# Properties

 Label 12-2e30-1.1-c7e6-0-0 Degree $12$ Conductor $1073741824$ Sign $1$ Analytic cond. $997794.$ Root an. cond. $3.16169$ Motivic weight $7$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 688·7-s + 5.10e3·9-s + 1.45e3·17-s + 1.29e3·23-s + 2.14e5·25-s + 8.92e4·31-s + 5.21e5·41-s − 1.56e6·47-s − 2.48e6·49-s + 3.51e6·63-s + 7.59e6·71-s + 2.08e6·73-s − 1.60e7·79-s + 1.12e7·81-s + 2.16e6·89-s − 1.08e6·97-s + 6.64e7·103-s − 4.62e6·113-s + 9.98e5·119-s + 6.46e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 7.40e6·153-s + ⋯
 L(s)  = 1 + 0.758·7-s + 2.33·9-s + 0.0716·17-s + 0.0222·23-s + 2.74·25-s + 0.538·31-s + 1.18·41-s − 2.20·47-s − 3.02·49-s + 1.76·63-s + 2.51·71-s + 0.628·73-s − 3.65·79-s + 2.34·81-s + 0.326·89-s − 0.121·97-s + 5.99·103-s − 0.301·113-s + 0.0543·119-s + 3.31·121-s + 0.167·153-s + 0.0168·161-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$2^{30}$$ Sign: $1$ Analytic conductor: $$997794.$$ Root analytic conductor: $$3.16169$$ Motivic weight: $$7$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{32} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 2^{30} ,\ ( \ : [7/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(4)$$ $$\approx$$ $$10.62333989$$ $$L(\frac12)$$ $$\approx$$ $$10.62333989$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 - 5102 T^{2} + 547829 p^{3} T^{4} - 397562596 p^{4} T^{6} + 547829 p^{17} T^{8} - 5102 p^{28} T^{10} + p^{42} T^{12}$$
5 $$1 - 214718 T^{2} + 1269826431 p^{2} T^{4} - 4608734769796 p^{4} T^{6} + 1269826431 p^{16} T^{8} - 214718 p^{28} T^{10} + p^{42} T^{12}$$
7 $$( 1 - 344 T + 1422245 T^{2} - 124668368 T^{3} + 1422245 p^{7} T^{4} - 344 p^{14} T^{5} + p^{21} T^{6} )^{2}$$
11 $$1 - 64629022 T^{2} + 2363172524483783 T^{4} -$$$$55\!\cdots\!04$$$$T^{6} + 2363172524483783 p^{14} T^{8} - 64629022 p^{28} T^{10} + p^{42} T^{12}$$
13 $$1 - 205410958 T^{2} + 1687860473809811 p T^{4} -$$$$16\!\cdots\!64$$$$T^{6} + 1687860473809811 p^{15} T^{8} - 205410958 p^{28} T^{10} + p^{42} T^{12}$$
17 $$( 1 - 726 T + 323301023 T^{2} - 9708009717300 T^{3} + 323301023 p^{7} T^{4} - 726 p^{14} T^{5} + p^{21} T^{6} )^{2}$$
19 $$1 - 2002416334 T^{2} + 2872909317854295863 T^{4} -$$$$28\!\cdots\!20$$$$T^{6} + 2872909317854295863 p^{14} T^{8} - 2002416334 p^{28} T^{10} + p^{42} T^{12}$$
23 $$( 1 - 648 T + 9708521717 T^{2} - 6547475963760 T^{3} + 9708521717 p^{7} T^{4} - 648 p^{14} T^{5} + p^{21} T^{6} )^{2}$$
29 $$1 - 47836636078 T^{2} +$$$$14\!\cdots\!79$$$$T^{4} -$$$$30\!\cdots\!44$$$$T^{6} +$$$$14\!\cdots\!79$$$$p^{14} T^{8} - 47836636078 p^{28} T^{10} + p^{42} T^{12}$$
31 $$( 1 - 1440 p T + 48354349725 T^{2} - 4324137771289408 T^{3} + 48354349725 p^{7} T^{4} - 1440 p^{15} T^{5} + p^{21} T^{6} )^{2}$$
37 $$1 - 78937168126 T^{2} +$$$$11\!\cdots\!51$$$$T^{4} -$$$$18\!\cdots\!48$$$$T^{6} +$$$$11\!\cdots\!51$$$$p^{14} T^{8} - 78937168126 p^{28} T^{10} + p^{42} T^{12}$$
41 $$( 1 - 260622 T + 351195126263 T^{2} - 83834535574878564 T^{3} + 351195126263 p^{7} T^{4} - 260622 p^{14} T^{5} + p^{21} T^{6} )^{2}$$
43 $$1 - 1505999929054 T^{2} +$$$$97\!\cdots\!71$$$$T^{4} -$$$$34\!\cdots\!32$$$$T^{6} +$$$$97\!\cdots\!71$$$$p^{14} T^{8} - 1505999929054 p^{28} T^{10} + p^{42} T^{12}$$
47 $$( 1 + 783216 T + 1470820452333 T^{2} + 778339576797138720 T^{3} + 1470820452333 p^{7} T^{4} + 783216 p^{14} T^{5} + p^{21} T^{6} )^{2}$$
53 $$1 - 3131635055902 T^{2} +$$$$64\!\cdots\!63$$$$T^{4} -$$$$86\!\cdots\!56$$$$T^{6} +$$$$64\!\cdots\!63$$$$p^{14} T^{8} - 3131635055902 p^{28} T^{10} + p^{42} T^{12}$$
59 $$1 - 8312323804862 T^{2} +$$$$38\!\cdots\!03$$$$T^{4} -$$$$11\!\cdots\!64$$$$T^{6} +$$$$38\!\cdots\!03$$$$p^{14} T^{8} - 8312323804862 p^{28} T^{10} + p^{42} T^{12}$$
61 $$1 - 14350504934382 T^{2} +$$$$96\!\cdots\!03$$$$T^{4} -$$$$38\!\cdots\!24$$$$T^{6} +$$$$96\!\cdots\!03$$$$p^{14} T^{8} - 14350504934382 p^{28} T^{10} + p^{42} T^{12}$$
67 $$1 - 35072892237678 T^{2} +$$$$51\!\cdots\!43$$$$T^{4} -$$$$41\!\cdots\!64$$$$T^{6} +$$$$51\!\cdots\!43$$$$p^{14} T^{8} - 35072892237678 p^{28} T^{10} + p^{42} T^{12}$$
71 $$( 1 - 3798552 T + 17820666583269 T^{2} - 72937737977373055056 T^{3} + 17820666583269 p^{7} T^{4} - 3798552 p^{14} T^{5} + p^{21} T^{6} )^{2}$$
73 $$( 1 - 1044782 T + 23207418111255 T^{2} - 22868192285089705636 T^{3} + 23207418111255 p^{7} T^{4} - 1044782 p^{14} T^{5} + p^{21} T^{6} )^{2}$$
79 $$( 1 + 8007952 T + 49828330384013 T^{2} +$$$$25\!\cdots\!96$$$$T^{3} + 49828330384013 p^{7} T^{4} + 8007952 p^{14} T^{5} + p^{21} T^{6} )^{2}$$
83 $$1 - 124932236904014 T^{2} +$$$$70\!\cdots\!51$$$$T^{4} -$$$$23\!\cdots\!92$$$$T^{6} +$$$$70\!\cdots\!51$$$$p^{14} T^{8} - 124932236904014 p^{28} T^{10} + p^{42} T^{12}$$
89 $$( 1 - 1084542 T + 130289056617383 T^{2} - 94766843887733093316 T^{3} + 130289056617383 p^{7} T^{4} - 1084542 p^{14} T^{5} + p^{21} T^{6} )^{2}$$
97 $$( 1 + 544154 T + 184934786492783 T^{2} + 86271378317799707180 T^{3} + 184934786492783 p^{7} T^{4} + 544154 p^{14} T^{5} + p^{21} T^{6} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$