# Properties

 Label 32-33e32-1.1-c2e16-0-0 Degree $32$ Conductor $3.912\times 10^{48}$ Sign $1$ Analytic cond. $3.61249\times 10^{23}$ Root an. cond. $5.44730$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 16·4-s − 8·7-s + 4·13-s + 103·16-s − 20·19-s + 178·25-s − 128·28-s + 28·31-s − 148·37-s + 272·43-s − 186·49-s + 64·52-s + 224·61-s + 374·64-s + 24·67-s − 4·73-s − 320·76-s + 216·79-s − 32·91-s − 44·97-s + 2.84e3·100-s − 228·103-s + 340·109-s − 824·112-s + 448·124-s + 127-s + 131-s + ⋯
 L(s)  = 1 + 4·4-s − 8/7·7-s + 4/13·13-s + 6.43·16-s − 1.05·19-s + 7.11·25-s − 4.57·28-s + 0.903·31-s − 4·37-s + 6.32·43-s − 3.79·49-s + 1.23·52-s + 3.67·61-s + 5.84·64-s + 0.358·67-s − 0.0547·73-s − 4.21·76-s + 2.73·79-s − 0.351·91-s − 0.453·97-s + 28.4·100-s − 2.21·103-s + 3.11·109-s − 7.35·112-s + 3.61·124-s + 0.00787·127-s + 0.00763·131-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$32$$ Conductor: $$3^{32} \cdot 11^{32}$$ Sign: $1$ Analytic conductor: $$3.61249\times 10^{23}$$ Root analytic conductor: $$5.44730$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1089} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(32,\ 3^{32} \cdot 11^{32} ,\ ( \ : [1]^{16} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$5.202995254$$ $$L(\frac12)$$ $$\approx$$ $$5.202995254$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
11 $$1$$
good2 $$1 - p^{4} T^{2} + 153 T^{4} - 587 p T^{6} + 1915 p^{2} T^{8} - 2701 p^{4} T^{10} + 217281 T^{12} - 497761 p T^{14} + 4170145 T^{16} - 497761 p^{5} T^{18} + 217281 p^{8} T^{20} - 2701 p^{16} T^{22} + 1915 p^{18} T^{24} - 587 p^{21} T^{26} + 153 p^{24} T^{28} - p^{32} T^{30} + p^{32} T^{32}$$
5 $$1 - 178 T^{2} + 14778 T^{4} - 770368 T^{6} + 28639243 T^{8} - 818668258 T^{10} + 19271651352 T^{12} - 83084625136 p T^{14} + 9588673069861 T^{16} - 83084625136 p^{5} T^{18} + 19271651352 p^{8} T^{20} - 818668258 p^{12} T^{22} + 28639243 p^{16} T^{24} - 770368 p^{20} T^{26} + 14778 p^{24} T^{28} - 178 p^{28} T^{30} + p^{32} T^{32}$$
7 $$( 1 + 4 T + 117 T^{2} + 806 T^{3} + 10368 T^{4} + 11132 p T^{5} + 726296 T^{6} + 4835322 T^{7} + 41663687 T^{8} + 4835322 p^{2} T^{9} + 726296 p^{4} T^{10} + 11132 p^{7} T^{11} + 10368 p^{8} T^{12} + 806 p^{10} T^{13} + 117 p^{12} T^{14} + 4 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
13 $$( 1 - 2 T + 718 T^{2} - 1888 T^{3} + 260375 T^{4} - 546188 T^{5} + 64887908 T^{6} - 90279418 T^{7} + 12344502880 T^{8} - 90279418 p^{2} T^{9} + 64887908 p^{4} T^{10} - 546188 p^{6} T^{11} + 260375 p^{8} T^{12} - 1888 p^{10} T^{13} + 718 p^{12} T^{14} - 2 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
17 $$1 - 2034 T^{2} + 2137099 T^{4} - 1555825258 T^{6} + 885576421975 T^{8} - 418087252540946 T^{10} + 168801749233965397 T^{12} - 59268558645523686770 T^{14} +$$$$18\!\cdots\!16$$$$T^{16} - 59268558645523686770 p^{4} T^{18} + 168801749233965397 p^{8} T^{20} - 418087252540946 p^{12} T^{22} + 885576421975 p^{16} T^{24} - 1555825258 p^{20} T^{26} + 2137099 p^{24} T^{28} - 2034 p^{28} T^{30} + p^{32} T^{32}$$
19 $$( 1 + 10 T + 1541 T^{2} + 12848 T^{3} + 1194081 T^{4} + 7050832 T^{5} + 617393333 T^{6} + 2562029742 T^{7} + 12972454372 p T^{8} + 2562029742 p^{2} T^{9} + 617393333 p^{4} T^{10} + 7050832 p^{6} T^{11} + 1194081 p^{8} T^{12} + 12848 p^{10} T^{13} + 1541 p^{12} T^{14} + 10 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
23 $$1 - 3898 T^{2} + 7470938 T^{4} - 9442830008 T^{6} + 8992724976405 T^{8} - 7035858281914756 T^{10} + 4797996886199722826 T^{12} -$$$$29\!\cdots\!26$$$$T^{14} +$$$$16\!\cdots\!40$$$$T^{16} -$$$$29\!\cdots\!26$$$$p^{4} T^{18} + 4797996886199722826 p^{8} T^{20} - 7035858281914756 p^{12} T^{22} + 8992724976405 p^{16} T^{24} - 9442830008 p^{20} T^{26} + 7470938 p^{24} T^{28} - 3898 p^{28} T^{30} + p^{32} T^{32}$$
29 $$1 - 7090 T^{2} + 24088979 T^{4} - 52860171052 T^{6} + 85979263288379 T^{8} - 113520722017723476 T^{10} +$$$$12\!\cdots\!45$$$$T^{12} -$$$$13\!\cdots\!54$$$$T^{14} +$$$$11\!\cdots\!32$$$$T^{16} -$$$$13\!\cdots\!54$$$$p^{4} T^{18} +$$$$12\!\cdots\!45$$$$p^{8} T^{20} - 113520722017723476 p^{12} T^{22} + 85979263288379 p^{16} T^{24} - 52860171052 p^{20} T^{26} + 24088979 p^{24} T^{28} - 7090 p^{28} T^{30} + p^{32} T^{32}$$
31 $$( 1 - 14 T + 3978 T^{2} - 62074 T^{3} + 6872087 T^{4} - 133270650 T^{5} + 7029590252 T^{6} - 182545665112 T^{7} + 6210443887425 T^{8} - 182545665112 p^{2} T^{9} + 7029590252 p^{4} T^{10} - 133270650 p^{6} T^{11} + 6872087 p^{8} T^{12} - 62074 p^{10} T^{13} + 3978 p^{12} T^{14} - 14 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
37 $$( 1 + 2 p T + 8842 T^{2} + 527868 T^{3} + 35589723 T^{4} + 1747694092 T^{5} + 86947356072 T^{6} + 3553006456782 T^{7} + 142971465181960 T^{8} + 3553006456782 p^{2} T^{9} + 86947356072 p^{4} T^{10} + 1747694092 p^{6} T^{11} + 35589723 p^{8} T^{12} + 527868 p^{10} T^{13} + 8842 p^{12} T^{14} + 2 p^{15} T^{15} + p^{16} T^{16} )^{2}$$
41 $$1 - 18838 T^{2} + 173358858 T^{4} - 1037317818784 T^{6} + 4528296295411213 T^{8} - 15321581034880516324 T^{10} +$$$$41\!\cdots\!18$$$$T^{12} -$$$$92\!\cdots\!66$$$$T^{14} +$$$$17\!\cdots\!40$$$$T^{16} -$$$$92\!\cdots\!66$$$$p^{4} T^{18} +$$$$41\!\cdots\!18$$$$p^{8} T^{20} - 15321581034880516324 p^{12} T^{22} + 4528296295411213 p^{16} T^{24} - 1037317818784 p^{20} T^{26} + 173358858 p^{24} T^{28} - 18838 p^{28} T^{30} + p^{32} T^{32}$$
43 $$( 1 - 136 T + 18164 T^{2} - 1483936 T^{3} + 115914165 T^{4} - 6877749720 T^{5} + 397184994126 T^{6} - 18797621494760 T^{7} + 882478921005452 T^{8} - 18797621494760 p^{2} T^{9} + 397184994126 p^{4} T^{10} - 6877749720 p^{6} T^{11} + 115914165 p^{8} T^{12} - 1483936 p^{10} T^{13} + 18164 p^{12} T^{14} - 136 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
47 $$1 - 14410 T^{2} + 112155039 T^{4} - 608867850142 T^{6} + 2578161157095895 T^{8} - 9021116111839284886 T^{10} +$$$$27\!\cdots\!77$$$$T^{12} -$$$$71\!\cdots\!70$$$$T^{14} +$$$$16\!\cdots\!56$$$$T^{16} -$$$$71\!\cdots\!70$$$$p^{4} T^{18} +$$$$27\!\cdots\!77$$$$p^{8} T^{20} - 9021116111839284886 p^{12} T^{22} + 2578161157095895 p^{16} T^{24} - 608867850142 p^{20} T^{26} + 112155039 p^{24} T^{28} - 14410 p^{28} T^{30} + p^{32} T^{32}$$
53 $$1 - 24436 T^{2} + 309860538 T^{4} - 2662731812674 T^{6} + 17249750715319795 T^{8} - 89091997635889422436 T^{10} +$$$$37\!\cdots\!16$$$$T^{12} -$$$$13\!\cdots\!92$$$$T^{14} +$$$$41\!\cdots\!85$$$$T^{16} -$$$$13\!\cdots\!92$$$$p^{4} T^{18} +$$$$37\!\cdots\!16$$$$p^{8} T^{20} - 89091997635889422436 p^{12} T^{22} + 17249750715319795 p^{16} T^{24} - 2662731812674 p^{20} T^{26} + 309860538 p^{24} T^{28} - 24436 p^{28} T^{30} + p^{32} T^{32}$$
59 $$1 - 20682 T^{2} + 198130262 T^{4} - 1153138487844 T^{6} + 4326529642531863 T^{8} - 8745369823524440418 T^{10} -$$$$95\!\cdots\!32$$$$T^{12} +$$$$16\!\cdots\!64$$$$T^{14} -$$$$21\!\cdots\!87$$$$p^{2} T^{16} +$$$$16\!\cdots\!64$$$$p^{4} T^{18} -$$$$95\!\cdots\!32$$$$p^{8} T^{20} - 8745369823524440418 p^{12} T^{22} + 4326529642531863 p^{16} T^{24} - 1153138487844 p^{20} T^{26} + 198130262 p^{24} T^{28} - 20682 p^{28} T^{30} + p^{32} T^{32}$$
61 $$( 1 - 112 T + 21493 T^{2} - 1326876 T^{3} + 134785133 T^{4} - 2241406816 T^{5} + 159075236043 T^{6} + 26423445075596 T^{7} - 717448526447500 T^{8} + 26423445075596 p^{2} T^{9} + 159075236043 p^{4} T^{10} - 2241406816 p^{6} T^{11} + 134785133 p^{8} T^{12} - 1326876 p^{10} T^{13} + 21493 p^{12} T^{14} - 112 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
67 $$( 1 - 12 T + 13613 T^{2} - 121480 T^{3} + 120754069 T^{4} - 1956334700 T^{5} + 777459351015 T^{6} - 12985935239032 T^{7} + 3804482475877080 T^{8} - 12985935239032 p^{2} T^{9} + 777459351015 p^{4} T^{10} - 1956334700 p^{6} T^{11} + 120754069 p^{8} T^{12} - 121480 p^{10} T^{13} + 13613 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
71 $$1 - 32664 T^{2} + 582373471 T^{4} - 103852571306 p T^{6} + 73712493122078731 T^{8} -$$$$61\!\cdots\!26$$$$T^{10} +$$$$43\!\cdots\!53$$$$T^{12} -$$$$26\!\cdots\!76$$$$T^{14} +$$$$14\!\cdots\!88$$$$T^{16} -$$$$26\!\cdots\!76$$$$p^{4} T^{18} +$$$$43\!\cdots\!53$$$$p^{8} T^{20} -$$$$61\!\cdots\!26$$$$p^{12} T^{22} + 73712493122078731 p^{16} T^{24} - 103852571306 p^{21} T^{26} + 582373471 p^{24} T^{28} - 32664 p^{28} T^{30} + p^{32} T^{32}$$
73 $$( 1 + 2 T + 28881 T^{2} - 79288 T^{3} + 398736745 T^{4} - 2373174904 T^{5} + 3498841626669 T^{6} - 25082690682734 T^{7} + 21777004472176804 T^{8} - 25082690682734 p^{2} T^{9} + 3498841626669 p^{4} T^{10} - 2373174904 p^{6} T^{11} + 398736745 p^{8} T^{12} - 79288 p^{10} T^{13} + 28881 p^{12} T^{14} + 2 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
79 $$( 1 - 108 T + 33145 T^{2} - 3235546 T^{3} + 561354580 T^{4} - 47468168324 T^{5} + 6091852673746 T^{6} - 437688075575774 T^{7} + 45380145174506261 T^{8} - 437688075575774 p^{2} T^{9} + 6091852673746 p^{4} T^{10} - 47468168324 p^{6} T^{11} + 561354580 p^{8} T^{12} - 3235546 p^{10} T^{13} + 33145 p^{12} T^{14} - 108 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
83 $$1 - 56836 T^{2} + 1507726937 T^{4} - 24632124674530 T^{6} + 277176756528405302 T^{8} -$$$$23\!\cdots\!24$$$$T^{10} +$$$$15\!\cdots\!72$$$$T^{12} -$$$$89\!\cdots\!78$$$$T^{14} +$$$$56\!\cdots\!61$$$$T^{16} -$$$$89\!\cdots\!78$$$$p^{4} T^{18} +$$$$15\!\cdots\!72$$$$p^{8} T^{20} -$$$$23\!\cdots\!24$$$$p^{12} T^{22} + 277176756528405302 p^{16} T^{24} - 24632124674530 p^{20} T^{26} + 1507726937 p^{24} T^{28} - 56836 p^{28} T^{30} + p^{32} T^{32}$$
89 $$1 - 50138 T^{2} + 1370519842 T^{4} - 26422369652440 T^{6} + 399704691200648693 T^{8} -$$$$50\!\cdots\!84$$$$T^{10} +$$$$54\!\cdots\!74$$$$T^{12} -$$$$51\!\cdots\!10$$$$T^{14} +$$$$43\!\cdots\!20$$$$T^{16} -$$$$51\!\cdots\!10$$$$p^{4} T^{18} +$$$$54\!\cdots\!74$$$$p^{8} T^{20} -$$$$50\!\cdots\!84$$$$p^{12} T^{22} + 399704691200648693 p^{16} T^{24} - 26422369652440 p^{20} T^{26} + 1370519842 p^{24} T^{28} - 50138 p^{28} T^{30} + p^{32} T^{32}$$
97 $$( 1 + 22 T + 49408 T^{2} + 115060 T^{3} + 1140430559 T^{4} - 13214673230 T^{5} + 16943162093090 T^{6} - 298782127218388 T^{7} + 183519623479606515 T^{8} - 298782127218388 p^{2} T^{9} + 16943162093090 p^{4} T^{10} - 13214673230 p^{6} T^{11} + 1140430559 p^{8} T^{12} + 115060 p^{10} T^{13} + 49408 p^{12} T^{14} + 22 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$