# Properties

 Degree 12 Conductor $3^{6} \cdot 7^{6}$ Sign $1$ Motivic weight 3 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s + 9·3-s − 11·5-s − 9·6-s − 13·7-s + 21·8-s + 27·9-s + 11·10-s − 35·11-s + 124·13-s + 13·14-s − 99·15-s − 31·16-s − 48·17-s − 27·18-s + 202·19-s − 117·21-s + 35·22-s − 216·23-s + 189·24-s + 183·25-s − 124·26-s − 54·27-s + 106·29-s + 99·30-s + 95·31-s − 64·32-s + ⋯
 L(s)  = 1 − 0.353·2-s + 1.73·3-s − 0.983·5-s − 0.612·6-s − 0.701·7-s + 0.928·8-s + 9-s + 0.347·10-s − 0.959·11-s + 2.64·13-s + 0.248·14-s − 1.70·15-s − 0.484·16-s − 0.684·17-s − 0.353·18-s + 2.43·19-s − 1.21·21-s + 0.339·22-s − 1.95·23-s + 1.60·24-s + 1.46·25-s − 0.935·26-s − 0.384·27-s + 0.678·29-s + 0.602·30-s + 0.550·31-s − 0.353·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 85766121 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 85766121 ^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$12$$ $$N$$ = $$85766121$$    =    $$3^{6} \cdot 7^{6}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{21} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(12,\ 85766121,\ (\ :[3/2]^{6}),\ 1)$ $L(2)$ $\approx$ $1.74113$ $L(\frac12)$ $\approx$ $1.74113$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 12. If $p \in \{3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad3 $$( 1 - p T + p^{2} T^{2} )^{3}$$
7 $$1 + 13 T + 236 T^{2} + 1735 p T^{3} + 236 p^{3} T^{4} + 13 p^{6} T^{5} + p^{9} T^{6}$$
good2 $$1 + T + T^{2} - 5 p^{2} T^{3} - 5 p T^{4} + p^{6} T^{5} + 265 p^{2} T^{6} + p^{9} T^{7} - 5 p^{7} T^{8} - 5 p^{11} T^{9} + p^{12} T^{10} + p^{15} T^{11} + p^{18} T^{12}$$
5 $$1 + 11 T - 62 T^{2} - 203 p T^{3} - 1208 p T^{4} - 54313 T^{5} + 121696 T^{6} - 54313 p^{3} T^{7} - 1208 p^{7} T^{8} - 203 p^{10} T^{9} - 62 p^{12} T^{10} + 11 p^{15} T^{11} + p^{18} T^{12}$$
11 $$1 + 35 T - 1400 T^{2} - 113593 T^{3} - 198940 T^{4} + 87110135 T^{5} + 3928586038 T^{6} + 87110135 p^{3} T^{7} - 198940 p^{6} T^{8} - 113593 p^{9} T^{9} - 1400 p^{12} T^{10} + 35 p^{15} T^{11} + p^{18} T^{12}$$
13 $$( 1 - 62 T + 7016 T^{2} - 253976 T^{3} + 7016 p^{3} T^{4} - 62 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
17 $$1 + 48 T - 10035 T^{2} - 125232 T^{3} + 74409318 T^{4} - 234420432 T^{5} - 437742983351 T^{6} - 234420432 p^{3} T^{7} + 74409318 p^{6} T^{8} - 125232 p^{9} T^{9} - 10035 p^{12} T^{10} + 48 p^{15} T^{11} + p^{18} T^{12}$$
19 $$1 - 202 T + 7946 T^{2} - 627636 T^{3} + 247297462 T^{4} - 17185599794 T^{5} + 349471935958 T^{6} - 17185599794 p^{3} T^{7} + 247297462 p^{6} T^{8} - 627636 p^{9} T^{9} + 7946 p^{12} T^{10} - 202 p^{15} T^{11} + p^{18} T^{12}$$
23 $$1 + 216 T + 10827 T^{2} + 387864 T^{3} + 53856198 T^{4} - 24653558952 T^{5} - 5413409425505 T^{6} - 24653558952 p^{3} T^{7} + 53856198 p^{6} T^{8} + 387864 p^{9} T^{9} + 10827 p^{12} T^{10} + 216 p^{15} T^{11} + p^{18} T^{12}$$
29 $$( 1 - 53 T + 52695 T^{2} - 3410210 T^{3} + 52695 p^{3} T^{4} - 53 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
31 $$1 - 95 T - 70347 T^{2} + 3756594 T^{3} + 3398738767 T^{4} - 83374434539 T^{5} - 110906046363338 T^{6} - 83374434539 p^{3} T^{7} + 3398738767 p^{6} T^{8} + 3756594 p^{9} T^{9} - 70347 p^{12} T^{10} - 95 p^{15} T^{11} + p^{18} T^{12}$$
37 $$1 + 262 T - 97404 T^{2} - 9678072 T^{3} + 12194182072 T^{4} + 680381910454 T^{5} - 605701122868778 T^{6} + 680381910454 p^{3} T^{7} + 12194182072 p^{6} T^{8} - 9678072 p^{9} T^{9} - 97404 p^{12} T^{10} + 262 p^{15} T^{11} + p^{18} T^{12}$$
41 $$( 1 - 244 T + 187983 T^{2} - 33933832 T^{3} + 187983 p^{3} T^{4} - 244 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
43 $$( 1 - 360 T + 166158 T^{2} - 38975294 T^{3} + 166158 p^{3} T^{4} - 360 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
47 $$1 - 210 T - 20853 T^{2} + 83809446 T^{3} - 12756928590 T^{4} - 2596137940074 T^{5} + 3698984470026571 T^{6} - 2596137940074 p^{3} T^{7} - 12756928590 p^{6} T^{8} + 83809446 p^{9} T^{9} - 20853 p^{12} T^{10} - 210 p^{15} T^{11} + p^{18} T^{12}$$
53 $$1 + 393 T - 211446 T^{2} - 23899125 T^{3} + 46453564620 T^{4} - 3425920762143 T^{5} - 9724787230272680 T^{6} - 3425920762143 p^{3} T^{7} + 46453564620 p^{6} T^{8} - 23899125 p^{9} T^{9} - 211446 p^{12} T^{10} + 393 p^{15} T^{11} + p^{18} T^{12}$$
59 $$1 + 1143 T + 557208 T^{2} + 118327563 T^{3} - 14314666608 T^{4} - 458696646099 p T^{5} - 16891447327378130 T^{6} - 458696646099 p^{4} T^{7} - 14314666608 p^{6} T^{8} + 118327563 p^{9} T^{9} + 557208 p^{12} T^{10} + 1143 p^{15} T^{11} + p^{18} T^{12}$$
61 $$1 - 70 T - 335143 T^{2} - 129510330 T^{3} + 42145697866 T^{4} + 25171752927730 T^{5} - 316289217432887 T^{6} + 25171752927730 p^{3} T^{7} + 42145697866 p^{6} T^{8} - 129510330 p^{9} T^{9} - 335143 p^{12} T^{10} - 70 p^{15} T^{11} + p^{18} T^{12}$$
67 $$1 - 628 T - 202942 T^{2} + 436381932 T^{3} - 77667044702 T^{4} - 1097444953952 p T^{5} + 16943668917790 p^{2} T^{6} - 1097444953952 p^{4} T^{7} - 77667044702 p^{6} T^{8} + 436381932 p^{9} T^{9} - 202942 p^{12} T^{10} - 628 p^{15} T^{11} + p^{18} T^{12}$$
71 $$( 1 - 318 T + 742929 T^{2} - 256167372 T^{3} + 742929 p^{3} T^{4} - 318 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
73 $$1 + 988 T - 186552 T^{2} - 102237300 T^{3} + 281568890272 T^{4} - 16988127696596 T^{5} - 164639785652996186 T^{6} - 16988127696596 p^{3} T^{7} + 281568890272 p^{6} T^{8} - 102237300 p^{9} T^{9} - 186552 p^{12} T^{10} + 988 p^{15} T^{11} + p^{18} T^{12}$$
79 $$1 + 861 T - 479895 T^{2} - 258646666 T^{3} + 325257480351 T^{4} - 27564282842211 T^{5} - 246706047980056146 T^{6} - 27564282842211 p^{3} T^{7} + 325257480351 p^{6} T^{8} - 258646666 p^{9} T^{9} - 479895 p^{12} T^{10} + 861 p^{15} T^{11} + p^{18} T^{12}$$
83 $$( 1 - 519 T + 1583745 T^{2} - 545598870 T^{3} + 1583745 p^{3} T^{4} - 519 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
89 $$1 + 1766 T + 725929 T^{2} - 728159446 T^{3} - 335534377858 T^{4} + 846551335831238 T^{5} + 1249625385561159997 T^{6} + 846551335831238 p^{3} T^{7} - 335534377858 p^{6} T^{8} - 728159446 p^{9} T^{9} + 725929 p^{12} T^{10} + 1766 p^{15} T^{11} + p^{18} T^{12}$$
97 $$( 1 - 19 T + 2168419 T^{2} + 10094878 T^{3} + 2168419 p^{3} T^{4} - 19 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}