# Properties

 Label 14-405e7-1.1-c3e7-0-1 Degree $14$ Conductor $1.787\times 10^{18}$ Sign $1$ Analytic cond. $4.44884\times 10^{9}$ Root an. cond. $4.88833$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s − 8·4-s + 35·5-s + 22·7-s − 18·8-s + 70·10-s + 23·11-s + 96·13-s + 44·14-s + 41·16-s + 161·17-s + 279·19-s − 280·20-s + 46·22-s + 96·23-s + 700·25-s + 192·26-s − 176·28-s − 296·29-s + 244·31-s + 8·32-s + 322·34-s + 770·35-s + 404·37-s + 558·38-s − 630·40-s − 47·41-s + ⋯
 L(s)  = 1 + 0.707·2-s − 4-s + 3.13·5-s + 1.18·7-s − 0.795·8-s + 2.21·10-s + 0.630·11-s + 2.04·13-s + 0.839·14-s + 0.640·16-s + 2.29·17-s + 3.36·19-s − 3.13·20-s + 0.445·22-s + 0.870·23-s + 28/5·25-s + 1.44·26-s − 1.18·28-s − 1.89·29-s + 1.41·31-s + 0.0441·32-s + 1.62·34-s + 3.71·35-s + 1.79·37-s + 2.38·38-s − 2.49·40-s − 0.179·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28} \cdot 5^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28} \cdot 5^{7}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$14$$ Conductor: $$3^{28} \cdot 5^{7}$$ Sign: $1$ Analytic conductor: $$4.44884\times 10^{9}$$ Root analytic conductor: $$4.88833$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{405} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(14,\ 3^{28} \cdot 5^{7} ,\ ( \ : [3/2]^{7} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$154.8007180$$ $$L(\frac12)$$ $$\approx$$ $$154.8007180$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$( 1 - p T )^{7}$$
good2 $$1 - p T + 3 p^{2} T^{2} - 11 p T^{3} + 63 T^{4} - 3 p^{2} T^{5} + 7 p^{3} T^{6} + 27 p^{5} T^{7} + 7 p^{6} T^{8} - 3 p^{8} T^{9} + 63 p^{9} T^{10} - 11 p^{13} T^{11} + 3 p^{17} T^{12} - p^{19} T^{13} + p^{21} T^{14}$$
7 $$1 - 22 T + 830 T^{2} - 19134 T^{3} + 426701 T^{4} - 7825194 T^{5} + 165254493 T^{6} - 347285588 p T^{7} + 165254493 p^{3} T^{8} - 7825194 p^{6} T^{9} + 426701 p^{9} T^{10} - 19134 p^{12} T^{11} + 830 p^{15} T^{12} - 22 p^{18} T^{13} + p^{21} T^{14}$$
11 $$1 - 23 T + 5151 T^{2} - 137494 T^{3} + 15125367 T^{4} - 32802843 p T^{5} + 29040094313 T^{6} - 601859871012 T^{7} + 29040094313 p^{3} T^{8} - 32802843 p^{7} T^{9} + 15125367 p^{9} T^{10} - 137494 p^{12} T^{11} + 5151 p^{15} T^{12} - 23 p^{18} T^{13} + p^{21} T^{14}$$
13 $$1 - 96 T + 12191 T^{2} - 828700 T^{3} + 67939809 T^{4} - 3670430048 T^{5} + 227859076023 T^{6} - 10004745881352 T^{7} + 227859076023 p^{3} T^{8} - 3670430048 p^{6} T^{9} + 67939809 p^{9} T^{10} - 828700 p^{12} T^{11} + 12191 p^{15} T^{12} - 96 p^{18} T^{13} + p^{21} T^{14}$$
17 $$1 - 161 T + 24747 T^{2} - 2620018 T^{3} + 242733069 T^{4} - 19012626063 T^{5} + 1428967389431 T^{6} - 5791825964988 p T^{7} + 1428967389431 p^{3} T^{8} - 19012626063 p^{6} T^{9} + 242733069 p^{9} T^{10} - 2620018 p^{12} T^{11} + 24747 p^{15} T^{12} - 161 p^{18} T^{13} + p^{21} T^{14}$$
19 $$1 - 279 T + 63455 T^{2} - 9627646 T^{3} + 1292727231 T^{4} - 139417672577 T^{5} + 13934197302465 T^{6} - 1188087181318116 T^{7} + 13934197302465 p^{3} T^{8} - 139417672577 p^{6} T^{9} + 1292727231 p^{9} T^{10} - 9627646 p^{12} T^{11} + 63455 p^{15} T^{12} - 279 p^{18} T^{13} + p^{21} T^{14}$$
23 $$1 - 96 T + 70730 T^{2} - 5128416 T^{3} + 2231256537 T^{4} - 129120969582 T^{5} + 1826857392719 p T^{6} - 1973532783601374 T^{7} + 1826857392719 p^{4} T^{8} - 129120969582 p^{6} T^{9} + 2231256537 p^{9} T^{10} - 5128416 p^{12} T^{11} + 70730 p^{15} T^{12} - 96 p^{18} T^{13} + p^{21} T^{14}$$
29 $$1 + 296 T + 154422 T^{2} + 29519614 T^{3} + 8749743807 T^{4} + 1202308386924 T^{5} + 280424950326131 T^{6} + 32104391095886106 T^{7} + 280424950326131 p^{3} T^{8} + 1202308386924 p^{6} T^{9} + 8749743807 p^{9} T^{10} + 29519614 p^{12} T^{11} + 154422 p^{15} T^{12} + 296 p^{18} T^{13} + p^{21} T^{14}$$
31 $$1 - 244 T + 138953 T^{2} - 21581892 T^{3} + 6820038797 T^{4} - 643944456252 T^{5} + 180689501328357 T^{6} - 12932535760119704 T^{7} + 180689501328357 p^{3} T^{8} - 643944456252 p^{6} T^{9} + 6820038797 p^{9} T^{10} - 21581892 p^{12} T^{11} + 138953 p^{15} T^{12} - 244 p^{18} T^{13} + p^{21} T^{14}$$
37 $$1 - 404 T + 7243 p T^{2} - 88433668 T^{3} + 32537859593 T^{4} - 8785146905212 T^{5} + 2408172804310695 T^{6} - 541287630181037208 T^{7} + 2408172804310695 p^{3} T^{8} - 8785146905212 p^{6} T^{9} + 32537859593 p^{9} T^{10} - 88433668 p^{12} T^{11} + 7243 p^{16} T^{12} - 404 p^{18} T^{13} + p^{21} T^{14}$$
41 $$1 + 47 T + 321096 T^{2} - 12021155 T^{3} + 45171631701 T^{4} - 4630735558038 T^{5} + 4063682212222409 T^{6} - 491807687844214791 T^{7} + 4063682212222409 p^{3} T^{8} - 4630735558038 p^{6} T^{9} + 45171631701 p^{9} T^{10} - 12021155 p^{12} T^{11} + 321096 p^{15} T^{12} + 47 p^{18} T^{13} + p^{21} T^{14}$$
43 $$1 - 525 T + 461651 T^{2} - 177053890 T^{3} + 91552352643 T^{4} - 27963570378443 T^{5} + 10910759641065633 T^{6} - 2737206494978679516 T^{7} + 10910759641065633 p^{3} T^{8} - 27963570378443 p^{6} T^{9} + 91552352643 p^{9} T^{10} - 177053890 p^{12} T^{11} + 461651 p^{15} T^{12} - 525 p^{18} T^{13} + p^{21} T^{14}$$
47 $$1 - 164 T + 458970 T^{2} - 110392306 T^{3} + 108169816881 T^{4} - 25717707674850 T^{5} + 17081669577666257 T^{6} - 3320863886913274956 T^{7} + 17081669577666257 p^{3} T^{8} - 25717707674850 p^{6} T^{9} + 108169816881 p^{9} T^{10} - 110392306 p^{12} T^{11} + 458970 p^{15} T^{12} - 164 p^{18} T^{13} + p^{21} T^{14}$$
53 $$1 - 506 T + 541623 T^{2} - 172442740 T^{3} + 132333548265 T^{4} - 35763427746966 T^{5} + 25492045997195975 T^{6} - 6232790423469661848 T^{7} + 25492045997195975 p^{3} T^{8} - 35763427746966 p^{6} T^{9} + 132333548265 p^{9} T^{10} - 172442740 p^{12} T^{11} + 541623 p^{15} T^{12} - 506 p^{18} T^{13} + p^{21} T^{14}$$
59 $$1 + 85 T + 331341 T^{2} + 32993294 T^{3} + 125055414789 T^{4} + 9577822572435 T^{5} + 29633014670734625 T^{6} + 2027875987639118244 T^{7} + 29633014670734625 p^{3} T^{8} + 9577822572435 p^{6} T^{9} + 125055414789 p^{9} T^{10} + 32993294 p^{12} T^{11} + 331341 p^{15} T^{12} + 85 p^{18} T^{13} + p^{21} T^{14}$$
61 $$1 - 828 T + 1137794 T^{2} - 655662598 T^{3} + 472897824279 T^{4} - 204252976588544 T^{5} + 110585022303384087 T^{6} - 44527407364853870622 T^{7} + 110585022303384087 p^{3} T^{8} - 204252976588544 p^{6} T^{9} + 472897824279 p^{9} T^{10} - 655662598 p^{12} T^{11} + 1137794 p^{15} T^{12} - 828 p^{18} T^{13} + p^{21} T^{14}$$
67 $$1 - 1093 T + 2017106 T^{2} - 1699166505 T^{3} + 1757369253407 T^{4} - 1166070992255454 T^{5} + 864466170577418349 T^{6} -$$$$45\!\cdots\!53$$$$T^{7} + 864466170577418349 p^{3} T^{8} - 1166070992255454 p^{6} T^{9} + 1757369253407 p^{9} T^{10} - 1699166505 p^{12} T^{11} + 2017106 p^{15} T^{12} - 1093 p^{18} T^{13} + p^{21} T^{14}$$
71 $$1 - 328 T + 1276365 T^{2} - 268163756 T^{3} + 813660594393 T^{4} - 105690507940488 T^{5} + 374340014427629213 T^{6} - 36698946645864837192 T^{7} + 374340014427629213 p^{3} T^{8} - 105690507940488 p^{6} T^{9} + 813660594393 p^{9} T^{10} - 268163756 p^{12} T^{11} + 1276365 p^{15} T^{12} - 328 p^{18} T^{13} + p^{21} T^{14}$$
73 $$1 - 2085 T + 4050563 T^{2} - 4984423486 T^{3} + 5631570806793 T^{4} - 4895773027898939 T^{5} + 3922468709989734339 T^{6} -$$$$25\!\cdots\!16$$$$T^{7} + 3922468709989734339 p^{3} T^{8} - 4895773027898939 p^{6} T^{9} + 5631570806793 p^{9} T^{10} - 4984423486 p^{12} T^{11} + 4050563 p^{15} T^{12} - 2085 p^{18} T^{13} + p^{21} T^{14}$$
79 $$1 - 2110 T + 3466397 T^{2} - 4074662340 T^{3} + 4200201567425 T^{4} - 3637140873668514 T^{5} + 2929970636702454981 T^{6} -$$$$20\!\cdots\!84$$$$T^{7} + 2929970636702454981 p^{3} T^{8} - 3637140873668514 p^{6} T^{9} + 4200201567425 p^{9} T^{10} - 4074662340 p^{12} T^{11} + 3466397 p^{15} T^{12} - 2110 p^{18} T^{13} + p^{21} T^{14}$$
83 $$1 - 1290 T + 3282374 T^{2} - 3191089158 T^{3} + 4600431414309 T^{4} - 3678070946162670 T^{5} + 3902415474964514617 T^{6} -$$$$26\!\cdots\!88$$$$T^{7} + 3902415474964514617 p^{3} T^{8} - 3678070946162670 p^{6} T^{9} + 4600431414309 p^{9} T^{10} - 3191089158 p^{12} T^{11} + 3282374 p^{15} T^{12} - 1290 p^{18} T^{13} + p^{21} T^{14}$$
89 $$1 + 3048 T + 5723102 T^{2} + 6323602086 T^{3} + 4534976524791 T^{4} + 780789931274556 T^{5} - 1899913405596725465 T^{6} -$$$$27\!\cdots\!50$$$$T^{7} - 1899913405596725465 p^{3} T^{8} + 780789931274556 p^{6} T^{9} + 4534976524791 p^{9} T^{10} + 6323602086 p^{12} T^{11} + 5723102 p^{15} T^{12} + 3048 p^{18} T^{13} + p^{21} T^{14}$$
97 $$1 - 1787 T + 4084243 T^{2} - 4789169974 T^{3} + 7296823674929 T^{4} - 7118143615887349 T^{5} + 8621884508580027771 T^{6} -$$$$72\!\cdots\!44$$$$T^{7} + 8621884508580027771 p^{3} T^{8} - 7118143615887349 p^{6} T^{9} + 7296823674929 p^{9} T^{10} - 4789169974 p^{12} T^{11} + 4084243 p^{15} T^{12} - 1787 p^{18} T^{13} + p^{21} T^{14}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$