# Properties

 Label 14-1323e7-1.1-c3e7-0-3 Degree $14$ Conductor $7.094\times 10^{21}$ Sign $-1$ Analytic cond. $1.76596\times 10^{13}$ Root an. cond. $8.83513$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $7$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s − 12·4-s − 5-s − 13·8-s + 10-s − 98·11-s + 124·13-s + 70·16-s + 30·17-s − 182·19-s + 12·20-s + 98·22-s + 6·23-s − 243·25-s − 124·26-s − 323·29-s − 26·31-s + 310·32-s − 30·34-s − 112·37-s + 182·38-s + 13·40-s − 524·41-s + 8·43-s + 1.17e3·44-s − 6·46-s + 288·47-s + ⋯
 L(s)  = 1 − 0.353·2-s − 3/2·4-s − 0.0894·5-s − 0.574·8-s + 0.0316·10-s − 2.68·11-s + 2.64·13-s + 1.09·16-s + 0.428·17-s − 2.19·19-s + 0.134·20-s + 0.949·22-s + 0.0543·23-s − 1.94·25-s − 0.935·26-s − 2.06·29-s − 0.150·31-s + 1.71·32-s − 0.151·34-s − 0.497·37-s + 0.776·38-s + 0.0513·40-s − 1.99·41-s + 0.0283·43-s + 4.02·44-s − 0.0192·46-s + 0.893·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{21} \cdot 7^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{21} \cdot 7^{14}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$14$$ Conductor: $$3^{21} \cdot 7^{14}$$ Sign: $-1$ Analytic conductor: $$1.76596\times 10^{13}$$ Root analytic conductor: $$8.83513$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$7$$ Selberg data: $$(14,\ 3^{21} \cdot 7^{14} ,\ ( \ : [3/2]^{7} ),\ -1 )$$

## 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$$F_p(T)$
bad3 $$1$$
7 $$1$$
good2 $$1 + T + 13 T^{2} + 19 p T^{3} + 137 T^{4} + 191 p T^{5} + 111 p^{4} T^{6} + 347 p^{3} T^{7} + 111 p^{7} T^{8} + 191 p^{7} T^{9} + 137 p^{9} T^{10} + 19 p^{13} T^{11} + 13 p^{15} T^{12} + p^{18} T^{13} + p^{21} T^{14}$$
5 $$1 + T + 244 T^{2} + 1061 T^{3} + 46088 T^{4} + 102697 T^{5} + 7045884 T^{6} + 15303883 T^{7} + 7045884 p^{3} T^{8} + 102697 p^{6} T^{9} + 46088 p^{9} T^{10} + 1061 p^{12} T^{11} + 244 p^{15} T^{12} + p^{18} T^{13} + p^{21} T^{14}$$
11 $$1 + 98 T + 10402 T^{2} + 661102 T^{3} + 41789666 T^{4} + 1991181686 T^{5} + 92085198666 T^{6} + 3419383371188 T^{7} + 92085198666 p^{3} T^{8} + 1991181686 p^{6} T^{9} + 41789666 p^{9} T^{10} + 661102 p^{12} T^{11} + 10402 p^{15} T^{12} + 98 p^{18} T^{13} + p^{21} T^{14}$$
13 $$1 - 124 T + 15025 T^{2} - 1013574 T^{3} + 70713900 T^{4} - 3353146350 T^{5} + 185662270219 T^{6} - 7665880866988 T^{7} + 185662270219 p^{3} T^{8} - 3353146350 p^{6} T^{9} + 70713900 p^{9} T^{10} - 1013574 p^{12} T^{11} + 15025 p^{15} T^{12} - 124 p^{18} T^{13} + p^{21} T^{14}$$
17 $$1 - 30 T + 775 p T^{2} - 288576 T^{3} + 66797568 T^{4} - 2749160034 T^{5} + 251451413719 T^{6} - 20159954844798 T^{7} + 251451413719 p^{3} T^{8} - 2749160034 p^{6} T^{9} + 66797568 p^{9} T^{10} - 288576 p^{12} T^{11} + 775 p^{16} T^{12} - 30 p^{18} T^{13} + p^{21} T^{14}$$
19 $$1 + 182 T + 29302 T^{2} + 3026172 T^{3} + 336281328 T^{4} + 31202794338 T^{5} + 3230844269698 T^{6} + 263706213505892 T^{7} + 3230844269698 p^{3} T^{8} + 31202794338 p^{6} T^{9} + 336281328 p^{9} T^{10} + 3026172 p^{12} T^{11} + 29302 p^{15} T^{12} + 182 p^{18} T^{13} + p^{21} T^{14}$$
23 $$1 - 6 T + 35855 T^{2} - 614214 T^{3} + 725136552 T^{4} - 19959011478 T^{5} + 491543247209 p T^{6} - 305416510826442 T^{7} + 491543247209 p^{4} T^{8} - 19959011478 p^{6} T^{9} + 725136552 p^{9} T^{10} - 614214 p^{12} T^{11} + 35855 p^{15} T^{12} - 6 p^{18} T^{13} + p^{21} T^{14}$$
29 $$1 + 323 T + 131380 T^{2} + 26433295 T^{3} + 6721509410 T^{4} + 36989250493 p T^{5} + 224084140161810 T^{6} + 30744417281128559 T^{7} + 224084140161810 p^{3} T^{8} + 36989250493 p^{7} T^{9} + 6721509410 p^{9} T^{10} + 26433295 p^{12} T^{11} + 131380 p^{15} T^{12} + 323 p^{18} T^{13} + p^{21} T^{14}$$
31 $$1 + 26 T + 124399 T^{2} - 4940326 T^{3} + 7382974102 T^{4} - 505783122130 T^{5} + 311065932374375 T^{6} - 19611903042307532 T^{7} + 311065932374375 p^{3} T^{8} - 505783122130 p^{6} T^{9} + 7382974102 p^{9} T^{10} - 4940326 p^{12} T^{11} + 124399 p^{15} T^{12} + 26 p^{18} T^{13} + p^{21} T^{14}$$
37 $$1 + 112 T + 245821 T^{2} + 26735158 T^{3} + 27401983396 T^{4} + 2831603027422 T^{5} + 1918236642638039 T^{6} + 178816583252903720 T^{7} + 1918236642638039 p^{3} T^{8} + 2831603027422 p^{6} T^{9} + 27401983396 p^{9} T^{10} + 26735158 p^{12} T^{11} + 245821 p^{15} T^{12} + 112 p^{18} T^{13} + p^{21} T^{14}$$
41 $$1 + 524 T + 9803 p T^{2} + 170338510 T^{3} + 70617228158 T^{4} + 24982111123952 T^{5} + 7347401476913313 T^{6} + 2168422196358646010 T^{7} + 7347401476913313 p^{3} T^{8} + 24982111123952 p^{6} T^{9} + 70617228158 p^{9} T^{10} + 170338510 p^{12} T^{11} + 9803 p^{16} T^{12} + 524 p^{18} T^{13} + p^{21} T^{14}$$
43 $$1 - 8 T + 353242 T^{2} + 9268682 T^{3} + 61414621286 T^{4} + 3001953039602 T^{5} + 6862933728305412 T^{6} + 351363943325456376 T^{7} + 6862933728305412 p^{3} T^{8} + 3001953039602 p^{6} T^{9} + 61414621286 p^{9} T^{10} + 9268682 p^{12} T^{11} + 353242 p^{15} T^{12} - 8 p^{18} T^{13} + p^{21} T^{14}$$
47 $$1 - 288 T + 450314 T^{2} - 67426956 T^{3} + 87334195302 T^{4} - 6539208203268 T^{5} + 11562318530585206 T^{6} - 570886399998335178 T^{7} + 11562318530585206 p^{3} T^{8} - 6539208203268 p^{6} T^{9} + 87334195302 p^{9} T^{10} - 67426956 p^{12} T^{11} + 450314 p^{15} T^{12} - 288 p^{18} T^{13} + p^{21} T^{14}$$
53 $$1 + 1353 T + 1639295 T^{2} + 1249499466 T^{3} + 869621721966 T^{4} + 463591988073504 T^{5} + 229217965106472457 T^{6} + 91610319140078074059 T^{7} + 229217965106472457 p^{3} T^{8} + 463591988073504 p^{6} T^{9} + 869621721966 p^{9} T^{10} + 1249499466 p^{12} T^{11} + 1639295 p^{15} T^{12} + 1353 p^{18} T^{13} + p^{21} T^{14}$$
59 $$1 - 165 T + 574820 T^{2} - 60298731 T^{3} + 194516457966 T^{4} - 16863172789689 T^{5} + 48115196810244664 T^{6} - 2898325322084853885 T^{7} + 48115196810244664 p^{3} T^{8} - 16863172789689 p^{6} T^{9} + 194516457966 p^{9} T^{10} - 60298731 p^{12} T^{11} + 574820 p^{15} T^{12} - 165 p^{18} T^{13} + p^{21} T^{14}$$
61 $$1 + 56 T + 1129930 T^{2} + 140067534 T^{3} + 608440256811 T^{4} + 87941190763644 T^{5} + 205764800561570935 T^{6} + 26845919604893843258 T^{7} + 205764800561570935 p^{3} T^{8} + 87941190763644 p^{6} T^{9} + 608440256811 p^{9} T^{10} + 140067534 p^{12} T^{11} + 1129930 p^{15} T^{12} + 56 p^{18} T^{13} + p^{21} T^{14}$$
67 $$1 + 988 T + 630817 T^{2} + 252424194 T^{3} + 280927015218 T^{4} + 188869245036312 T^{5} + 88728243512682457 T^{6} + 26050094387218060762 T^{7} + 88728243512682457 p^{3} T^{8} + 188869245036312 p^{6} T^{9} + 280927015218 p^{9} T^{10} + 252424194 p^{12} T^{11} + 630817 p^{15} T^{12} + 988 p^{18} T^{13} + p^{21} T^{14}$$
71 $$1 + 792 T + 1000694 T^{2} + 537655860 T^{3} + 423192481410 T^{4} + 212806715723784 T^{5} + 176788376355568834 T^{6} + 84887120455713214074 T^{7} + 176788376355568834 p^{3} T^{8} + 212806715723784 p^{6} T^{9} + 423192481410 p^{9} T^{10} + 537655860 p^{12} T^{11} + 1000694 p^{15} T^{12} + 792 p^{18} T^{13} + p^{21} T^{14}$$
73 $$1 + 1487 T + 2524966 T^{2} + 2482459397 T^{3} + 2523687153172 T^{4} + 1935920533900367 T^{5} + 1491035958084571202 T^{6} +$$$$93\!\cdots\!19$$$$T^{7} + 1491035958084571202 p^{3} T^{8} + 1935920533900367 p^{6} T^{9} + 2523687153172 p^{9} T^{10} + 2482459397 p^{12} T^{11} + 2524966 p^{15} T^{12} + 1487 p^{18} T^{13} + p^{21} T^{14}$$
79 $$1 + 1273 T + 1075087 T^{2} + 315987008 T^{3} + 210372034826 T^{4} + 25718384145704 T^{5} + 17792609044986825 T^{6} - 76748980395578922381 T^{7} + 17792609044986825 p^{3} T^{8} + 25718384145704 p^{6} T^{9} + 210372034826 p^{9} T^{10} + 315987008 p^{12} T^{11} + 1075087 p^{15} T^{12} + 1273 p^{18} T^{13} + p^{21} T^{14}$$
83 $$1 + 1170 T + 2704394 T^{2} + 2771006274 T^{3} + 3451944870390 T^{4} + 3145137763387590 T^{5} + 2842201839927787894 T^{6} +$$$$22\!\cdots\!80$$$$T^{7} + 2842201839927787894 p^{3} T^{8} + 3145137763387590 p^{6} T^{9} + 3451944870390 p^{9} T^{10} + 2771006274 p^{12} T^{11} + 2704394 p^{15} T^{12} + 1170 p^{18} T^{13} + p^{21} T^{14}$$
89 $$1 - 1058 T + 2035984 T^{2} - 1406330494 T^{3} + 2233417382573 T^{4} - 1228585627908860 T^{5} + 1858532021147586657 T^{6} -$$$$94\!\cdots\!42$$$$T^{7} + 1858532021147586657 p^{3} T^{8} - 1228585627908860 p^{6} T^{9} + 2233417382573 p^{9} T^{10} - 1406330494 p^{12} T^{11} + 2035984 p^{15} T^{12} - 1058 p^{18} T^{13} + p^{21} T^{14}$$
97 $$1 - 3730 T + 9595483 T^{2} - 17663935492 T^{3} + 27514714160281 T^{4} - 36077823529689454 T^{5} + 42061807913396213387 T^{6} -$$$$42\!\cdots\!68$$$$T^{7} + 42061807913396213387 p^{3} T^{8} - 36077823529689454 p^{6} T^{9} + 27514714160281 p^{9} T^{10} - 17663935492 p^{12} T^{11} + 9595483 p^{15} T^{12} - 3730 p^{18} T^{13} + p^{21} T^{14}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$