# Properties

 Label 32-405e16-1.1-c3e16-0-1 Degree $32$ Conductor $5.239\times 10^{41}$ Sign $1$ Analytic cond. $1.13015\times 10^{22}$ 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 + 37·4-s + 3·5-s + 90·11-s + 613·16-s − 4·19-s + 111·20-s − 31·25-s − 516·29-s + 38·31-s + 576·41-s + 3.33e3·44-s + 2.74e3·49-s + 270·55-s − 2.20e3·59-s + 20·61-s + 6.04e3·64-s + 2.95e3·71-s − 148·76-s + 218·79-s + 1.83e3·80-s − 4.07e3·89-s − 12·95-s − 1.14e3·100-s + 7.13e3·101-s − 958·109-s − 1.90e4·116-s − 7.27e3·121-s + ⋯
 L(s)  = 1 + 37/8·4-s + 0.268·5-s + 2.46·11-s + 9.57·16-s − 0.0482·19-s + 1.24·20-s − 0.247·25-s − 3.30·29-s + 0.220·31-s + 2.19·41-s + 11.4·44-s + 8.00·49-s + 0.661·55-s − 4.85·59-s + 0.0419·61-s + 11.8·64-s + 4.93·71-s − 0.223·76-s + 0.310·79-s + 2.57·80-s − 4.85·89-s − 0.0129·95-s − 1.14·100-s + 7.02·101-s − 0.841·109-s − 15.2·116-s − 5.46·121-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{64} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{64} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$32$$ Conductor: $$3^{64} \cdot 5^{16}$$ Sign: $1$ Analytic conductor: $$1.13015\times 10^{22}$$ 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: $$(32,\ 3^{64} \cdot 5^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$395.8557987$$ $$L(\frac12)$$ $$\approx$$ $$395.8557987$$ $$L(\frac{5}{2})$$ 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 - 3 T + 8 p T^{2} + 1533 T^{3} - 5696 T^{4} - 3717 p T^{5} + 30664 p^{2} T^{6} - 141909 p^{3} T^{7} + 23902 p^{4} T^{8} - 141909 p^{6} T^{9} + 30664 p^{8} T^{10} - 3717 p^{10} T^{11} - 5696 p^{12} T^{12} + 1533 p^{15} T^{13} + 8 p^{19} T^{14} - 3 p^{21} T^{15} + p^{24} T^{16}$$
good2 $$1 - 37 T^{2} + 189 p^{2} T^{4} - 1417 p^{3} T^{6} + 140999 T^{8} - 1566027 T^{10} + 3965011 p^{2} T^{12} - 2288161 p^{6} T^{14} + 4806027 p^{8} T^{16} - 2288161 p^{12} T^{18} + 3965011 p^{14} T^{20} - 1566027 p^{18} T^{22} + 140999 p^{24} T^{24} - 1417 p^{33} T^{26} + 189 p^{38} T^{28} - 37 p^{42} T^{30} + p^{48} T^{32}$$
7 $$1 - 2746 T^{2} + 3920535 T^{4} - 3799787198 T^{6} + 2786049777317 T^{8} - 1640205583267248 T^{10} + 115126196667169306 p T^{12} -$$$$33\!\cdots\!92$$$$T^{14} +$$$$12\!\cdots\!90$$$$T^{16} -$$$$33\!\cdots\!92$$$$p^{6} T^{18} + 115126196667169306 p^{13} T^{20} - 1640205583267248 p^{18} T^{22} + 2786049777317 p^{24} T^{24} - 3799787198 p^{30} T^{26} + 3920535 p^{36} T^{28} - 2746 p^{42} T^{30} + p^{48} T^{32}$$
11 $$( 1 - 45 T + 6673 T^{2} - 258714 T^{3} + 22094686 T^{4} - 764410356 T^{5} + 4377741821 p T^{6} - 1476648574551 T^{7} + 75234818756890 T^{8} - 1476648574551 p^{3} T^{9} + 4377741821 p^{7} T^{10} - 764410356 p^{9} T^{11} + 22094686 p^{12} T^{12} - 258714 p^{15} T^{13} + 6673 p^{18} T^{14} - 45 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
13 $$1 - 16669 T^{2} + 131836626 T^{4} - 664100547179 T^{6} + 2449602067304840 T^{8} - 7361875938870845289 T^{10} +$$$$19\!\cdots\!74$$$$T^{12} -$$$$49\!\cdots\!35$$$$T^{14} +$$$$11\!\cdots\!38$$$$T^{16} -$$$$49\!\cdots\!35$$$$p^{6} T^{18} +$$$$19\!\cdots\!74$$$$p^{12} T^{20} - 7361875938870845289 p^{18} T^{22} + 2449602067304840 p^{24} T^{24} - 664100547179 p^{30} T^{26} + 131836626 p^{36} T^{28} - 16669 p^{42} T^{30} + p^{48} T^{32}$$
17 $$1 - 2450 p T^{2} + 902277741 T^{4} - 13395318881222 T^{6} + 151327515243116846 T^{8} -$$$$13\!\cdots\!06$$$$T^{10} +$$$$10\!\cdots\!23$$$$T^{12} -$$$$64\!\cdots\!62$$$$T^{14} +$$$$11\!\cdots\!22$$$$p^{2} T^{16} -$$$$64\!\cdots\!62$$$$p^{6} T^{18} +$$$$10\!\cdots\!23$$$$p^{12} T^{20} -$$$$13\!\cdots\!06$$$$p^{18} T^{22} + 151327515243116846 p^{24} T^{24} - 13395318881222 p^{30} T^{26} + 902277741 p^{36} T^{28} - 2450 p^{43} T^{30} + p^{48} T^{32}$$
19 $$( 1 + 2 T + 30531 T^{2} + 541942 T^{3} + 465624254 T^{4} + 13840800234 T^{5} + 4769220969385 T^{6} + 167543587475774 T^{7} + 36901908228236058 T^{8} + 167543587475774 p^{3} T^{9} + 4769220969385 p^{6} T^{10} + 13840800234 p^{9} T^{11} + 465624254 p^{12} T^{12} + 541942 p^{15} T^{13} + 30531 p^{18} T^{14} + 2 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
23 $$1 - 71542 T^{2} + 2977162083 T^{4} - 90987915764606 T^{6} + 2211781935528224105 T^{8} -$$$$44\!\cdots\!08$$$$T^{10} +$$$$77\!\cdots\!42$$$$T^{12} -$$$$11\!\cdots\!84$$$$T^{14} +$$$$15\!\cdots\!22$$$$T^{16} -$$$$11\!\cdots\!84$$$$p^{6} T^{18} +$$$$77\!\cdots\!42$$$$p^{12} T^{20} -$$$$44\!\cdots\!08$$$$p^{18} T^{22} + 2211781935528224105 p^{24} T^{24} - 90987915764606 p^{30} T^{26} + 2977162083 p^{36} T^{28} - 71542 p^{42} T^{30} + p^{48} T^{32}$$
29 $$( 1 + 258 T + 114367 T^{2} + 26481270 T^{3} + 7156276849 T^{4} + 1361396125908 T^{5} + 293841763913878 T^{6} + 47439998673210312 T^{7} + 8428524038498605474 T^{8} + 47439998673210312 p^{3} T^{9} + 293841763913878 p^{6} T^{10} + 1361396125908 p^{9} T^{11} + 7156276849 p^{12} T^{12} + 26481270 p^{15} T^{13} + 114367 p^{18} T^{14} + 258 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
31 $$( 1 - 19 T + 100674 T^{2} + 1895179 T^{3} + 5575730000 T^{4} + 206828855631 T^{5} + 229438602271414 T^{6} + 11942468233871849 T^{7} + 7393722570659007774 T^{8} + 11942468233871849 p^{3} T^{9} + 229438602271414 p^{6} T^{10} + 206828855631 p^{9} T^{11} + 5575730000 p^{12} T^{12} + 1895179 p^{15} T^{13} + 100674 p^{18} T^{14} - 19 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
37 $$1 - 440548 T^{2} + 93516072384 T^{4} - 12630824927332076 T^{6} +$$$$12\!\cdots\!32$$$$T^{8} -$$$$89\!\cdots\!76$$$$T^{10} +$$$$52\!\cdots\!88$$$$T^{12} -$$$$27\!\cdots\!16$$$$T^{14} +$$$$13\!\cdots\!50$$$$T^{16} -$$$$27\!\cdots\!16$$$$p^{6} T^{18} +$$$$52\!\cdots\!88$$$$p^{12} T^{20} -$$$$89\!\cdots\!76$$$$p^{18} T^{22} +$$$$12\!\cdots\!32$$$$p^{24} T^{24} - 12630824927332076 p^{30} T^{26} + 93516072384 p^{36} T^{28} - 440548 p^{42} T^{30} + p^{48} T^{32}$$
41 $$( 1 - 288 T + 280552 T^{2} - 73221588 T^{3} + 40717856062 T^{4} - 9351582084468 T^{5} + 4102782756503920 T^{6} - 863614938864737844 T^{7} +$$$$31\!\cdots\!03$$$$T^{8} - 863614938864737844 p^{3} T^{9} + 4102782756503920 p^{6} T^{10} - 9351582084468 p^{9} T^{11} + 40717856062 p^{12} T^{12} - 73221588 p^{15} T^{13} + 280552 p^{18} T^{14} - 288 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
43 $$1 - 950095 T^{2} + 437299902657 T^{4} - 129824459037449726 T^{6} +$$$$27\!\cdots\!76$$$$T^{8} -$$$$46\!\cdots\!08$$$$T^{10} +$$$$60\!\cdots\!71$$$$T^{12} -$$$$64\!\cdots\!11$$$$T^{14} +$$$$56\!\cdots\!34$$$$T^{16} -$$$$64\!\cdots\!11$$$$p^{6} T^{18} +$$$$60\!\cdots\!71$$$$p^{12} T^{20} -$$$$46\!\cdots\!08$$$$p^{18} T^{22} +$$$$27\!\cdots\!76$$$$p^{24} T^{24} - 129824459037449726 p^{30} T^{26} + 437299902657 p^{36} T^{28} - 950095 p^{42} T^{30} + p^{48} T^{32}$$
47 $$1 - 1066654 T^{2} + 532421607303 T^{4} - 164828062009030154 T^{6} +$$$$35\!\cdots\!61$$$$T^{8} -$$$$56\!\cdots\!48$$$$T^{10} +$$$$70\!\cdots\!38$$$$T^{12} -$$$$74\!\cdots\!20$$$$T^{14} +$$$$76\!\cdots\!14$$$$T^{16} -$$$$74\!\cdots\!20$$$$p^{6} T^{18} +$$$$70\!\cdots\!38$$$$p^{12} T^{20} -$$$$56\!\cdots\!48$$$$p^{18} T^{22} +$$$$35\!\cdots\!61$$$$p^{24} T^{24} - 164828062009030154 p^{30} T^{26} + 532421607303 p^{36} T^{28} - 1066654 p^{42} T^{30} + p^{48} T^{32}$$
53 $$1 - 1817908 T^{2} + 1611006757428 T^{4} - 922312347044964140 T^{6} +$$$$38\!\cdots\!04$$$$T^{8} -$$$$12\!\cdots\!20$$$$T^{10} +$$$$30\!\cdots\!20$$$$T^{12} -$$$$61\!\cdots\!04$$$$T^{14} +$$$$10\!\cdots\!54$$$$T^{16} -$$$$61\!\cdots\!04$$$$p^{6} T^{18} +$$$$30\!\cdots\!20$$$$p^{12} T^{20} -$$$$12\!\cdots\!20$$$$p^{18} T^{22} +$$$$38\!\cdots\!04$$$$p^{24} T^{24} - 922312347044964140 p^{30} T^{26} + 1611006757428 p^{36} T^{28} - 1817908 p^{42} T^{30} + p^{48} T^{32}$$
59 $$( 1 + 1101 T + 1632325 T^{2} + 1246231302 T^{3} + 1091627498134 T^{4} + 645042288302388 T^{5} + 420293547846430531 T^{6} +$$$$20\!\cdots\!11$$$$T^{7} +$$$$10\!\cdots\!66$$$$T^{8} +$$$$20\!\cdots\!11$$$$p^{3} T^{9} + 420293547846430531 p^{6} T^{10} + 645042288302388 p^{9} T^{11} + 1091627498134 p^{12} T^{12} + 1246231302 p^{15} T^{13} + 1632325 p^{18} T^{14} + 1101 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
61 $$( 1 - 10 T + 1065681 T^{2} - 77244494 T^{3} + 572514786419 T^{4} - 60269722986600 T^{5} + 206056746724365154 T^{6} - 23165304849108125200 T^{7} +$$$$54\!\cdots\!08$$$$T^{8} - 23165304849108125200 p^{3} T^{9} + 206056746724365154 p^{6} T^{10} - 60269722986600 p^{9} T^{11} + 572514786419 p^{12} T^{12} - 77244494 p^{15} T^{13} + 1065681 p^{18} T^{14} - 10 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
67 $$1 - 2656108 T^{2} + 3532638993804 T^{4} - 3143729963654001860 T^{6} +$$$$21\!\cdots\!78$$$$T^{8} -$$$$11\!\cdots\!88$$$$T^{10} +$$$$50\!\cdots\!76$$$$T^{12} -$$$$19\!\cdots\!72$$$$T^{14} +$$$$62\!\cdots\!47$$$$T^{16} -$$$$19\!\cdots\!72$$$$p^{6} T^{18} +$$$$50\!\cdots\!76$$$$p^{12} T^{20} -$$$$11\!\cdots\!88$$$$p^{18} T^{22} +$$$$21\!\cdots\!78$$$$p^{24} T^{24} - 3143729963654001860 p^{30} T^{26} + 3532638993804 p^{36} T^{28} - 2656108 p^{42} T^{30} + p^{48} T^{32}$$
71 $$( 1 - 1476 T + 3187450 T^{2} - 3360993624 T^{3} + 4161371889196 T^{4} - 3386869531237728 T^{5} + 3025874711031382510 T^{6} -$$$$19\!\cdots\!64$$$$T^{7} +$$$$13\!\cdots\!82$$$$T^{8} -$$$$19\!\cdots\!64$$$$p^{3} T^{9} + 3025874711031382510 p^{6} T^{10} - 3386869531237728 p^{9} T^{11} + 4161371889196 p^{12} T^{12} - 3360993624 p^{15} T^{13} + 3187450 p^{18} T^{14} - 1476 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
73 $$1 - 3404446 T^{2} + 6126306632589 T^{4} - 7521328741695950774 T^{6} +$$$$69\!\cdots\!86$$$$T^{8} -$$$$51\!\cdots\!42$$$$T^{10} +$$$$31\!\cdots\!39$$$$T^{12} -$$$$15\!\cdots\!62$$$$T^{14} +$$$$66\!\cdots\!30$$$$T^{16} -$$$$15\!\cdots\!62$$$$p^{6} T^{18} +$$$$31\!\cdots\!39$$$$p^{12} T^{20} -$$$$51\!\cdots\!42$$$$p^{18} T^{22} +$$$$69\!\cdots\!86$$$$p^{24} T^{24} - 7521328741695950774 p^{30} T^{26} + 6126306632589 p^{36} T^{28} - 3404446 p^{42} T^{30} + p^{48} T^{32}$$
79 $$( 1 - 109 T + 2801466 T^{2} - 331495223 T^{3} + 3830980169408 T^{4} - 446725089992151 T^{5} + 3301239660918451750 T^{6} -$$$$34\!\cdots\!69$$$$T^{7} +$$$$19\!\cdots\!22$$$$T^{8} -$$$$34\!\cdots\!69$$$$p^{3} T^{9} + 3301239660918451750 p^{6} T^{10} - 446725089992151 p^{9} T^{11} + 3830980169408 p^{12} T^{12} - 331495223 p^{15} T^{13} + 2801466 p^{18} T^{14} - 109 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
83 $$1 - 6737518 T^{2} + 256659614721 p T^{4} - 42075161735524670174 T^{6} +$$$$58\!\cdots\!81$$$$T^{8} -$$$$61\!\cdots\!72$$$$T^{10} +$$$$51\!\cdots\!38$$$$T^{12} -$$$$35\!\cdots\!76$$$$T^{14} +$$$$21\!\cdots\!14$$$$T^{16} -$$$$35\!\cdots\!76$$$$p^{6} T^{18} +$$$$51\!\cdots\!38$$$$p^{12} T^{20} -$$$$61\!\cdots\!72$$$$p^{18} T^{22} +$$$$58\!\cdots\!81$$$$p^{24} T^{24} - 42075161735524670174 p^{30} T^{26} + 256659614721 p^{37} T^{28} - 6737518 p^{42} T^{30} + p^{48} T^{32}$$
89 $$( 1 + 2037 T + 5372956 T^{2} + 7622227437 T^{3} + 11878232772349 T^{4} + 13022692821688500 T^{5} + 15169726987524966634 T^{6} +$$$$13\!\cdots\!46$$$$T^{7} +$$$$12\!\cdots\!80$$$$T^{8} +$$$$13\!\cdots\!46$$$$p^{3} T^{9} + 15169726987524966634 p^{6} T^{10} + 13022692821688500 p^{9} T^{11} + 11878232772349 p^{12} T^{12} + 7622227437 p^{15} T^{13} + 5372956 p^{18} T^{14} + 2037 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
97 $$1 - 6325147 T^{2} + 19957023240705 T^{4} - 41151654463376937002 T^{6} +$$$$61\!\cdots\!84$$$$T^{8} -$$$$71\!\cdots\!96$$$$T^{10} +$$$$66\!\cdots\!35$$$$T^{12} -$$$$55\!\cdots\!75$$$$T^{14} +$$$$48\!\cdots\!30$$$$T^{16} -$$$$55\!\cdots\!75$$$$p^{6} T^{18} +$$$$66\!\cdots\!35$$$$p^{12} T^{20} -$$$$71\!\cdots\!96$$$$p^{18} T^{22} +$$$$61\!\cdots\!84$$$$p^{24} T^{24} - 41151654463376937002 p^{30} T^{26} + 19957023240705 p^{36} T^{28} - 6325147 p^{42} T^{30} + p^{48} T^{32}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$