106 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}$$
37 $$1 - 8 T + 142 T^{2} - 872 T^{3} + 9911 T^{4} - 52016 T^{5} + 449252 T^{6} - 52016 p T^{7} + 9911 p^{2} T^{8} - 872 p^{3} T^{9} + 142 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}$$
41 $$1 - 32 T + 597 T^{2} - 7760 T^{3} + 78494 T^{4} - 647040 T^{5} + 4497049 T^{6} - 647040 p T^{7} + 78494 p^{2} T^{8} - 7760 p^{3} T^{9} + 597 p^{4} T^{10} - 32 p^{5} T^{11} + p^{6} T^{12}$$
43 $$1 - 4 T + 157 T^{2} - 160 T^{3} + 9938 T^{4} + 11060 T^{5} + 442193 T^{6} + 11060 p T^{7} + 9938 p^{2} T^{8} - 160 p^{3} T^{9} + 157 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}$$
47 $$1 - 6 T + 217 T^{2} - 1142 T^{3} + 22146 T^{4} - 96502 T^{5} + 1330941 T^{6} - 96502 p T^{7} + 22146 p^{2} T^{8} - 1142 p^{3} T^{9} + 217 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}$$
53 $$1 - 8 T + 142 T^{2} - 624 T^{3} + 8055 T^{4} - 33704 T^{5} + 443124 T^{6} - 33704 p T^{7} + 8055 p^{2} T^{8} - 624 p^{3} T^{9} + 142 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}$$
59 $$1 + 4 T + 106 T^{2} + 804 T^{3} + 11415 T^{4} + 59440 T^{5} + 798588 T^{6} + 59440 p T^{7} + 11415 p^{2} T^{8} + 804 p^{3} T^{9} + 106 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}$$
61 $$1 - 16 T + 245 T^{2} - 2240 T^{3} + 24374 T^{4} - 199632 T^{5} + 1856313 T^{6} - 199632 p T^{7} + 24374 p^{2} T^{8} - 2240 p^{3} T^{9} + 245 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12}$$
67 $$1 - 2 T + 49 T^{2} - 458 T^{3} + 11018 T^{4} - 49970 T^{5} + 322277 T^{6} - 49970 p T^{7} + 11018 p^{2} T^{8} - 458 p^{3} T^{9} + 49 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}$$
71 $$1 - 28 T + 546 T^{2} - 7900 T^{3} + 98591 T^{4} - 1012032 T^{5} + 9200236 T^{6} - 1012032 p T^{7} + 98591 p^{2} T^{8} - 7900 p^{3} T^{9} + 546 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12}$$
73 $$1 - 16 T + 246 T^{2} - 2768 T^{3} + 28991 T^{4} - 286368 T^{5} + 2601844 T^{6} - 286368 p T^{7} + 28991 p^{2} T^{8} - 2768 p^{3} T^{9} + 246 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12}$$
79 $$1 + 342 T^{2} + 288 T^{3} + 55791 T^{4} + 50976 T^{5} + 5551220 T^{6} + 50976 p T^{7} + 55791 p^{2} T^{8} + 288 p^{3} T^{9} + 342 p^{4} T^{10} + p^{6} T^{12}$$
83 $$1 + 12 T + 210 T^{2} + 1028 T^{3} + 22311 T^{4} + 168072 T^{5} + 2881772 T^{6} + 168072 p T^{7} + 22311 p^{2} T^{8} + 1028 p^{3} T^{9} + 210 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12}$$
89 $$1 - 18 T + 253 T^{2} - 1798 T^{3} + 8058 T^{4} + 72934 T^{5} - 1088091 T^{6} + 72934 p T^{7} + 8058 p^{2} T^{8} - 1798 p^{3} T^{9} + 253 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12}$$
97 $$1 + 342 T^{2} + 1080 T^{3} + 57231 T^{4} + 248616 T^{5} + 6595652 T^{6} + 248616 p T^{7} + 57231 p^{2} T^{8} + 1080 p^{3} T^{9} + 342 p^{4} T^{10} + p^{6} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$