LMFDB - L-function 32-384e16-1.1-c4e16-0-1

Dirichlet series

L(s) = 1

− 8·3^-s + 80·7^-s + 32·9^-s + 816·19^-s − 640·21^-s + 4.00e3·25^-s + 8·27^-s + 592·31^-s + 2.24e3·37^-s + 368·43^-s − 1.40e4·49^-s − 6.52e3·57^-s − 3.52e3·61^-s + 2.56e3·63^-s − 3.53e3·67^-s + 3.68e3·73^-s − 3.20e4·75^-s − 1.44e4·79^-s − 732·81^-s − 4.73e3·93^-s + 3.26e3·97^-s + 1.56e4·103^-s + 1.88e4·109^-s − 1.79e4·111^-s + 1.00e5·121^-s + 127^-s − 2.94e3·129^-s + ⋯

L(s) = 1

− 8/9·3^-s + 1.63·7^-s + 0.395·9^-s + 2.26·19^-s − 1.45·21^-s + 32/5·25^-s + 0.0109·27^-s + 0.616·31^-s + 1.63·37^-s + 0.199·43^-s − 5.83·49^-s − 2.00·57^-s − 0.945·61^-s + 0.644·63^-s − 0.787·67^-s + 0.690·73^-s − 5.68·75^-s − 2.31·79^-s − 0.111·81^-s − 0.547·93^-s + 0.346·97^-s + 1.47·103^-s + 1.58·109^-s − 1.45·111^-s + 6.88·121^-s + 6.20e−5·127^-s − 0.176·129^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{112} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{112} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$32$
Conductor:	$2^{112} \cdot 3^{16}$
Sign:	$1$
Analytic conductor:	$3.79847\times 10^{25}$
Root analytic conductor:	$6.30032$
Motivic weight:	$4$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(32,\ 2^{112} \cdot 3^{16} ,\ ( \ : [2]^{16} ),\ 1 )$

Particular Values

$L(\frac{5}{2})$	$\approx$	$0.08532872465$
$L(\frac12)$	$\approx$	$0.08532872465$
$L(3)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 + 8 T + 32 T^{2} - 8 T^{3} - 140 p T^{4} + 872 p^{4} T^{5} + 14240 p^{3} T^{6} - 25192 p^{4} T^{7} - 47818 p^{6} T^{8} - 25192 p^{8} T^{9} + 14240 p^{11} T^{10} + 872 p^{16} T^{11} - 140 p^{17} T^{12} - 8 p^{20} T^{13} + 32 p^{24} T^{14} + 8 p^{28} T^{15} + p^{32} T^{16} $
good	5	$ 1 - 32 p^{3} T^{2} + 8360184 T^{4} - 11812615648 T^{6} + 12480057659164 T^{8} - 10476970039805088 T^{10} + 7380786397552340168 T^{12} - $$46\!\cdots\!44$$ T^{14} + $$28\!\cdots\!54$$ T^{16} - $$46\!\cdots\!44$$ p^{8} T^{18} + 7380786397552340168 p^{16} T^{20} - 10476970039805088 p^{24} T^{22} + 12480057659164 p^{32} T^{24} - 11812615648 p^{40} T^{26} + 8360184 p^{48} T^{28} - 32 p^{59} T^{30} + p^{64} T^{32} $
	7	$ ( 1 - 40 T + 192 p^{2} T^{2} - 486104 T^{3} + 7121924 p T^{4} - 2620332552 T^{5} + 190024111936 T^{6} - 9010112390392 T^{7} + 531157393675206 T^{8} - 9010112390392 p^{4} T^{9} + 190024111936 p^{8} T^{10} - 2620332552 p^{12} T^{11} + 7121924 p^{17} T^{12} - 486104 p^{20} T^{13} + 192 p^{26} T^{14} - 40 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	11	$ 1 - 100800 T^{2} + 5118269624 T^{4} - 175491900862272 T^{6} + 4603745418499112092 T^{8} - $$99\!\cdots\!52$$ T^{10} + $$18\!\cdots\!04$$ T^{12} - $$31\!\cdots\!24$$ T^{14} + $$48\!\cdots\!70$$ T^{16} - $$31\!\cdots\!24$$ p^{8} T^{18} + $$18\!\cdots\!04$$ p^{16} T^{20} - $$99\!\cdots\!52$$ p^{24} T^{22} + 4603745418499112092 p^{32} T^{24} - 175491900862272 p^{40} T^{26} + 5118269624 p^{48} T^{28} - 100800 p^{56} T^{30} + p^{64} T^{32} $
	13	$ ( 1 + 121400 T^{2} - 4006400 T^{3} + 7618414876 T^{4} - 353957181952 T^{5} + 338581225522824 T^{6} - 15008146429838336 T^{7} + 11237343777845368518 T^{8} - 15008146429838336 p^{4} T^{9} + 338581225522824 p^{8} T^{10} - 353957181952 p^{12} T^{11} + 7618414876 p^{16} T^{12} - 4006400 p^{20} T^{13} + 121400 p^{24} T^{14} + p^{32} T^{16} )^{2} $
	17	$ 1 - 642640 T^{2} + 226355525112 T^{4} - 55337621505254128 T^{6} + $$10\!\cdots\!80$$ T^{8} - $$15\!\cdots\!08$$ T^{10} + $$19\!\cdots\!08$$ T^{12} - $$12\!\cdots\!32$$ p T^{14} + $$18\!\cdots\!74$$ T^{16} - $$12\!\cdots\!32$$ p^{9} T^{18} + $$19\!\cdots\!08$$ p^{16} T^{20} - $$15\!\cdots\!08$$ p^{24} T^{22} + $$10\!\cdots\!80$$ p^{32} T^{24} - 55337621505254128 p^{40} T^{26} + 226355525112 p^{48} T^{28} - 642640 p^{56} T^{30} + p^{64} T^{32} $
	19	$ ( 1 - 408 T + 31808 p T^{2} - 208638248 T^{3} + 187463867548 T^{4} - 59631391409656 T^{5} + 39100536590819136 T^{6} - 11111807264627709512 T^{7} + $$58\!\cdots\!10$$ T^{8} - 11111807264627709512 p^{4} T^{9} + 39100536590819136 p^{8} T^{10} - 59631391409656 p^{12} T^{11} + 187463867548 p^{16} T^{12} - 208638248 p^{20} T^{13} + 31808 p^{25} T^{14} - 408 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	23	$ 1 - 2254800 T^{2} + 2573091649784 T^{4} - 1981999052617967472 T^{6} + $$11\!\cdots\!60$$ T^{8} - $$55\!\cdots\!20$$ T^{10} + $$22\!\cdots\!96$$ T^{12} - $$76\!\cdots\!76$$ T^{14} + $$22\!\cdots\!58$$ T^{16} - $$76\!\cdots\!76$$ p^{8} T^{18} + $$22\!\cdots\!96$$ p^{16} T^{20} - $$55\!\cdots\!20$$ p^{24} T^{22} + $$11\!\cdots\!60$$ p^{32} T^{24} - 1981999052617967472 p^{40} T^{26} + 2573091649784 p^{48} T^{28} - 2254800 p^{56} T^{30} + p^{64} T^{32} $
	29	$ 1 - 5491872 T^{2} + 15764191778552 T^{4} - 31325649647455309024 T^{6} + $$47\!\cdots\!52$$ T^{8} - $$59\!\cdots\!64$$ T^{10} + $$62\!\cdots\!64$$ T^{12} - $$55\!\cdots\!48$$ T^{14} + $$41\!\cdots\!58$$ T^{16} - $$55\!\cdots\!48$$ p^{8} T^{18} + $$62\!\cdots\!64$$ p^{16} T^{20} - $$59\!\cdots\!64$$ p^{24} T^{22} + $$47\!\cdots\!52$$ p^{32} T^{24} - 31325649647455309024 p^{40} T^{26} + 15764191778552 p^{48} T^{28} - 5491872 p^{56} T^{30} + p^{64} T^{32} $
	31	$ ( 1 - 296 T + 3866784 T^{2} - 1538703896 T^{3} + 8503750154716 T^{4} - 3150111482099784 T^{5} + 12751886092108801376 T^{6} - $$43\!\cdots\!92$$ T^{7} + $$13\!\cdots\!02$$ T^{8} - $$43\!\cdots\!92$$ p^{4} T^{9} + 12751886092108801376 p^{8} T^{10} - 3150111482099784 p^{12} T^{11} + 8503750154716 p^{16} T^{12} - 1538703896 p^{20} T^{13} + 3866784 p^{24} T^{14} - 296 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	37	$ ( 1 - 1120 T + 7803000 T^{2} - 6601566752 T^{3} + 32472276403868 T^{4} - 25036831654581600 T^{5} + 95073193505141492296 T^{6} - $$63\!\cdots\!08$$ T^{7} + $$20\!\cdots\!54$$ T^{8} - $$63\!\cdots\!08$$ p^{4} T^{9} + 95073193505141492296 p^{8} T^{10} - 25036831654581600 p^{12} T^{11} + 32472276403868 p^{16} T^{12} - 6601566752 p^{20} T^{13} + 7803000 p^{24} T^{14} - 1120 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	41	$ 1 - 24552528 T^{2} + 320354816296952 T^{4} - $$28\!\cdots\!20$$ T^{6} + $$19\!\cdots\!88$$ T^{8} - $$10\!\cdots\!08$$ T^{10} + $$10\!\cdots\!44$$ p T^{12} - $$16\!\cdots\!04$$ T^{14} + $$50\!\cdots\!34$$ T^{16} - $$16\!\cdots\!04$$ p^{8} T^{18} + $$10\!\cdots\!44$$ p^{17} T^{20} - $$10\!\cdots\!08$$ p^{24} T^{22} + $$19\!\cdots\!88$$ p^{32} T^{24} - $$28\!\cdots\!20$$ p^{40} T^{26} + 320354816296952 p^{48} T^{28} - 24552528 p^{56} T^{30} + p^{64} T^{32} $
	43	$ ( 1 - 184 T + 284352 p T^{2} + 6391691384 T^{3} + 75471585750940 T^{4} + 84561136965478632 T^{5} + $$32\!\cdots\!60$$ T^{6} + $$49\!\cdots\!44$$ T^{7} + $$11\!\cdots\!58$$ T^{8} + $$49\!\cdots\!44$$ p^{4} T^{9} + $$32\!\cdots\!60$$ p^{8} T^{10} + 84561136965478632 p^{12} T^{11} + 75471585750940 p^{16} T^{12} + 6391691384 p^{20} T^{13} + 284352 p^{25} T^{14} - 184 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	47	$ 1 - 34208272 T^{2} + 638007471980664 T^{4} - $$83\!\cdots\!52$$ T^{6} + $$85\!\cdots\!08$$ T^{8} - $$72\!\cdots\!36$$ T^{10} + $$51\!\cdots\!08$$ T^{12} - $$30\!\cdots\!20$$ T^{14} + $$16\!\cdots\!02$$ T^{16} - $$30\!\cdots\!20$$ p^{8} T^{18} + $$51\!\cdots\!08$$ p^{16} T^{20} - $$72\!\cdots\!36$$ p^{24} T^{22} + $$85\!\cdots\!08$$ p^{32} T^{24} - $$83\!\cdots\!52$$ p^{40} T^{26} + 638007471980664 p^{48} T^{28} - 34208272 p^{56} T^{30} + p^{64} T^{32} $
	53	$ 1 - 80081056 T^{2} + 3065183500117752 T^{4} - $$14\!\cdots\!72$$ p T^{6} + $$13\!\cdots\!68$$ T^{8} - $$18\!\cdots\!72$$ T^{10} + $$21\!\cdots\!52$$ T^{12} - $$20\!\cdots\!36$$ T^{14} + $$17\!\cdots\!18$$ T^{16} - $$20\!\cdots\!36$$ p^{8} T^{18} + $$21\!\cdots\!52$$ p^{16} T^{20} - $$18\!\cdots\!72$$ p^{24} T^{22} + $$13\!\cdots\!68$$ p^{32} T^{24} - $$14\!\cdots\!72$$ p^{41} T^{26} + 3065183500117752 p^{48} T^{28} - 80081056 p^{56} T^{30} + p^{64} T^{32} $
	59	$ 1 - 83753088 T^{2} + 3510247343477816 T^{4} - $$10\!\cdots\!40$$ T^{6} + $$22\!\cdots\!72$$ T^{8} - $$42\!\cdots\!16$$ T^{10} + $$68\!\cdots\!20$$ T^{12} - $$99\!\cdots\!12$$ T^{14} + $$12\!\cdots\!78$$ T^{16} - $$99\!\cdots\!12$$ p^{8} T^{18} + $$68\!\cdots\!20$$ p^{16} T^{20} - $$42\!\cdots\!16$$ p^{24} T^{22} + $$22\!\cdots\!72$$ p^{32} T^{24} - $$10\!\cdots\!40$$ p^{40} T^{26} + 3510247343477816 p^{48} T^{28} - 83753088 p^{56} T^{30} + p^{64} T^{32} $
	61	$ ( 1 + 1760 T + 47001336 T^{2} + 128816847264 T^{3} + 1100233580112540 T^{4} + 4292167156382071776 T^{5} + $$18\!\cdots\!40$$ T^{6} + $$86\!\cdots\!00$$ T^{7} + $$25\!\cdots\!50$$ T^{8} + $$86\!\cdots\!00$$ p^{4} T^{9} + $$18\!\cdots\!40$$ p^{8} T^{10} + 4292167156382071776 p^{12} T^{11} + 1100233580112540 p^{16} T^{12} + 128816847264 p^{20} T^{13} + 47001336 p^{24} T^{14} + 1760 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	67	$ ( 1 + 1768 T + 68567520 T^{2} + 214377613080 T^{3} + 2960039920647132 T^{4} + 8809586189871070536 T^{5} + $$93\!\cdots\!12$$ T^{6} + $$25\!\cdots\!08$$ T^{7} + $$21\!\cdots\!78$$ T^{8} + $$25\!\cdots\!08$$ p^{4} T^{9} + $$93\!\cdots\!12$$ p^{8} T^{10} + 8809586189871070536 p^{12} T^{11} + 2960039920647132 p^{16} T^{12} + 214377613080 p^{20} T^{13} + 68567520 p^{24} T^{14} + 1768 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	71	$ 1 - 116630736 T^{2} + 7001190284309240 T^{4} - $$29\!\cdots\!00$$ T^{6} + $$95\!\cdots\!32$$ T^{8} - $$23\!\cdots\!76$$ T^{10} + $$43\!\cdots\!24$$ T^{12} - $$60\!\cdots\!48$$ T^{14} + $$98\!\cdots\!50$$ T^{16} - $$60\!\cdots\!48$$ p^{8} T^{18} + $$43\!\cdots\!24$$ p^{16} T^{20} - $$23\!\cdots\!76$$ p^{24} T^{22} + $$95\!\cdots\!32$$ p^{32} T^{24} - $$29\!\cdots\!00$$ p^{40} T^{26} + 7001190284309240 p^{48} T^{28} - 116630736 p^{56} T^{30} + p^{64} T^{32} $
	73	$ ( 1 - 1840 T + 49889016 T^{2} + 15253020784 T^{3} + 1855084280061212 T^{4} + 4235840312198043600 T^{5} + $$36\!\cdots\!00$$ T^{6} + $$20\!\cdots\!72$$ T^{7} + $$12\!\cdots\!62$$ T^{8} + $$20\!\cdots\!72$$ p^{4} T^{9} + $$36\!\cdots\!00$$ p^{8} T^{10} + 4235840312198043600 p^{12} T^{11} + 1855084280061212 p^{16} T^{12} + 15253020784 p^{20} T^{13} + 49889016 p^{24} T^{14} - 1840 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	79	$ ( 1 + 7224 T + 196384352 T^{2} + 1025749762696 T^{3} + 18221658583719772 T^{4} + 74606095198943756696 T^{5} + $$10\!\cdots\!16$$ T^{6} + $$37\!\cdots\!80$$ T^{7} + $$48\!\cdots\!58$$ T^{8} + $$37\!\cdots\!80$$ p^{4} T^{9} + $$10\!\cdots\!16$$ p^{8} T^{10} + 74606095198943756696 p^{12} T^{11} + 18221658583719772 p^{16} T^{12} + 1025749762696 p^{20} T^{13} + 196384352 p^{24} T^{14} + 7224 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	83	$ 1 - 204959680 T^{2} + 20014208587471032 T^{4} - $$12\!\cdots\!88$$ T^{6} + $$65\!\cdots\!56$$ T^{8} - $$38\!\cdots\!36$$ T^{10} + $$23\!\cdots\!72$$ T^{12} - $$12\!\cdots\!72$$ T^{14} + $$61\!\cdots\!54$$ T^{16} - $$12\!\cdots\!72$$ p^{8} T^{18} + $$23\!\cdots\!72$$ p^{16} T^{20} - $$38\!\cdots\!36$$ p^{24} T^{22} + $$65\!\cdots\!56$$ p^{32} T^{24} - $$12\!\cdots\!88$$ p^{40} T^{26} + 20014208587471032 p^{48} T^{28} - 204959680 p^{56} T^{30} + p^{64} T^{32} $
	89	$ 1 - 282902928 T^{2} + 46397949161852792 T^{4} - $$59\!\cdots\!08$$ T^{6} + $$62\!\cdots\!48$$ T^{8} - $$58\!\cdots\!32$$ T^{10} + $$47\!\cdots\!88$$ T^{12} - $$34\!\cdots\!00$$ T^{14} + $$22\!\cdots\!22$$ T^{16} - $$34\!\cdots\!00$$ p^{8} T^{18} + $$47\!\cdots\!88$$ p^{16} T^{20} - $$58\!\cdots\!32$$ p^{24} T^{22} + $$62\!\cdots\!48$$ p^{32} T^{24} - $$59\!\cdots\!08$$ p^{40} T^{26} + 46397949161852792 p^{48} T^{28} - 282902928 p^{56} T^{30} + p^{64} T^{32} $
	97	$ ( 1 - 1632 T + 323922296 T^{2} - 364139830304 T^{3} + 46869361751343388 T^{4} - 54960712317698626912 T^{5} + $$42\!\cdots\!16$$ T^{6} - $$68\!\cdots\!60$$ T^{7} + $$34\!\cdots\!02$$ T^{8} - $$68\!\cdots\!60$$ p^{4} T^{9} + $$42\!\cdots\!16$$ p^{8} T^{10} - 54960712317698626912 p^{12} T^{11} + 46869361751343388 p^{16} T^{12} - 364139830304 p^{20} T^{13} + 323922296 p^{24} T^{14} - 1632 p^{28} T^{15} + p^{32} T^{16} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−2.32458048487894615598399578576, −2.30983344304185358718436969548, −2.22936180719508515386425263104, −2.06252416898407955368989584682, −2.02360834998366908052067788433, −1.97071169620690148836275291670, −1.88554549250929118000781693305, −1.65593822434705847537621653801, −1.57147533259244726649121180670, −1.55796590773195879480148859158, −1.42900850243718062830157344838, −1.28116904295484217030579502967, −1.26872842880251755164345769072, −1.19896975834138343945315419714, −1.09106748077117273132267123094, −0.937208082102262637432986970944, −0.890889777604678480609781961915, −0.878473132977076926871698765718, −0.849093468015425998021770226142, −0.66002124624473258429684388643, −0.49269834415615655011761355081, −0.37360425790386157686043567818, −0.35350638757344865679687626752, −0.07788758608182126775474019558, −0.01426244226729586722237205582, 0.01426244226729586722237205582, 0.07788758608182126775474019558, 0.35350638757344865679687626752, 0.37360425790386157686043567818, 0.49269834415615655011761355081, 0.66002124624473258429684388643, 0.849093468015425998021770226142, 0.878473132977076926871698765718, 0.890889777604678480609781961915, 0.937208082102262637432986970944, 1.09106748077117273132267123094, 1.19896975834138343945315419714, 1.26872842880251755164345769072, 1.28116904295484217030579502967, 1.42900850243718062830157344838, 1.55796590773195879480148859158, 1.57147533259244726649121180670, 1.65593822434705847537621653801, 1.88554549250929118000781693305, 1.97071169620690148836275291670, 2.02360834998366908052067788433, 2.06252416898407955368989584682, 2.22936180719508515386425263104, 2.30983344304185358718436969548, 2.32458048487894615598399578576

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{5}{2})\)	\(\approx\)	\(0.08532872465\)
\(L(\frac12)\)	\(\approx\)	\(0.08532872465\)
\(L(3)\)		not available
\(L(1)\)		not available

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