LMFDB - L-function 16-867e8-1.1-c5e8-0-0

Dirichlet series

L(s) = 1

+ 3·2^-s − 72·3^-s − 30·4^-s − 216·6^-s + 18·7^-s − 96·8^-s + 2.91e3·9^-s + 966·11^-s + 2.16e3·12^-s + 382·13^-s + 54·14^-s + 25·16^-s + 8.74e3·18^-s + 4.52e3·19^-s − 1.29e3·21^-s + 2.89e3·22^-s − 240·23^-s + 6.91e3·24^-s − 5.29e3·25^-s + 1.14e3·26^-s − 8.74e4·27^-s − 540·28^-s − 6.07e3·29^-s − 1.72e4·31^-s + 5.22e3·32^-s − 6.95e4·33^-s − 8.74e4·36^-s + ⋯

L(s) = 1

+ 0.530·2^-s − 4.61·3^-s − 0.937·4^-s − 2.44·6^-s + 0.138·7^-s − 0.530·8^-s + 12·9^-s + 2.40·11^-s + 4.33·12^-s + 0.626·13^-s + 0.0736·14^-s + 0.0244·16^-s + 6.36·18^-s + 2.87·19^-s − 0.641·21^-s + 1.27·22^-s − 0.0946·23^-s + 2.44·24^-s − 1.69·25^-s + 0.332·26^-s − 23.0·27^-s − 0.130·28^-s − 1.34·29^-s − 3.22·31^-s + 0.901·32^-s − 11.1·33^-s − 11.2·36^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$3^{8} \cdot 17^{16}$
Sign:	$1$
Analytic conductor:	$1.39777\times 10^{17}$
Root analytic conductor:	$11.7920$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 3^{8} \cdot 17^{16} ,\ ( \ : [5/2]^{8} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$6.399210097$
$L(\frac12)$	$\approx$	$6.399210097$
$L(\frac{7}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ ( 1 + p^{2} T )^{8} $
	17	$ 1 $
good	2	$ 1 - 3 T + 39 T^{2} - 111 T^{3} + 595 p T^{4} - 519 p^{4} T^{5} + 1715 p^{4} T^{6} - 17625 p^{4} T^{7} - 3479 p^{5} T^{8} - 17625 p^{9} T^{9} + 1715 p^{14} T^{10} - 519 p^{19} T^{11} + 595 p^{21} T^{12} - 111 p^{25} T^{13} + 39 p^{30} T^{14} - 3 p^{35} T^{15} + p^{40} T^{16} $
	5	$ 1 + 5291 T^{2} - 7548 p^{2} T^{3} + 13383846 T^{4} - 43565724 p^{2} T^{5} + 72900129181 T^{6} - 120784634328 p^{2} T^{7} + 323679949132306 T^{8} - 120784634328 p^{7} T^{9} + 72900129181 p^{10} T^{10} - 43565724 p^{17} T^{11} + 13383846 p^{20} T^{12} - 7548 p^{27} T^{13} + 5291 p^{30} T^{14} + p^{40} T^{16} $
	7	$ 1 - 18 T + 6189 p T^{2} - 670806 T^{3} + 1449020513 T^{4} - 3240524232 p T^{5} + 36363757157166 T^{6} - 496345337365140 T^{7} + 656892756014483466 T^{8} - 496345337365140 p^{5} T^{9} + 36363757157166 p^{10} T^{10} - 3240524232 p^{16} T^{11} + 1449020513 p^{20} T^{12} - 670806 p^{25} T^{13} + 6189 p^{31} T^{14} - 18 p^{35} T^{15} + p^{40} T^{16} $
	11	$ 1 - 966 T + 1170505 T^{2} - 744278466 T^{3} + 555893309318 T^{4} - 280389015578394 T^{5} + 160709887094503647 T^{6} - 66990353995776268350 T^{7} + $$31\!\cdots\!10$$ T^{8} - 66990353995776268350 p^{5} T^{9} + 160709887094503647 p^{10} T^{10} - 280389015578394 p^{15} T^{11} + 555893309318 p^{20} T^{12} - 744278466 p^{25} T^{13} + 1170505 p^{30} T^{14} - 966 p^{35} T^{15} + p^{40} T^{16} $
	13	$ 1 - 382 T + 1004807 T^{2} - 22273414 T^{3} + 445411305161 T^{4} + 185932949084640 T^{5} + 127624826893392198 T^{6} + $$13\!\cdots\!60$$ T^{7} + $$28\!\cdots\!98$$ p T^{8} + $$13\!\cdots\!60$$ p^{5} T^{9} + 127624826893392198 p^{10} T^{10} + 185932949084640 p^{15} T^{11} + 445411305161 p^{20} T^{12} - 22273414 p^{25} T^{13} + 1004807 p^{30} T^{14} - 382 p^{35} T^{15} + p^{40} T^{16} $
	19	$ 1 - 4526 T + 16700303 T^{2} - 36527271910 T^{3} + 64832353487369 T^{4} - 59436581787748492 T^{5} + 163977345069041298 p T^{6} + $$18\!\cdots\!32$$ T^{7} - $$34\!\cdots\!70$$ T^{8} + $$18\!\cdots\!32$$ p^{5} T^{9} + 163977345069041298 p^{11} T^{10} - 59436581787748492 p^{15} T^{11} + 64832353487369 p^{20} T^{12} - 36527271910 p^{25} T^{13} + 16700303 p^{30} T^{14} - 4526 p^{35} T^{15} + p^{40} T^{16} $
	23	$ 1 + 240 T + 1070320 p T^{2} + 4223567760 T^{3} + 15591216346308 p T^{4} + 56596648963071504 T^{5} + $$35\!\cdots\!12$$ T^{6} + $$49\!\cdots\!24$$ T^{7} + $$26\!\cdots\!38$$ T^{8} + $$49\!\cdots\!24$$ p^{5} T^{9} + $$35\!\cdots\!12$$ p^{10} T^{10} + 56596648963071504 p^{15} T^{11} + 15591216346308 p^{21} T^{12} + 4223567760 p^{25} T^{13} + 1070320 p^{31} T^{14} + 240 p^{35} T^{15} + p^{40} T^{16} $
	29	$ 1 + 6072 T + 105010587 T^{2} + 568361280852 T^{3} + 205127274825238 p T^{4} + 27574537084016816868 T^{5} + $$21\!\cdots\!05$$ T^{6} + $$84\!\cdots\!80$$ T^{7} + $$52\!\cdots\!30$$ T^{8} + $$84\!\cdots\!80$$ p^{5} T^{9} + $$21\!\cdots\!05$$ p^{10} T^{10} + 27574537084016816868 p^{15} T^{11} + 205127274825238 p^{21} T^{12} + 568361280852 p^{25} T^{13} + 105010587 p^{30} T^{14} + 6072 p^{35} T^{15} + p^{40} T^{16} $
	31	$ 1 + 17278 T + 273140870 T^{2} + 2552159109028 T^{3} + 22012463290492184 T^{4} + $$13\!\cdots\!22$$ T^{5} + $$81\!\cdots\!88$$ T^{6} + $$38\!\cdots\!10$$ T^{7} + $$22\!\cdots\!65$$ T^{8} + $$38\!\cdots\!10$$ p^{5} T^{9} + $$81\!\cdots\!88$$ p^{10} T^{10} + $$13\!\cdots\!22$$ p^{15} T^{11} + 22012463290492184 p^{20} T^{12} + 2552159109028 p^{25} T^{13} + 273140870 p^{30} T^{14} + 17278 p^{35} T^{15} + p^{40} T^{16} $
	37	$ 1 - 4682 T + 249639899 T^{2} - 3009857493410 T^{3} + 33158969603034845 T^{4} - $$50\!\cdots\!64$$ T^{5} + $$45\!\cdots\!66$$ T^{6} - $$41\!\cdots\!80$$ T^{7} + $$42\!\cdots\!14$$ T^{8} - $$41\!\cdots\!80$$ p^{5} T^{9} + $$45\!\cdots\!66$$ p^{10} T^{10} - $$50\!\cdots\!64$$ p^{15} T^{11} + 33158969603034845 p^{20} T^{12} - 3009857493410 p^{25} T^{13} + 249639899 p^{30} T^{14} - 4682 p^{35} T^{15} + p^{40} T^{16} $
	41	$ 1 + 15204 T + 605422812 T^{2} + 9232164988140 T^{3} + 194393132330802116 T^{4} + $$25\!\cdots\!16$$ T^{5} + $$39\!\cdots\!92$$ T^{6} + $$44\!\cdots\!40$$ T^{7} + $$55\!\cdots\!58$$ T^{8} + $$44\!\cdots\!40$$ p^{5} T^{9} + $$39\!\cdots\!92$$ p^{10} T^{10} + $$25\!\cdots\!16$$ p^{15} T^{11} + 194393132330802116 p^{20} T^{12} + 9232164988140 p^{25} T^{13} + 605422812 p^{30} T^{14} + 15204 p^{35} T^{15} + p^{40} T^{16} $
	43	$ 1 - 7278 T + 524985567 T^{2} - 4294025018562 T^{3} + 130461383215254665 T^{4} - $$13\!\cdots\!60$$ T^{5} + $$20\!\cdots\!18$$ T^{6} - $$27\!\cdots\!96$$ T^{7} + $$29\!\cdots\!74$$ T^{8} - $$27\!\cdots\!96$$ p^{5} T^{9} + $$20\!\cdots\!18$$ p^{10} T^{10} - $$13\!\cdots\!60$$ p^{15} T^{11} + 130461383215254665 p^{20} T^{12} - 4294025018562 p^{25} T^{13} + 524985567 p^{30} T^{14} - 7278 p^{35} T^{15} + p^{40} T^{16} $
	47	$ 1 - 39768 T + 1192259296 T^{2} - 27713656510200 T^{3} + 640113323787000572 T^{4} - $$12\!\cdots\!72$$ T^{5} + $$23\!\cdots\!88$$ T^{6} - $$39\!\cdots\!28$$ T^{7} + $$62\!\cdots\!50$$ T^{8} - $$39\!\cdots\!28$$ p^{5} T^{9} + $$23\!\cdots\!88$$ p^{10} T^{10} - $$12\!\cdots\!72$$ p^{15} T^{11} + 640113323787000572 p^{20} T^{12} - 27713656510200 p^{25} T^{13} + 1192259296 p^{30} T^{14} - 39768 p^{35} T^{15} + p^{40} T^{16} $
	53	$ 1 - 18756 T + 2775286819 T^{2} - 45321256558620 T^{3} + 3550215782111636630 T^{4} - $$49\!\cdots\!40$$ T^{5} + $$27\!\cdots\!61$$ T^{6} - $$32\!\cdots\!88$$ T^{7} + $$13\!\cdots\!58$$ T^{8} - $$32\!\cdots\!88$$ p^{5} T^{9} + $$27\!\cdots\!61$$ p^{10} T^{10} - $$49\!\cdots\!40$$ p^{15} T^{11} + 3550215782111636630 p^{20} T^{12} - 45321256558620 p^{25} T^{13} + 2775286819 p^{30} T^{14} - 18756 p^{35} T^{15} + p^{40} T^{16} $
	59	$ 1 - 80826 T + 5856031197 T^{2} - 287600221285890 T^{3} + 13021290942574125818 T^{4} - $$48\!\cdots\!22$$ T^{5} + $$16\!\cdots\!71$$ T^{6} - $$50\!\cdots\!34$$ T^{7} + $$14\!\cdots\!86$$ T^{8} - $$50\!\cdots\!34$$ p^{5} T^{9} + $$16\!\cdots\!71$$ p^{10} T^{10} - $$48\!\cdots\!22$$ p^{15} T^{11} + 13021290942574125818 p^{20} T^{12} - 287600221285890 p^{25} T^{13} + 5856031197 p^{30} T^{14} - 80826 p^{35} T^{15} + p^{40} T^{16} $
	61	$ 1 + 9386 T + 2393407919 T^{2} + 793380425734 T^{3} + 3170856742889284401 T^{4} - $$44\!\cdots\!68$$ T^{5} + $$25\!\cdots\!38$$ T^{6} - $$82\!\cdots\!48$$ T^{7} + $$18\!\cdots\!10$$ T^{8} - $$82\!\cdots\!48$$ p^{5} T^{9} + $$25\!\cdots\!38$$ p^{10} T^{10} - $$44\!\cdots\!68$$ p^{15} T^{11} + 3170856742889284401 p^{20} T^{12} + 793380425734 p^{25} T^{13} + 2393407919 p^{30} T^{14} + 9386 p^{35} T^{15} + p^{40} T^{16} $
	67	$ 1 + 21254 T + 4390935339 T^{2} + 148558098567546 T^{3} + 10855147206248455269 T^{4} + $$39\!\cdots\!76$$ T^{5} + $$19\!\cdots\!02$$ T^{6} + $$72\!\cdots\!44$$ T^{7} + $$27\!\cdots\!70$$ T^{8} + $$72\!\cdots\!44$$ p^{5} T^{9} + $$19\!\cdots\!02$$ p^{10} T^{10} + $$39\!\cdots\!76$$ p^{15} T^{11} + 10855147206248455269 p^{20} T^{12} + 148558098567546 p^{25} T^{13} + 4390935339 p^{30} T^{14} + 21254 p^{35} T^{15} + p^{40} T^{16} $
	71	$ 1 - 75072 T + 8522724408 T^{2} - 407863613193408 T^{3} + 26991229921460674076 T^{4} - $$98\!\cdots\!96$$ T^{5} + $$53\!\cdots\!96$$ T^{6} - $$18\!\cdots\!92$$ T^{7} + $$96\!\cdots\!54$$ T^{8} - $$18\!\cdots\!92$$ p^{5} T^{9} + $$53\!\cdots\!96$$ p^{10} T^{10} - $$98\!\cdots\!96$$ p^{15} T^{11} + 26991229921460674076 p^{20} T^{12} - 407863613193408 p^{25} T^{13} + 8522724408 p^{30} T^{14} - 75072 p^{35} T^{15} + p^{40} T^{16} $
	73	$ 1 - 44910 T + 12130258237 T^{2} - 534610655597750 T^{3} + 72124073876772388982 T^{4} - $$28\!\cdots\!74$$ T^{5} + $$26\!\cdots\!83$$ T^{6} - $$92\!\cdots\!66$$ T^{7} + $$67\!\cdots\!10$$ T^{8} - $$92\!\cdots\!66$$ p^{5} T^{9} + $$26\!\cdots\!83$$ p^{10} T^{10} - $$28\!\cdots\!74$$ p^{15} T^{11} + 72124073876772388982 p^{20} T^{12} - 534610655597750 p^{25} T^{13} + 12130258237 p^{30} T^{14} - 44910 p^{35} T^{15} + p^{40} T^{16} $
	79	$ 1 - 13300 T + 13217381307 T^{2} - 3780201277420 p T^{3} + 82686829326629880010 T^{4} - $$24\!\cdots\!52$$ T^{5} + $$35\!\cdots\!81$$ T^{6} - $$11\!\cdots\!88$$ T^{7} + $$12\!\cdots\!98$$ T^{8} - $$11\!\cdots\!88$$ p^{5} T^{9} + $$35\!\cdots\!81$$ p^{10} T^{10} - $$24\!\cdots\!52$$ p^{15} T^{11} + 82686829326629880010 p^{20} T^{12} - 3780201277420 p^{26} T^{13} + 13217381307 p^{30} T^{14} - 13300 p^{35} T^{15} + p^{40} T^{16} $
	83	$ 1 - 254064 T + 46784741220 T^{2} - 6481010462524560 T^{3} + $$74\!\cdots\!60$$ T^{4} - $$73\!\cdots\!72$$ T^{5} + $$62\!\cdots\!80$$ T^{6} - $$46\!\cdots\!88$$ T^{7} + $$31\!\cdots\!78$$ T^{8} - $$46\!\cdots\!88$$ p^{5} T^{9} + $$62\!\cdots\!80$$ p^{10} T^{10} - $$73\!\cdots\!72$$ p^{15} T^{11} + $$74\!\cdots\!60$$ p^{20} T^{12} - 6481010462524560 p^{25} T^{13} + 46784741220 p^{30} T^{14} - 254064 p^{35} T^{15} + p^{40} T^{16} $
	89	$ 1 + 56796 T + 37794014672 T^{2} + 2144375300393124 T^{3} + $$64\!\cdots\!56$$ T^{4} + $$35\!\cdots\!16$$ T^{5} + $$66\!\cdots\!00$$ T^{6} + $$32\!\cdots\!88$$ T^{7} + $$45\!\cdots\!30$$ T^{8} + $$32\!\cdots\!88$$ p^{5} T^{9} + $$66\!\cdots\!00$$ p^{10} T^{10} + $$35\!\cdots\!16$$ p^{15} T^{11} + $$64\!\cdots\!56$$ p^{20} T^{12} + 2144375300393124 p^{25} T^{13} + 37794014672 p^{30} T^{14} + 56796 p^{35} T^{15} + p^{40} T^{16} $
	97	$ 1 - 25828 T + 30125578028 T^{2} - 648915969241208 T^{3} + $$56\!\cdots\!58$$ T^{4} - $$10\!\cdots\!40$$ T^{5} + $$71\!\cdots\!44$$ T^{6} - $$11\!\cdots\!76$$ T^{7} + $$70\!\cdots\!23$$ T^{8} - $$11\!\cdots\!76$$ p^{5} T^{9} + $$71\!\cdots\!44$$ p^{10} T^{10} - $$10\!\cdots\!40$$ p^{15} T^{11} + $$56\!\cdots\!58$$ p^{20} T^{12} - 648915969241208 p^{25} T^{13} + 30125578028 p^{30} T^{14} - 25828 p^{35} T^{15} + p^{40} T^{16} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−3.70756162459722433000244937981, −3.53322167043882369597214634154, −3.44373123264926568706274813803, −3.43396024268516529668290664368, −3.18670743551087393352070408949, −3.07003039582880869846736406700, −2.98293127614227296259704900797, −2.40683784646171944555465847662, −2.31934751523855652049411315203, −2.28231816610892914907211945309, −2.08033899490311456946893876252, −1.97002560911852633230375341313, −1.65626979152205287236611003116, −1.59159820517714327626258933769, −1.54959929412719671294608098718, −1.52661036695959576985799593094, −1.26761924292947614754079082426, −0.952164852767185724546183277195, −0.879359673009523743272521346994, −0.69099713743236837904679022020, −0.60539891540637294045149489917, −0.57726598567730174767629304702, −0.51788538309405254711909506119, −0.36238550361072424481972698717, −0.18404476427601701080797431299, 0.18404476427601701080797431299, 0.36238550361072424481972698717, 0.51788538309405254711909506119, 0.57726598567730174767629304702, 0.60539891540637294045149489917, 0.69099713743236837904679022020, 0.879359673009523743272521346994, 0.952164852767185724546183277195, 1.26761924292947614754079082426, 1.52661036695959576985799593094, 1.54959929412719671294608098718, 1.59159820517714327626258933769, 1.65626979152205287236611003116, 1.97002560911852633230375341313, 2.08033899490311456946893876252, 2.28231816610892914907211945309, 2.31934751523855652049411315203, 2.40683784646171944555465847662, 2.98293127614227296259704900797, 3.07003039582880869846736406700, 3.18670743551087393352070408949, 3.43396024268516529668290664368, 3.44373123264926568706274813803, 3.53322167043882369597214634154, 3.70756162459722433000244937981

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(6.399210097\)
\(L(\frac12)\)	\(\approx\)	\(6.399210097\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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