LMFDB - L-function 16-867e8-1.1-c5e8-0-2

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

$L(3)$	$\approx$	$625.4337028$
$L(\frac12)$	$\approx$	$625.4337028$
$L(\frac{7}{2})$		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} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−3.70322010870508820379532278109, −3.47849807320139870303519003184, −3.10729616376263076215693568074, −3.10007480358780561297809364680, −3.07425830917949673137075264927, −2.76784372441144515943165514574, −2.75239308926235315793770322679, −2.69224730878026099380221439188, −2.59329204604615187999045381558, −2.55291817437851964505730560908, −2.37646243561960782041274631108, −2.33022591560534758915821225211, −2.10439617187150381633543153813, −1.70313815253110710973434874103, −1.68899317365590378976413024591, −1.67234923579805363146143888229, −1.40215101638394000644307615372, −1.21619791561736697417993904042, −1.17738688689892914536326145898, −0.954479566430089819928034961532, −0.64214240358445404603680613801, −0.63419915514984880005546454608, −0.58527654978199931940324288382, −0.38547639613593142888854490578, −0.32753932977356345322269702782, 0.32753932977356345322269702782, 0.38547639613593142888854490578, 0.58527654978199931940324288382, 0.63419915514984880005546454608, 0.64214240358445404603680613801, 0.954479566430089819928034961532, 1.17738688689892914536326145898, 1.21619791561736697417993904042, 1.40215101638394000644307615372, 1.67234923579805363146143888229, 1.68899317365590378976413024591, 1.70313815253110710973434874103, 2.10439617187150381633543153813, 2.33022591560534758915821225211, 2.37646243561960782041274631108, 2.55291817437851964505730560908, 2.59329204604615187999045381558, 2.69224730878026099380221439188, 2.75239308926235315793770322679, 2.76784372441144515943165514574, 3.07425830917949673137075264927, 3.10007480358780561297809364680, 3.10729616376263076215693568074, 3.47849807320139870303519003184, 3.70322010870508820379532278109

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(625.4337028\)
\(L(\frac12)\)	\(\approx\)	\(625.4337028\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more