LMFDB - L-function 32-138e16-1.1-c4e16-0-0

Dirichlet series

L(s) = 1

+ 64·4^-s + 216·9^-s − 208·13^-s + 2.30e3·16^-s + 840·23^-s + 5.52e3·25^-s + 3.60e3·29^-s + 224·31^-s + 1.38e4·36^-s − 6.14e3·41^-s + 8.88e3·47^-s + 1.22e4·49^-s − 1.33e4·52^-s − 1.82e4·59^-s + 6.14e4·64^-s − 3.00e4·71^-s + 9.53e3·73^-s + 2.62e4·81^-s + 5.37e4·92^-s + 3.53e5·100^-s − 2.43e4·101^-s + 2.30e5·116^-s − 4.49e4·117^-s + 1.68e5·121^-s + 1.43e4·124^-s + 127^-s + 131^-s + ⋯

L(s) = 1

+ 4·4^-s + 8/3·9^-s − 1.23·13^-s + 9·16^-s + 1.58·23^-s + 8.84·25^-s + 4.28·29^-s + 0.233·31^-s + 32/3·36^-s − 3.65·41^-s + 4.01·47^-s + 5.10·49^-s − 4.92·52^-s − 5.23·59^-s + 15·64^-s − 5.96·71^-s + 1.78·73^-s + 4·81^-s + 6.35·92^-s + 35.3·100^-s − 2.39·101^-s + 17.1·116^-s − 3.28·117^-s + 11.4·121^-s + 0.932·124^-s + 6.20e−5·127^-s + 5.82e−5·131^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$32$
Conductor:	$2^{16} \cdot 3^{16} \cdot 23^{16}$
Sign:	$1$
Analytic conductor:	$2.94018\times 10^{18}$
Root analytic conductor:	$3.77691$
Motivic weight:	$4$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(32,\ 2^{16} \cdot 3^{16} \cdot 23^{16} ,\ ( \ : [2]^{16} ),\ 1 )$

Particular Values

$L(\frac{5}{2})$	$\approx$	$1263.246957$
$L(\frac12)$	$\approx$	$1263.246957$
$L(3)$		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 - p^{3} T^{2} )^{8} $
	3	$ ( 1 - p^{3} T^{2} )^{8} $
	23	$ 1 - 840 T + 44360 T^{2} + 178554216 T^{3} - 1696567452 p T^{4} - 72654864648 p^{2} T^{5} + 106410410872 p^{4} T^{6} + 108058738152 p^{6} T^{7} - 277670302714 p^{8} T^{8} + 108058738152 p^{10} T^{9} + 106410410872 p^{12} T^{10} - 72654864648 p^{14} T^{11} - 1696567452 p^{17} T^{12} + 178554216 p^{20} T^{13} + 44360 p^{24} T^{14} - 840 p^{28} T^{15} + p^{32} T^{16} $
good	5	$ 1 - 5528 T^{2} + 3129704 p T^{4} - 6012520328 p T^{6} + 8736367746604 p T^{8} - 10139095849378456 p T^{10} + 9692547211097431768 p T^{12} - $$38\!\cdots\!56$$ T^{14} + $$26\!\cdots\!34$$ T^{16} - $$38\!\cdots\!56$$ p^{8} T^{18} + 9692547211097431768 p^{17} T^{20} - 10139095849378456 p^{25} T^{22} + 8736367746604 p^{33} T^{24} - 6012520328 p^{41} T^{26} + 3129704 p^{49} T^{28} - 5528 p^{56} T^{30} + p^{64} T^{32} $
	7	$ 1 - 1752 p T^{2} + 95429088 T^{4} - 77651741256 p T^{6} + 2501110400967004 T^{8} - 1377178074658493208 p T^{10} + $$65\!\cdots\!08$$ p^{2} T^{12} - $$27\!\cdots\!68$$ p^{3} T^{14} + $$98\!\cdots\!66$$ p^{4} T^{16} - $$27\!\cdots\!68$$ p^{11} T^{18} + $$65\!\cdots\!08$$ p^{18} T^{20} - 1377178074658493208 p^{25} T^{22} + 2501110400967004 p^{32} T^{24} - 77651741256 p^{41} T^{26} + 95429088 p^{48} T^{28} - 1752 p^{57} T^{30} + p^{64} T^{32} $
	11	$ 1 - 168240 T^{2} + 13751591232 T^{4} - 727033719630480 T^{6} + 27909354212020346140 T^{8} - $$82\!\cdots\!52$$ T^{10} + $$17\!\cdots\!88$$ p T^{12} - $$38\!\cdots\!68$$ T^{14} + $$61\!\cdots\!82$$ T^{16} - $$38\!\cdots\!68$$ p^{8} T^{18} + $$17\!\cdots\!88$$ p^{17} T^{20} - $$82\!\cdots\!52$$ p^{24} T^{22} + 27909354212020346140 p^{32} T^{24} - 727033719630480 p^{40} T^{26} + 13751591232 p^{48} T^{28} - 168240 p^{56} T^{30} + p^{64} T^{32} $
	13	$ ( 1 + 8 p T + 51232 T^{2} + 6296440 T^{3} + 2494759004 T^{4} + 304277234600 T^{5} + 5613356563040 p T^{6} + 10743243892930936 T^{7} + 2599367041660495558 T^{8} + 10743243892930936 p^{4} T^{9} + 5613356563040 p^{9} T^{10} + 304277234600 p^{12} T^{11} + 2494759004 p^{16} T^{12} + 6296440 p^{20} T^{13} + 51232 p^{24} T^{14} + 8 p^{29} T^{15} + p^{32} T^{16} )^{2} $
	17	$ 1 - 871256 T^{2} + 371509209856 T^{4} - 103258341468464008 T^{6} + $$21\!\cdots\!92$$ T^{8} - $$33\!\cdots\!52$$ T^{10} + $$43\!\cdots\!08$$ T^{12} - $$46\!\cdots\!88$$ T^{14} + $$41\!\cdots\!70$$ T^{16} - $$46\!\cdots\!88$$ p^{8} T^{18} + $$43\!\cdots\!08$$ p^{16} T^{20} - $$33\!\cdots\!52$$ p^{24} T^{22} + $$21\!\cdots\!92$$ p^{32} T^{24} - 103258341468464008 p^{40} T^{26} + 371509209856 p^{48} T^{28} - 871256 p^{56} T^{30} + p^{64} T^{32} $
	19	$ 1 - 984168 T^{2} + 523401930600 T^{4} - 194685284481286104 T^{6} + $$55\!\cdots\!04$$ T^{8} - $$13\!\cdots\!36$$ T^{10} + $$25\!\cdots\!56$$ T^{12} - $$41\!\cdots\!20$$ T^{14} + $$58\!\cdots\!30$$ T^{16} - $$41\!\cdots\!20$$ p^{8} T^{18} + $$25\!\cdots\!56$$ p^{16} T^{20} - $$13\!\cdots\!36$$ p^{24} T^{22} + $$55\!\cdots\!04$$ p^{32} T^{24} - 194685284481286104 p^{40} T^{26} + 523401930600 p^{48} T^{28} - 984168 p^{56} T^{30} + p^{64} T^{32} $
	29	$ ( 1 - 1800 T + 3077520 T^{2} - 3480834168 T^{3} + 3768698994940 T^{4} - 117506802952584 p T^{5} + 3066589885220833392 T^{6} - $$26\!\cdots\!68$$ T^{7} + $$22\!\cdots\!54$$ T^{8} - $$26\!\cdots\!68$$ p^{4} T^{9} + 3066589885220833392 p^{8} T^{10} - 117506802952584 p^{13} T^{11} + 3768698994940 p^{16} T^{12} - 3480834168 p^{20} T^{13} + 3077520 p^{24} T^{14} - 1800 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	31	$ ( 1 - 112 T + 3419760 T^{2} + 304290448 T^{3} + 209484367172 p T^{4} + 1706316920005488 T^{5} + 8249565596849829520 T^{6} + $$30\!\cdots\!44$$ T^{7} + $$84\!\cdots\!42$$ T^{8} + $$30\!\cdots\!44$$ p^{4} T^{9} + 8249565596849829520 p^{8} T^{10} + 1706316920005488 p^{12} T^{11} + 209484367172 p^{17} T^{12} + 304290448 p^{20} T^{13} + 3419760 p^{24} T^{14} - 112 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	37	$ 1 - 12568592 T^{2} + 82846351016320 T^{4} - $$38\!\cdots\!36$$ T^{6} + $$13\!\cdots\!12$$ T^{8} - $$41\!\cdots\!72$$ T^{10} + $$10\!\cdots\!56$$ T^{12} - $$23\!\cdots\!84$$ T^{14} + $$47\!\cdots\!10$$ T^{16} - $$23\!\cdots\!84$$ p^{8} T^{18} + $$10\!\cdots\!56$$ p^{16} T^{20} - $$41\!\cdots\!72$$ p^{24} T^{22} + $$13\!\cdots\!12$$ p^{32} T^{24} - $$38\!\cdots\!36$$ p^{40} T^{26} + 82846351016320 p^{48} T^{28} - 12568592 p^{56} T^{30} + p^{64} T^{32} $
	41	$ ( 1 + 3072 T + 20150752 T^{2} + 53025897792 T^{3} + 184116878567260 T^{4} + 408892695611865408 T^{5} + $$99\!\cdots\!96$$ T^{6} + $$18\!\cdots\!08$$ T^{7} + $$34\!\cdots\!38$$ T^{8} + $$18\!\cdots\!08$$ p^{4} T^{9} + $$99\!\cdots\!96$$ p^{8} T^{10} + 408892695611865408 p^{12} T^{11} + 184116878567260 p^{16} T^{12} + 53025897792 p^{20} T^{13} + 20150752 p^{24} T^{14} + 3072 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	43	$ 1 - 16821384 T^{2} + 138348240958440 T^{4} - $$80\!\cdots\!12$$ T^{6} + $$39\!\cdots\!04$$ T^{8} - $$17\!\cdots\!80$$ T^{10} + $$73\!\cdots\!24$$ T^{12} - $$28\!\cdots\!88$$ T^{14} + $$10\!\cdots\!86$$ T^{16} - $$28\!\cdots\!88$$ p^{8} T^{18} + $$73\!\cdots\!24$$ p^{16} T^{20} - $$17\!\cdots\!80$$ p^{24} T^{22} + $$39\!\cdots\!04$$ p^{32} T^{24} - $$80\!\cdots\!12$$ p^{40} T^{26} + 138348240958440 p^{48} T^{28} - 16821384 p^{56} T^{30} + p^{64} T^{32} $
	47	$ ( 1 - 4440 T + 29315624 T^{2} - 100696742760 T^{3} + 401513294421660 T^{4} - 1129672588229791800 T^{5} + $$34\!\cdots\!64$$ T^{6} - $$80\!\cdots\!88$$ T^{7} + $$20\!\cdots\!62$$ T^{8} - $$80\!\cdots\!88$$ p^{4} T^{9} + $$34\!\cdots\!64$$ p^{8} T^{10} - 1129672588229791800 p^{12} T^{11} + 401513294421660 p^{16} T^{12} - 100696742760 p^{20} T^{13} + 29315624 p^{24} T^{14} - 4440 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	53	$ 1 - 81820088 T^{2} + 3309365513309896 T^{4} - $$87\!\cdots\!40$$ T^{6} + $$17\!\cdots\!96$$ T^{8} - $$26\!\cdots\!92$$ T^{10} + $$32\!\cdots\!12$$ T^{12} - $$33\!\cdots\!56$$ T^{14} + $$28\!\cdots\!66$$ T^{16} - $$33\!\cdots\!56$$ p^{8} T^{18} + $$32\!\cdots\!12$$ p^{16} T^{20} - $$26\!\cdots\!92$$ p^{24} T^{22} + $$17\!\cdots\!96$$ p^{32} T^{24} - $$87\!\cdots\!40$$ p^{40} T^{26} + 3309365513309896 p^{48} T^{28} - 81820088 p^{56} T^{30} + p^{64} T^{32} $
	59	$ ( 1 + 9120 T + 95529056 T^{2} + 519996534816 T^{3} + 3052542375336924 T^{4} + 11445951667572554976 T^{5} + $$49\!\cdots\!36$$ T^{6} + $$14\!\cdots\!96$$ T^{7} + $$60\!\cdots\!46$$ T^{8} + $$14\!\cdots\!96$$ p^{4} T^{9} + $$49\!\cdots\!36$$ p^{8} T^{10} + 11445951667572554976 p^{12} T^{11} + 3052542375336924 p^{16} T^{12} + 519996534816 p^{20} T^{13} + 95529056 p^{24} T^{14} + 9120 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	61	$ 1 - 90694608 T^{2} + 4239836803554048 T^{4} - $$13\!\cdots\!44$$ T^{6} + $$35\!\cdots\!56$$ T^{8} - $$78\!\cdots\!40$$ T^{10} + $$14\!\cdots\!20$$ T^{12} - $$24\!\cdots\!76$$ T^{14} + $$36\!\cdots\!50$$ T^{16} - $$24\!\cdots\!76$$ p^{8} T^{18} + $$14\!\cdots\!20$$ p^{16} T^{20} - $$78\!\cdots\!40$$ p^{24} T^{22} + $$35\!\cdots\!56$$ p^{32} T^{24} - $$13\!\cdots\!44$$ p^{40} T^{26} + 4239836803554048 p^{48} T^{28} - 90694608 p^{56} T^{30} + p^{64} T^{32} $
	67	$ 1 - 90670504 T^{2} + 5117920319415528 T^{4} - $$22\!\cdots\!40$$ T^{6} + $$82\!\cdots\!84$$ T^{8} - $$25\!\cdots\!44$$ T^{10} + $$69\!\cdots\!00$$ T^{12} - $$16\!\cdots\!00$$ T^{14} + $$35\!\cdots\!46$$ T^{16} - $$16\!\cdots\!00$$ p^{8} T^{18} + $$69\!\cdots\!00$$ p^{16} T^{20} - $$25\!\cdots\!44$$ p^{24} T^{22} + $$82\!\cdots\!84$$ p^{32} T^{24} - $$22\!\cdots\!40$$ p^{40} T^{26} + 5117920319415528 p^{48} T^{28} - 90670504 p^{56} T^{30} + p^{64} T^{32} $
	71	$ ( 1 + 15024 T + 226140656 T^{2} + 2169472467312 T^{3} + 19727900044841532 T^{4} + $$14\!\cdots\!28$$ T^{5} + $$96\!\cdots\!76$$ T^{6} + $$55\!\cdots\!36$$ T^{7} + $$30\!\cdots\!22$$ T^{8} + $$55\!\cdots\!36$$ p^{4} T^{9} + $$96\!\cdots\!76$$ p^{8} T^{10} + $$14\!\cdots\!28$$ p^{12} T^{11} + 19727900044841532 p^{16} T^{12} + 2169472467312 p^{20} T^{13} + 226140656 p^{24} T^{14} + 15024 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	73	$ ( 1 - 4768 T + 171463104 T^{2} - 717876323552 T^{3} + 13617150666587804 T^{4} - 49055683680190864800 T^{5} + $$66\!\cdots\!52$$ T^{6} - $$20\!\cdots\!88$$ T^{7} + $$30\!\cdots\!82$$ p T^{8} - $$20\!\cdots\!88$$ p^{4} T^{9} + $$66\!\cdots\!52$$ p^{8} T^{10} - 49055683680190864800 p^{12} T^{11} + 13617150666587804 p^{16} T^{12} - 717876323552 p^{20} T^{13} + 171463104 p^{24} T^{14} - 4768 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	79	$ 1 - 308756264 T^{2} + 43204410727054816 T^{4} - $$36\!\cdots\!80$$ T^{6} + $$21\!\cdots\!72$$ T^{8} - $$10\!\cdots\!60$$ T^{10} + $$47\!\cdots\!20$$ T^{12} - $$23\!\cdots\!64$$ T^{14} + $$10\!\cdots\!02$$ T^{16} - $$23\!\cdots\!64$$ p^{8} T^{18} + $$47\!\cdots\!20$$ p^{16} T^{20} - $$10\!\cdots\!60$$ p^{24} T^{22} + $$21\!\cdots\!72$$ p^{32} T^{24} - $$36\!\cdots\!80$$ p^{40} T^{26} + 43204410727054816 p^{48} T^{28} - 308756264 p^{56} T^{30} + p^{64} T^{32} $
	83	$ 1 - 128547888 T^{2} + 11030550917402304 T^{4} - $$10\!\cdots\!00$$ T^{6} + $$70\!\cdots\!12$$ T^{8} - $$40\!\cdots\!76$$ T^{10} + $$24\!\cdots\!92$$ T^{12} - $$12\!\cdots\!68$$ T^{14} + $$59\!\cdots\!22$$ T^{16} - $$12\!\cdots\!68$$ p^{8} T^{18} + $$24\!\cdots\!92$$ p^{16} T^{20} - $$40\!\cdots\!76$$ p^{24} T^{22} + $$70\!\cdots\!12$$ p^{32} T^{24} - $$10\!\cdots\!00$$ p^{40} T^{26} + 11030550917402304 p^{48} T^{28} - 128547888 p^{56} T^{30} + p^{64} T^{32} $
	89	$ 1 - 310558296 T^{2} + 52328438071681344 T^{4} - $$64\!\cdots\!56$$ T^{6} + $$65\!\cdots\!96$$ T^{8} - $$58\!\cdots\!48$$ T^{10} + $$46\!\cdots\!64$$ T^{12} - $$33\!\cdots\!08$$ T^{14} + $$22\!\cdots\!90$$ T^{16} - $$33\!\cdots\!08$$ p^{8} T^{18} + $$46\!\cdots\!64$$ p^{16} T^{20} - $$58\!\cdots\!48$$ p^{24} T^{22} + $$65\!\cdots\!96$$ p^{32} T^{24} - $$64\!\cdots\!56$$ p^{40} T^{26} + 52328438071681344 p^{48} T^{28} - 310558296 p^{56} T^{30} + p^{64} T^{32} $
	97	$ 1 - 696373712 T^{2} + 250204479362725624 T^{4} - $$60\!\cdots\!88$$ T^{6} + $$11\!\cdots\!68$$ T^{8} - $$16\!\cdots\!88$$ T^{10} + $$20\!\cdots\!64$$ T^{12} - $$22\!\cdots\!64$$ T^{14} + $$21\!\cdots\!14$$ T^{16} - $$22\!\cdots\!64$$ p^{8} T^{18} + $$20\!\cdots\!64$$ p^{16} T^{20} - $$16\!\cdots\!88$$ p^{24} T^{22} + $$11\!\cdots\!68$$ p^{32} T^{24} - $$60\!\cdots\!88$$ p^{40} T^{26} + 250204479362725624 p^{48} T^{28} - 696373712 p^{56} T^{30} + p^{64} T^{32} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−2.93137078550774891333636637351, −2.92139308390534978537524561265, −2.88214121437045072192469025603, −2.82493221657629260712462137673, −2.67354479248026111582792100149, −2.64439888155844314279465244270, −2.47681518747390284069564045058, −2.40316586916300488642324918005, −2.24618904416173389805607292927, −1.95386461946138755303533298285, −1.93284183572454206921938894349, −1.83426960345629751181697801682, −1.81067295512030827155426342800, −1.49763274861329333310038374561, −1.46116192743396777635584728906, −1.41687642221805143420716538564, −1.24905621008589371217489535835, −1.01673628048019865270155541166, −0.963779725728799674125977155099, −0.904654454708554381950284696790, −0.828661423770063129617577076967, −0.77484949072739970587906029318, −0.56922183975629538996676461930, −0.42771448406921908060802707692, −0.31199494381736314259877208393, 0.31199494381736314259877208393, 0.42771448406921908060802707692, 0.56922183975629538996676461930, 0.77484949072739970587906029318, 0.828661423770063129617577076967, 0.904654454708554381950284696790, 0.963779725728799674125977155099, 1.01673628048019865270155541166, 1.24905621008589371217489535835, 1.41687642221805143420716538564, 1.46116192743396777635584728906, 1.49763274861329333310038374561, 1.81067295512030827155426342800, 1.83426960345629751181697801682, 1.93284183572454206921938894349, 1.95386461946138755303533298285, 2.24618904416173389805607292927, 2.40316586916300488642324918005, 2.47681518747390284069564045058, 2.64439888155844314279465244270, 2.67354479248026111582792100149, 2.82493221657629260712462137673, 2.88214121437045072192469025603, 2.92139308390534978537524561265, 2.93137078550774891333636637351

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 $\F_{q}$

\(L(\frac{5}{2})\)	\(\approx\)	\(1263.246957\)
\(L(\frac12)\)	\(\approx\)	\(1263.246957\)
\(L(3)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more