LMFDB - L-function 16-336e8-1.1-c6e8-0-8

Dirichlet series

L(s) = 1

+ 108·3^-s + 462·5^-s − 580·7^-s + 6.31e3·9^-s − 1.80e3·11^-s + 4.98e4·15^-s + 9.56e3·17^-s − 2.30e4·19^-s − 6.26e4·21^-s − 2.40e3·23^-s + 9.18e4·25^-s + 2.62e5·27^-s − 2.48e3·29^-s − 1.48e5·31^-s − 1.95e5·33^-s − 2.67e5·35^-s − 8.40e4·37^-s − 9.29e4·43^-s + 2.91e6·45^-s + 3.23e5·47^-s + 1.63e5·49^-s + 1.03e6·51^-s + 3.58e5·53^-s − 8.34e5·55^-s − 2.48e6·57^-s − 7.19e5·59^-s + 4.21e5·61^-s + ⋯

L(s) = 1

+ 4·3^-s + 3.69·5^-s − 1.69·7^-s + 26/3·9^-s − 1.35·11^-s + 14.7·15^-s + 1.94·17^-s − 3.35·19^-s − 6.76·21^-s − 0.197·23^-s + 5.87·25^-s + 40/3·27^-s − 0.101·29^-s − 4.98·31^-s − 5.42·33^-s − 6.24·35^-s − 1.65·37^-s − 1.16·43^-s + 32.0·45^-s + 3.11·47^-s + 1.39·49^-s + 7.78·51^-s + 2.40·53^-s − 5.01·55^-s − 13.4·57^-s − 3.50·59^-s + 1.85·61^-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

$L(\frac{7}{2})$	$\approx$	$337.7009735$
$L(\frac12)$	$\approx$	$337.7009735$
$L(4)$		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 + 580 T + 24646 p T^{2} + 127600 p^{3} T^{3} + 73027 p^{6} T^{4} + 127600 p^{9} T^{5} + 24646 p^{13} T^{6} + 580 p^{18} T^{7} + p^{24} T^{8} $
good	5	$ 1 - 462 T + 121591 T^{2} - 23304666 T^{3} + 3663451549 T^{4} - 17853486588 p^{2} T^{5} + 1697558359186 p^{2} T^{6} - 5617362787032 p^{4} T^{7} + 581774881299754 p^{4} T^{8} - 5617362787032 p^{10} T^{9} + 1697558359186 p^{14} T^{10} - 17853486588 p^{20} T^{11} + 3663451549 p^{24} T^{12} - 23304666 p^{30} T^{13} + 121591 p^{36} T^{14} - 462 p^{42} T^{15} + p^{48} T^{16} $
	11	$ 1 + 1806 T + 343139 T^{2} + 64134102 p T^{3} + 17940955739 p T^{4} - 3348273478860 p^{3} T^{5} + 3975088885585982894 T^{6} + $$12\!\cdots\!88$$ T^{7} + $$80\!\cdots\!02$$ T^{8} + $$12\!\cdots\!88$$ p^{6} T^{9} + 3975088885585982894 p^{12} T^{10} - 3348273478860 p^{21} T^{11} + 17940955739 p^{25} T^{12} + 64134102 p^{31} T^{13} + 343139 p^{36} T^{14} + 1806 p^{42} T^{15} + p^{48} T^{16} $
	13	$ 1 - 20741762 T^{2} + 250646566781089 T^{4} - $$19\!\cdots\!50$$ T^{6} + $$11\!\cdots\!92$$ T^{8} - $$19\!\cdots\!50$$ p^{12} T^{10} + 250646566781089 p^{24} T^{12} - 20741762 p^{36} T^{14} + p^{48} T^{16} $
	17	$ 1 - 9564 T + 89393680 T^{2} - 563354489472 T^{3} + 2845522364345362 T^{4} - 6080256459438545484 T^{5} - $$22\!\cdots\!76$$ T^{6} + $$25\!\cdots\!28$$ T^{7} - $$15\!\cdots\!73$$ T^{8} + $$25\!\cdots\!28$$ p^{6} T^{9} - $$22\!\cdots\!76$$ p^{12} T^{10} - 6080256459438545484 p^{18} T^{11} + 2845522364345362 p^{24} T^{12} - 563354489472 p^{30} T^{13} + 89393680 p^{36} T^{14} - 9564 p^{42} T^{15} + p^{48} T^{16} $
	19	$ 1 + 23022 T + 361976623 T^{2} + 4266110012490 T^{3} + 44167009488306445 T^{4} + $$41\!\cdots\!00$$ T^{5} + $$35\!\cdots\!10$$ T^{6} + $$27\!\cdots\!08$$ T^{7} + $$19\!\cdots\!58$$ T^{8} + $$27\!\cdots\!08$$ p^{6} T^{9} + $$35\!\cdots\!10$$ p^{12} T^{10} + $$41\!\cdots\!00$$ p^{18} T^{11} + 44167009488306445 p^{24} T^{12} + 4266110012490 p^{30} T^{13} + 361976623 p^{36} T^{14} + 23022 p^{42} T^{15} + p^{48} T^{16} $
	23	$ 1 + 2400 T - 340709188 T^{2} + 1754150275392 T^{3} + 61912758108945610 T^{4} - $$51\!\cdots\!92$$ T^{5} - $$49\!\cdots\!80$$ T^{6} + $$20\!\cdots\!60$$ p T^{7} + $$40\!\cdots\!67$$ T^{8} + $$20\!\cdots\!60$$ p^{7} T^{9} - $$49\!\cdots\!80$$ p^{12} T^{10} - $$51\!\cdots\!92$$ p^{18} T^{11} + 61912758108945610 p^{24} T^{12} + 1754150275392 p^{30} T^{13} - 340709188 p^{36} T^{14} + 2400 p^{42} T^{15} + p^{48} T^{16} $
	29	$ ( 1 + 1242 T + 1090691725 T^{2} + 1943067761982 T^{3} + 828942231558495912 T^{4} + 1943067761982 p^{6} T^{5} + 1090691725 p^{12} T^{6} + 1242 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	31	$ 1 + 148416 T + 13015135150 T^{2} + 841919264803968 T^{3} + 44373678311034269425 T^{4} + $$19\!\cdots\!52$$ T^{5} + $$78\!\cdots\!42$$ T^{6} + $$27\!\cdots\!12$$ T^{7} + $$87\!\cdots\!64$$ T^{8} + $$27\!\cdots\!12$$ p^{6} T^{9} + $$78\!\cdots\!42$$ p^{12} T^{10} + $$19\!\cdots\!52$$ p^{18} T^{11} + 44373678311034269425 p^{24} T^{12} + 841919264803968 p^{30} T^{13} + 13015135150 p^{36} T^{14} + 148416 p^{42} T^{15} + p^{48} T^{16} $
	37	$ 1 + 84046 T - 95144565 T^{2} - 203068161754894 T^{3} - 5662159274758793959 T^{4} + $$30\!\cdots\!80$$ T^{5} + $$40\!\cdots\!22$$ T^{6} - $$10\!\cdots\!40$$ T^{7} - $$24\!\cdots\!74$$ T^{8} - $$10\!\cdots\!40$$ p^{6} T^{9} + $$40\!\cdots\!22$$ p^{12} T^{10} + $$30\!\cdots\!80$$ p^{18} T^{11} - 5662159274758793959 p^{24} T^{12} - 203068161754894 p^{30} T^{13} - 95144565 p^{36} T^{14} + 84046 p^{42} T^{15} + p^{48} T^{16} $
	41	$ 1 - 22413338264 T^{2} + $$26\!\cdots\!52$$ T^{4} - $$21\!\cdots\!16$$ T^{6} + $$11\!\cdots\!18$$ T^{8} - $$21\!\cdots\!16$$ p^{12} T^{10} + $$26\!\cdots\!52$$ p^{24} T^{12} - 22413338264 p^{36} T^{14} + p^{48} T^{16} $
	43	$ ( 1 + 46486 T + 20861480377 T^{2} + 817839869460550 T^{3} + $$18\!\cdots\!56$$ T^{4} + 817839869460550 p^{6} T^{5} + 20861480377 p^{12} T^{6} + 46486 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	47	$ 1 - 323124 T + 74587753216 T^{2} - 12855395740416576 T^{3} + $$18\!\cdots\!54$$ T^{4} - $$24\!\cdots\!60$$ T^{5} + $$28\!\cdots\!28$$ T^{6} - $$31\!\cdots\!36$$ T^{7} + $$33\!\cdots\!75$$ T^{8} - $$31\!\cdots\!36$$ p^{6} T^{9} + $$28\!\cdots\!28$$ p^{12} T^{10} - $$24\!\cdots\!60$$ p^{18} T^{11} + $$18\!\cdots\!54$$ p^{24} T^{12} - 12855395740416576 p^{30} T^{13} + 74587753216 p^{36} T^{14} - 323124 p^{42} T^{15} + p^{48} T^{16} $
	53	$ 1 - 358086 T + 67654849271 T^{2} - 10791940432837962 T^{3} + $$12\!\cdots\!49$$ T^{4} - $$82\!\cdots\!52$$ T^{5} + $$27\!\cdots\!50$$ p T^{6} - $$27\!\cdots\!52$$ T^{7} + $$35\!\cdots\!02$$ T^{8} - $$27\!\cdots\!52$$ p^{6} T^{9} + $$27\!\cdots\!50$$ p^{13} T^{10} - $$82\!\cdots\!52$$ p^{18} T^{11} + $$12\!\cdots\!49$$ p^{24} T^{12} - 10791940432837962 p^{30} T^{13} + 67654849271 p^{36} T^{14} - 358086 p^{42} T^{15} + p^{48} T^{16} $
	59	$ 1 + 719382 T + 395234742835 T^{2} + 160228856063524314 T^{3} + $$56\!\cdots\!17$$ T^{4} + $$16\!\cdots\!92$$ T^{5} + $$45\!\cdots\!26$$ T^{6} + $$10\!\cdots\!04$$ T^{7} + $$23\!\cdots\!02$$ T^{8} + $$10\!\cdots\!04$$ p^{6} T^{9} + $$45\!\cdots\!26$$ p^{12} T^{10} + $$16\!\cdots\!92$$ p^{18} T^{11} + $$56\!\cdots\!17$$ p^{24} T^{12} + 160228856063524314 p^{30} T^{13} + 395234742835 p^{36} T^{14} + 719382 p^{42} T^{15} + p^{48} T^{16} $
	61	$ 1 - 421536 T + 103907366020 T^{2} - 18832752930327168 T^{3} + $$15\!\cdots\!58$$ T^{4} - $$45\!\cdots\!72$$ T^{5} - $$21\!\cdots\!32$$ T^{6} + $$10\!\cdots\!96$$ T^{7} - $$28\!\cdots\!73$$ T^{8} + $$10\!\cdots\!96$$ p^{6} T^{9} - $$21\!\cdots\!32$$ p^{12} T^{10} - $$45\!\cdots\!72$$ p^{18} T^{11} + $$15\!\cdots\!58$$ p^{24} T^{12} - 18832752930327168 p^{30} T^{13} + 103907366020 p^{36} T^{14} - 421536 p^{42} T^{15} + p^{48} T^{16} $
	67	$ 1 + 267010 T - 152513452041 T^{2} - 61154210757796450 T^{3} + $$10\!\cdots\!47$$ p T^{4} + $$47\!\cdots\!60$$ T^{5} - $$12\!\cdots\!10$$ T^{6} - $$14\!\cdots\!80$$ T^{7} + $$13\!\cdots\!62$$ T^{8} - $$14\!\cdots\!80$$ p^{6} T^{9} - $$12\!\cdots\!10$$ p^{12} T^{10} + $$47\!\cdots\!60$$ p^{18} T^{11} + $$10\!\cdots\!47$$ p^{25} T^{12} - 61154210757796450 p^{30} T^{13} - 152513452041 p^{36} T^{14} + 267010 p^{42} T^{15} + p^{48} T^{16} $
	71	$ ( 1 - 232332 T + 385365830296 T^{2} - 62567607151798020 T^{3} + $$67\!\cdots\!14$$ T^{4} - 62567607151798020 p^{6} T^{5} + 385365830296 p^{12} T^{6} - 232332 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	73	$ 1 - 1944486 T + 2298456740059 T^{2} - 2018599705351076922 T^{3} + $$14\!\cdots\!81$$ T^{4} - $$87\!\cdots\!48$$ T^{5} + $$46\!\cdots\!10$$ T^{6} - $$21\!\cdots\!04$$ T^{7} + $$88\!\cdots\!42$$ T^{8} - $$21\!\cdots\!04$$ p^{6} T^{9} + $$46\!\cdots\!10$$ p^{12} T^{10} - $$87\!\cdots\!48$$ p^{18} T^{11} + $$14\!\cdots\!81$$ p^{24} T^{12} - 2018599705351076922 p^{30} T^{13} + 2298456740059 p^{36} T^{14} - 1944486 p^{42} T^{15} + p^{48} T^{16} $
	79	$ 1 - 685904 T - 133173773202 T^{2} + 95757006040253216 T^{3} + $$16\!\cdots\!81$$ T^{4} + $$21\!\cdots\!20$$ T^{5} - $$10\!\cdots\!90$$ T^{6} - $$45\!\cdots\!20$$ p T^{7} + $$25\!\cdots\!20$$ T^{8} - $$45\!\cdots\!20$$ p^{7} T^{9} - $$10\!\cdots\!90$$ p^{12} T^{10} + $$21\!\cdots\!20$$ p^{18} T^{11} + $$16\!\cdots\!81$$ p^{24} T^{12} + 95757006040253216 p^{30} T^{13} - 133173773202 p^{36} T^{14} - 685904 p^{42} T^{15} + p^{48} T^{16} $
	83	$ 1 - 1431268238810 T^{2} + $$10\!\cdots\!57$$ T^{4} - $$56\!\cdots\!70$$ T^{6} + $$21\!\cdots\!48$$ T^{8} - $$56\!\cdots\!70$$ p^{12} T^{10} + $$10\!\cdots\!57$$ p^{24} T^{12} - 1431268238810 p^{36} T^{14} + p^{48} T^{16} $
	89	$ 1 - 4130604 T + 8816525059216 T^{2} - 12925604134667650176 T^{3} + $$14\!\cdots\!86$$ T^{4} - $$13\!\cdots\!20$$ T^{5} + $$10\!\cdots\!12$$ T^{6} - $$72\!\cdots\!80$$ T^{7} + $$50\!\cdots\!39$$ T^{8} - $$72\!\cdots\!80$$ p^{6} T^{9} + $$10\!\cdots\!12$$ p^{12} T^{10} - $$13\!\cdots\!20$$ p^{18} T^{11} + $$14\!\cdots\!86$$ p^{24} T^{12} - 12925604134667650176 p^{30} T^{13} + 8816525059216 p^{36} T^{14} - 4130604 p^{42} T^{15} + p^{48} T^{16} $
	97	$ 1 - 4008470100050 T^{2} + $$86\!\cdots\!01$$ T^{4} - $$11\!\cdots\!30$$ T^{6} + $$11\!\cdots\!20$$ T^{8} - $$11\!\cdots\!30$$ p^{12} T^{10} + $$86\!\cdots\!01$$ p^{24} T^{12} - 4008470100050 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

−3.94820906606913939489458530107, −3.74147930243281583037805518402, −3.56442926398324197005883601414, −3.54181757399513506008182340987, −3.32348026671491195383363120766, −3.30586560838272770917034463839, −3.29097426991182555766096482196, −2.82104423943087157617728077028, −2.74114687963180021090301434843, −2.44074054026770662567500067967, −2.37362046641755401747587378355, −2.37059064419993134514859202618, −2.29059869027890811723183142241, −2.24831614452905568056639052895, −1.79386368659961870409909513670, −1.76683653057053384164533717451, −1.75526113336205906860092649121, −1.69006846960349958253500636440, −1.53404525525762185280715726229, −1.27960425546007117639387032654, −0.69710703629165300941890278480, −0.58734024668201646613110959383, −0.53445566931600181750675819076, −0.45058623985018949228873426485, −0.35444726760883804682018252660, 0.35444726760883804682018252660, 0.45058623985018949228873426485, 0.53445566931600181750675819076, 0.58734024668201646613110959383, 0.69710703629165300941890278480, 1.27960425546007117639387032654, 1.53404525525762185280715726229, 1.69006846960349958253500636440, 1.75526113336205906860092649121, 1.76683653057053384164533717451, 1.79386368659961870409909513670, 2.24831614452905568056639052895, 2.29059869027890811723183142241, 2.37059064419993134514859202618, 2.37362046641755401747587378355, 2.44074054026770662567500067967, 2.74114687963180021090301434843, 2.82104423943087157617728077028, 3.29097426991182555766096482196, 3.30586560838272770917034463839, 3.32348026671491195383363120766, 3.54181757399513506008182340987, 3.56442926398324197005883601414, 3.74147930243281583037805518402, 3.94820906606913939489458530107

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 $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(337.7009735\)
\(L(\frac12)\)	\(\approx\)	\(337.7009735\)
\(L(4)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more