LMFDB - L-function 16-336e8-1.1-c6e8-0-1

Dirichlet series

L(s) = 1

− 108·3^-s + 210·5^-s + 608·7^-s + 6.31e3·9^-s − 2.05e3·11^-s − 2.26e4·15^-s − 1.12e4·17^-s − 2.18e4·19^-s − 6.56e4·21^-s − 1.55e4·23^-s − 1.24e4·25^-s − 2.62e5·27^-s + 3.53e4·29^-s + 5.10e4·31^-s + 2.22e5·33^-s + 1.27e5·35^-s + 2.02e4·37^-s − 3.87e5·43^-s + 1.32e6·45^-s + 5.52e4·47^-s + 4.59e4·49^-s + 1.21e6·51^-s − 3.36e5·53^-s − 4.32e5·55^-s + 2.35e6·57^-s + 5.60e5·59^-s + 8.50e5·61^-s + ⋯

L(s) = 1

− 4·3^-s + 1.67·5^-s + 1.77·7^-s + 26/3·9^-s − 1.54·11^-s − 6.71·15^-s − 2.28·17^-s − 3.18·19^-s − 7.09·21^-s − 1.27·23^-s − 0.798·25^-s − 13.3·27^-s + 1.44·29^-s + 1.71·31^-s + 6.18·33^-s + 2.97·35^-s + 0.400·37^-s − 4.87·43^-s + 14.5·45^-s + 0.531·47^-s + 0.390·49^-s + 9.15·51^-s − 2.25·53^-s − 2.59·55^-s + 12.7·57^-s + 2.72·59^-s + 3.74·61^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$2^{32} \cdot 3^{8} \cdot 7^{8}$
Sign:	$1$
Analytic conductor:	$1.27454\times 10^{15}$
Root analytic conductor:	$8.79193$
Motivic weight:	$6$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{32} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : [3]^{8} ),\ 1 )$

Particular Values

$L(\frac{7}{2})$	$\approx$	$0.05470201176$
$L(\frac12)$	$\approx$	$0.05470201176$
$L(4)$		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 + p^{3} T + p^{5} T^{2} )^{4} $
	7	$ 1 - 608 T + 46246 p T^{2} - 407216 p^{3} T^{3} + 391627 p^{6} T^{4} - 407216 p^{9} T^{5} + 46246 p^{13} T^{6} - 608 p^{18} T^{7} + p^{24} T^{8} $
good	5	$ 1 - 42 p T + 2263 p^{2} T^{2} - 2814 p^{5} T^{3} + 2487157 p^{4} T^{4} - 3203172 p^{7} T^{5} + 2284784602 p^{6} T^{6} - 67984944072 p^{7} T^{7} + 1597655137114 p^{8} T^{8} - 67984944072 p^{13} T^{9} + 2284784602 p^{18} T^{10} - 3203172 p^{25} T^{11} + 2487157 p^{28} T^{12} - 2814 p^{35} T^{13} + 2263 p^{38} T^{14} - 42 p^{43} T^{15} + p^{48} T^{16} $
	11	$ 1 + 2058 T - 1227829 T^{2} - 7406515002 T^{3} - 3386967710903 T^{4} + 9662922109061604 T^{5} + 10529065850650150862 T^{6} - $$28\!\cdots\!72$$ T^{7} - $$13\!\cdots\!22$$ T^{8} - $$28\!\cdots\!72$$ p^{6} T^{9} + 10529065850650150862 p^{12} T^{10} + 9662922109061604 p^{18} T^{11} - 3386967710903 p^{24} T^{12} - 7406515002 p^{30} T^{13} - 1227829 p^{36} T^{14} + 2058 p^{42} T^{15} + p^{48} T^{16} $
	13	$ 1 - 19302338 T^{2} + 180296894192257 T^{4} - $$11\!\cdots\!26$$ T^{6} + $$63\!\cdots\!40$$ T^{8} - $$11\!\cdots\!26$$ p^{12} T^{10} + 180296894192257 p^{24} T^{12} - 19302338 p^{36} T^{14} + p^{48} T^{16} $
	17	$ 1 + 11244 T + 80398792 T^{2} + 430153612320 T^{3} + 2164758128646850 T^{4} + 4541903683526051964 T^{5} - $$19\!\cdots\!96$$ T^{6} - $$24\!\cdots\!68$$ T^{7} - $$12\!\cdots\!85$$ T^{8} - $$24\!\cdots\!68$$ p^{6} T^{9} - $$19\!\cdots\!96$$ p^{12} T^{10} + 4541903683526051964 p^{18} T^{11} + 2164758128646850 p^{24} T^{12} + 430153612320 p^{30} T^{13} + 80398792 p^{36} T^{14} + 11244 p^{42} T^{15} + p^{48} T^{16} $
	19	$ 1 + 21834 T + 334034863 T^{2} + 3823723158174 T^{3} + 36160700332505293 T^{4} + $$31\!\cdots\!60$$ T^{5} + $$26\!\cdots\!74$$ T^{6} + $$20\!\cdots\!60$$ T^{7} + $$15\!\cdots\!38$$ T^{8} + $$20\!\cdots\!60$$ p^{6} T^{9} + $$26\!\cdots\!74$$ p^{12} T^{10} + $$31\!\cdots\!60$$ p^{18} T^{11} + 36160700332505293 p^{24} T^{12} + 3823723158174 p^{30} T^{13} + 334034863 p^{36} T^{14} + 21834 p^{42} T^{15} + p^{48} T^{16} $
	23	$ 1 + 15504 T - 59161180 T^{2} - 2401961274912 T^{3} - 24147460491672806 T^{4} - $$26\!\cdots\!60$$ T^{5} - $$29\!\cdots\!32$$ T^{6} + $$21\!\cdots\!72$$ p T^{7} + $$78\!\cdots\!75$$ T^{8} + $$21\!\cdots\!72$$ p^{7} T^{9} - $$29\!\cdots\!32$$ p^{12} T^{10} - $$26\!\cdots\!60$$ p^{18} T^{11} - 24147460491672806 p^{24} T^{12} - 2401961274912 p^{30} T^{13} - 59161180 p^{36} T^{14} + 15504 p^{42} T^{15} + p^{48} T^{16} $
	29	$ ( 1 - 17658 T + 1822197325 T^{2} - 25912698371262 T^{3} + 1521927334843462824 T^{4} - 25912698371262 p^{6} T^{5} + 1822197325 p^{12} T^{6} - 17658 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	31	$ 1 - 51060 T + 2333720014 T^{2} - 74786500242840 T^{3} + 1960005684273765769 T^{4} - $$19\!\cdots\!28$$ T^{5} - $$12\!\cdots\!18$$ T^{6} + $$67\!\cdots\!48$$ T^{7} - $$25\!\cdots\!56$$ T^{8} + $$67\!\cdots\!48$$ p^{6} T^{9} - $$12\!\cdots\!18$$ p^{12} T^{10} - $$19\!\cdots\!28$$ p^{18} T^{11} + 1960005684273765769 p^{24} T^{12} - 74786500242840 p^{30} T^{13} + 2333720014 p^{36} T^{14} - 51060 p^{42} T^{15} + p^{48} T^{16} $
	37	$ 1 - 20282 T + 677653755 T^{2} - 1668594165814 T^{3} - 4995910852509553975 T^{4} + $$21\!\cdots\!32$$ T^{5} - $$19\!\cdots\!82$$ T^{6} - $$28\!\cdots\!56$$ T^{7} - $$35\!\cdots\!42$$ T^{8} - $$28\!\cdots\!56$$ p^{6} T^{9} - $$19\!\cdots\!82$$ p^{12} T^{10} + $$21\!\cdots\!32$$ p^{18} T^{11} - 4995910852509553975 p^{24} T^{12} - 1668594165814 p^{30} T^{13} + 677653755 p^{36} T^{14} - 20282 p^{42} T^{15} + p^{48} T^{16} $
	41	$ 1 - 776685208 p T^{2} + $$46\!\cdots\!40$$ T^{4} - $$40\!\cdots\!92$$ T^{6} + $$23\!\cdots\!74$$ T^{8} - $$40\!\cdots\!92$$ p^{12} T^{10} + $$46\!\cdots\!40$$ p^{24} T^{12} - 776685208 p^{37} T^{14} + p^{48} T^{16} $
	43	$ ( 1 + 193906 T + 33812347033 T^{2} + 3788015706390418 T^{3} + $$34\!\cdots\!60$$ T^{4} + 3788015706390418 p^{6} T^{5} + 33812347033 p^{12} T^{6} + 193906 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	47	$ 1 - 55212 T + 39123235288 T^{2} - 2103969958291680 T^{3} + $$89\!\cdots\!62$$ T^{4} - $$43\!\cdots\!76$$ T^{5} + $$14\!\cdots\!96$$ T^{6} - $$62\!\cdots\!84$$ T^{7} + $$17\!\cdots\!87$$ T^{8} - $$62\!\cdots\!84$$ p^{6} T^{9} + $$14\!\cdots\!96$$ p^{12} T^{10} - $$43\!\cdots\!76$$ p^{18} T^{11} + $$89\!\cdots\!62$$ p^{24} T^{12} - 2103969958291680 p^{30} T^{13} + 39123235288 p^{36} T^{14} - 55212 p^{42} T^{15} + p^{48} T^{16} $
	53	$ 1 + 336174 T - 12007524289 T^{2} - 5190536822683134 T^{3} + $$31\!\cdots\!65$$ T^{4} + $$41\!\cdots\!32$$ T^{5} - $$65\!\cdots\!78$$ T^{6} + $$10\!\cdots\!60$$ T^{7} + $$29\!\cdots\!10$$ T^{8} + $$10\!\cdots\!60$$ p^{6} T^{9} - $$65\!\cdots\!78$$ p^{12} T^{10} + $$41\!\cdots\!32$$ p^{18} T^{11} + $$31\!\cdots\!65$$ p^{24} T^{12} - 5190536822683134 p^{30} T^{13} - 12007524289 p^{36} T^{14} + 336174 p^{42} T^{15} + p^{48} T^{16} $
	59	$ 1 - 560454 T + 235422959347 T^{2} - 73262582735044650 T^{3} + $$18\!\cdots\!57$$ T^{4} - $$33\!\cdots\!52$$ T^{5} + $$49\!\cdots\!14$$ T^{6} - $$54\!\cdots\!32$$ T^{7} + $$70\!\cdots\!22$$ T^{8} - $$54\!\cdots\!32$$ p^{6} T^{9} + $$49\!\cdots\!14$$ p^{12} T^{10} - $$33\!\cdots\!52$$ p^{18} T^{11} + $$18\!\cdots\!57$$ p^{24} T^{12} - 73262582735044650 p^{30} T^{13} + 235422959347 p^{36} T^{14} - 560454 p^{42} T^{15} + p^{48} T^{16} $
	61	$ 1 - 850728 T + 507030047380 T^{2} - 226109894199149856 T^{3} + $$86\!\cdots\!42$$ T^{4} - $$28\!\cdots\!64$$ T^{5} + $$84\!\cdots\!76$$ T^{6} - $$22\!\cdots\!20$$ T^{7} + $$52\!\cdots\!87$$ T^{8} - $$22\!\cdots\!20$$ p^{6} T^{9} + $$84\!\cdots\!76$$ p^{12} T^{10} - $$28\!\cdots\!64$$ p^{18} T^{11} + $$86\!\cdots\!42$$ p^{24} T^{12} - 226109894199149856 p^{30} T^{13} + 507030047380 p^{36} T^{14} - 850728 p^{42} T^{15} + p^{48} T^{16} $
	67	$ 1 - 947882 T + 318595843623 T^{2} - 72319663255205590 T^{3} + $$28\!\cdots\!57$$ T^{4} - $$62\!\cdots\!12$$ T^{5} - $$10\!\cdots\!02$$ T^{6} + $$49\!\cdots\!96$$ T^{7} - $$68\!\cdots\!70$$ T^{8} + $$49\!\cdots\!96$$ p^{6} T^{9} - $$10\!\cdots\!02$$ p^{12} T^{10} - $$62\!\cdots\!12$$ p^{18} T^{11} + $$28\!\cdots\!57$$ p^{24} T^{12} - 72319663255205590 p^{30} T^{13} + 318595843623 p^{36} T^{14} - 947882 p^{42} T^{15} + p^{48} T^{16} $
	71	$ ( 1 + 73596 T + 148848570688 T^{2} - 28477618711517772 T^{3} + $$10\!\cdots\!26$$ T^{4} - 28477618711517772 p^{6} T^{5} + 148848570688 p^{12} T^{6} + 73596 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	73	$ 1 + 533034 T + 690700987771 T^{2} + 317684305006699446 T^{3} + $$26\!\cdots\!37$$ T^{4} + $$10\!\cdots\!00$$ T^{5} + $$65\!\cdots\!58$$ T^{6} + $$22\!\cdots\!48$$ T^{7} + $$11\!\cdots\!38$$ T^{8} + $$22\!\cdots\!48$$ p^{6} T^{9} + $$65\!\cdots\!58$$ p^{12} T^{10} + $$10\!\cdots\!00$$ p^{18} T^{11} + $$26\!\cdots\!37$$ p^{24} T^{12} + 317684305006699446 p^{30} T^{13} + 690700987771 p^{36} T^{14} + 533034 p^{42} T^{15} + p^{48} T^{16} $
	79	$ 1 - 6260 T - 784672265154 T^{2} + 41346838804133528 T^{3} + $$35\!\cdots\!21$$ T^{4} - $$20\!\cdots\!00$$ T^{5} - $$11\!\cdots\!26$$ T^{6} + $$23\!\cdots\!92$$ T^{7} + $$30\!\cdots\!12$$ T^{8} + $$23\!\cdots\!92$$ p^{6} T^{9} - $$11\!\cdots\!26$$ p^{12} T^{10} - $$20\!\cdots\!00$$ p^{18} T^{11} + $$35\!\cdots\!21$$ p^{24} T^{12} + 41346838804133528 p^{30} T^{13} - 784672265154 p^{36} T^{14} - 6260 p^{42} T^{15} + p^{48} T^{16} $
	83	$ 1 - 597905785082 T^{2} + $$27\!\cdots\!37$$ T^{4} - $$10\!\cdots\!06$$ T^{6} + $$44\!\cdots\!24$$ T^{8} - $$10\!\cdots\!06$$ p^{12} T^{10} + $$27\!\cdots\!37$$ p^{24} T^{12} - 597905785082 p^{36} T^{14} + p^{48} T^{16} $
	89	$ 1 - 413460 T + 9936525424 T^{2} + 19451859028104960 T^{3} - $$28\!\cdots\!82$$ T^{4} + $$13\!\cdots\!60$$ T^{5} + $$10\!\cdots\!84$$ T^{6} - $$31\!\cdots\!40$$ T^{7} + $$37\!\cdots\!27$$ T^{8} - $$31\!\cdots\!40$$ p^{6} T^{9} + $$10\!\cdots\!84$$ p^{12} T^{10} + $$13\!\cdots\!60$$ p^{18} T^{11} - $$28\!\cdots\!82$$ p^{24} T^{12} + 19451859028104960 p^{30} T^{13} + 9936525424 p^{36} T^{14} - 413460 p^{42} T^{15} + p^{48} T^{16} $
	97	$ 1 + 825365486446 T^{2} + $$28\!\cdots\!33$$ T^{4} + $$16\!\cdots\!74$$ T^{6} + $$30\!\cdots\!88$$ T^{8} + $$16\!\cdots\!74$$ p^{12} T^{10} + $$28\!\cdots\!33$$ p^{24} T^{12} + 825365486446 p^{36} T^{14} + p^{48} 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

−4.43769550986084484423201756962, −4.04600775760174805029596375937, −3.73473392974416623218931195310, −3.67813880682703442486696273127, −3.64649101105955930103242814574, −3.43230258321002983905118604106, −3.15307979587169245163951080025, −2.73788992772056106413714003463, −2.73729738310347396424375747746, −2.57366831971785191194307074325, −2.32410193358477134266941220407, −1.97000724635430372484084828460, −1.96977008273411030581430529070, −1.89495188298140719323778105020, −1.86181309665869193692649625954, −1.82967986548234150672358251090, −1.79348038192828929090694387275, −1.23332124115431659996854890873, −1.02655098520065039107888128647, −0.72134907398649632509769308732, −0.69940839274835253977069717131, −0.69364944797848598183906588812, −0.45346445064374010231728096423, −0.17139298942582227151914372636, −0.04204054354827005875215397158, 0.04204054354827005875215397158, 0.17139298942582227151914372636, 0.45346445064374010231728096423, 0.69364944797848598183906588812, 0.69940839274835253977069717131, 0.72134907398649632509769308732, 1.02655098520065039107888128647, 1.23332124115431659996854890873, 1.79348038192828929090694387275, 1.82967986548234150672358251090, 1.86181309665869193692649625954, 1.89495188298140719323778105020, 1.96977008273411030581430529070, 1.97000724635430372484084828460, 2.32410193358477134266941220407, 2.57366831971785191194307074325, 2.73729738310347396424375747746, 2.73788992772056106413714003463, 3.15307979587169245163951080025, 3.43230258321002983905118604106, 3.64649101105955930103242814574, 3.67813880682703442486696273127, 3.73473392974416623218931195310, 4.04600775760174805029596375937, 4.43769550986084484423201756962

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 $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(0.05470201176\)
\(L(\frac12)\)	\(\approx\)	\(0.05470201176\)
\(L(4)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more