LMFDB - L-function 16-26e8-1.1-c7e8-0-0

Dirichlet series

L(s) = 1

+ 32·2^-s + 384·4^-s + 556·5^-s − 548·7^-s + 1.76e3·9^-s + 1.77e4·10^-s − 7.39e3·11^-s − 2.58e4·13^-s − 1.75e4·14^-s − 6.14e4·16^-s + 2.83e4·17^-s + 5.65e4·18^-s − 9.98e4·19^-s + 2.13e5·20^-s − 2.36e5·22^-s − 3.33e4·23^-s − 7.10e4·25^-s − 8.26e5·26^-s + 7.08e4·27^-s − 2.10e5·28^-s + 9.31e4·29^-s + 6.22e5·31^-s − 7.86e5·32^-s + 9.06e5·34^-s − 3.04e5·35^-s + 6.78e5·36^-s − 9.63e3·37^-s + ⋯

L(s) = 1

+ 2.82·2^-s + 3·4^-s + 1.98·5^-s − 0.603·7^-s + 0.807·9^-s + 5.62·10^-s − 1.67·11^-s − 3.25·13^-s − 1.70·14^-s − 3.75·16^-s + 1.39·17^-s + 2.28·18^-s − 3.34·19^-s + 5.96·20^-s − 4.73·22^-s − 0.572·23^-s − 0.909·25^-s − 9.21·26^-s + 0.692·27^-s − 1.81·28^-s + 0.709·29^-s + 3.75·31^-s − 4.24·32^-s + 3.95·34^-s − 1.20·35^-s + 2.42·36^-s − 0.0312·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$2^{8} \cdot 13^{8}$
Sign:	$1$
Analytic conductor:	$1.89368\times 10^{7}$
Root analytic conductor:	$2.84991$
Motivic weight:	$7$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{8} \cdot 13^{8} ,\ ( \ : [7/2]^{8} ),\ 1 )$

Particular Values

$L(4)$	$\approx$	$12.74656108$
$L(\frac12)$	$\approx$	$12.74656108$
$L(\frac{9}{2})$		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 + p^{6} T^{2} )^{4} $
	13	$ 1 + 1986 p T + 1099853 p^{2} T^{2} - 476713734 p^{3} T^{3} - 4584001548 p^{6} T^{4} - 476713734 p^{10} T^{5} + 1099853 p^{16} T^{6} + 1986 p^{22} T^{7} + p^{28} T^{8} $
good	3	$ 1 - 589 p T^{2} - 2624 p^{3} T^{3} - 247679 p^{2} T^{4} + 1729216 p^{4} T^{5} + 107431462 p^{4} T^{6} - 115526848 p^{6} T^{7} - 16347737066 p^{6} T^{8} - 115526848 p^{13} T^{9} + 107431462 p^{18} T^{10} + 1729216 p^{25} T^{11} - 247679 p^{30} T^{12} - 2624 p^{38} T^{13} - 589 p^{43} T^{14} + p^{56} T^{16} $
	5	$ ( 1 - 278 T + 151453 T^{2} - 10092158 p T^{3} + 507231524 p^{2} T^{4} - 10092158 p^{8} T^{5} + 151453 p^{14} T^{6} - 278 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	7	$ 1 + 548 T - 1138035 T^{2} - 1222097964 T^{3} + 107471171841 T^{4} + 727605589451640 T^{5} + 478925092223085470 T^{6} - $$12\!\cdots\!20$$ T^{7} - $$37\!\cdots\!50$$ T^{8} - $$12\!\cdots\!20$$ p^{7} T^{9} + 478925092223085470 p^{14} T^{10} + 727605589451640 p^{21} T^{11} + 107471171841 p^{28} T^{12} - 1222097964 p^{35} T^{13} - 1138035 p^{42} T^{14} + 548 p^{49} T^{15} + p^{56} T^{16} $
	11	$ 1 + 672 p T - 15015695 T^{2} - 209572703216 T^{3} + 202098762294505 T^{4} + 3132174508300450192 T^{5} - $$54\!\cdots\!50$$ T^{6} - $$18\!\cdots\!12$$ T^{7} + $$15\!\cdots\!46$$ T^{8} - $$18\!\cdots\!12$$ p^{7} T^{9} - $$54\!\cdots\!50$$ p^{14} T^{10} + 3132174508300450192 p^{21} T^{11} + 202098762294505 p^{28} T^{12} - 209572703216 p^{35} T^{13} - 15015695 p^{42} T^{14} + 672 p^{50} T^{15} + p^{56} T^{16} $
	17	$ 1 - 28316 T - 563111602 T^{2} + 7342663983176 T^{3} + 488082653800903993 T^{4} + $$16\!\cdots\!36$$ T^{5} - $$26\!\cdots\!30$$ T^{6} + $$65\!\cdots\!24$$ T^{7} + $$75\!\cdots\!36$$ T^{8} + $$65\!\cdots\!24$$ p^{7} T^{9} - $$26\!\cdots\!30$$ p^{14} T^{10} + $$16\!\cdots\!36$$ p^{21} T^{11} + 488082653800903993 p^{28} T^{12} + 7342663983176 p^{35} T^{13} - 563111602 p^{42} T^{14} - 28316 p^{49} T^{15} + p^{56} T^{16} $
	19	$ 1 + 99888 T + 3675226065 T^{2} + 82754874802800 T^{3} + 3147787540762584233 T^{4} + $$61\!\cdots\!92$$ p T^{5} + $$20\!\cdots\!82$$ T^{6} + $$26\!\cdots\!72$$ T^{7} + $$88\!\cdots\!06$$ T^{8} + $$26\!\cdots\!72$$ p^{7} T^{9} + $$20\!\cdots\!82$$ p^{14} T^{10} + $$61\!\cdots\!92$$ p^{22} T^{11} + 3147787540762584233 p^{28} T^{12} + 82754874802800 p^{35} T^{13} + 3675226065 p^{42} T^{14} + 99888 p^{49} T^{15} + p^{56} T^{16} $
	23	$ 1 + 33388 T - 3382348979 T^{2} - 387071200641348 T^{3} - 4201788242576114527 T^{4} + $$10\!\cdots\!32$$ T^{5} + $$76\!\cdots\!90$$ T^{6} - $$44\!\cdots\!88$$ p T^{7} - $$28\!\cdots\!86$$ T^{8} - $$44\!\cdots\!88$$ p^{8} T^{9} + $$76\!\cdots\!90$$ p^{14} T^{10} + $$10\!\cdots\!32$$ p^{21} T^{11} - 4201788242576114527 p^{28} T^{12} - 387071200641348 p^{35} T^{13} - 3382348979 p^{42} T^{14} + 33388 p^{49} T^{15} + p^{56} T^{16} $
	29	$ 1 - 93140 T - 883636642 p T^{2} + 2243727152264568 T^{3} + 80055595218391895489 T^{4} + $$16\!\cdots\!12$$ T^{5} - $$54\!\cdots\!38$$ T^{6} - $$50\!\cdots\!24$$ T^{7} + $$23\!\cdots\!96$$ T^{8} - $$50\!\cdots\!24$$ p^{7} T^{9} - $$54\!\cdots\!38$$ p^{14} T^{10} + $$16\!\cdots\!12$$ p^{21} T^{11} + 80055595218391895489 p^{28} T^{12} + 2243727152264568 p^{35} T^{13} - 883636642 p^{43} T^{14} - 93140 p^{49} T^{15} + p^{56} T^{16} $
	31	$ ( 1 - 311160 T + 140644653260 T^{2} - 26779795197492120 T^{3} + $$62\!\cdots\!06$$ T^{4} - 26779795197492120 p^{7} T^{5} + 140644653260 p^{14} T^{6} - 311160 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	37	$ 1 + 9636 T - 360499434930 T^{2} - 281458294795032 T^{3} + $$79\!\cdots\!93$$ T^{4} - $$11\!\cdots\!92$$ T^{5} - $$11\!\cdots\!42$$ T^{6} + $$57\!\cdots\!48$$ T^{7} + $$12\!\cdots\!52$$ T^{8} + $$57\!\cdots\!48$$ p^{7} T^{9} - $$11\!\cdots\!42$$ p^{14} T^{10} - $$11\!\cdots\!92$$ p^{21} T^{11} + $$79\!\cdots\!93$$ p^{28} T^{12} - 281458294795032 p^{35} T^{13} - 360499434930 p^{42} T^{14} + 9636 p^{49} T^{15} + p^{56} T^{16} $
	41	$ 1 - 82892 T - 546055663858 T^{2} + 146083811772340136 T^{3} + $$16\!\cdots\!97$$ T^{4} - $$49\!\cdots\!72$$ T^{5} - $$26\!\cdots\!70$$ T^{6} + $$51\!\cdots\!24$$ T^{7} + $$39\!\cdots\!32$$ T^{8} + $$51\!\cdots\!24$$ p^{7} T^{9} - $$26\!\cdots\!70$$ p^{14} T^{10} - $$49\!\cdots\!72$$ p^{21} T^{11} + $$16\!\cdots\!97$$ p^{28} T^{12} + 146083811772340136 p^{35} T^{13} - 546055663858 p^{42} T^{14} - 82892 p^{49} T^{15} + p^{56} T^{16} $
	43	$ 1 + 569264 T - 351575602143 T^{2} - 362392294979896560 T^{3} + $$28\!\cdots\!45$$ T^{4} + $$10\!\cdots\!36$$ T^{5} + $$36\!\cdots\!86$$ T^{6} - $$17\!\cdots\!80$$ T^{7} - $$17\!\cdots\!06$$ T^{8} - $$17\!\cdots\!80$$ p^{7} T^{9} + $$36\!\cdots\!86$$ p^{14} T^{10} + $$10\!\cdots\!36$$ p^{21} T^{11} + $$28\!\cdots\!45$$ p^{28} T^{12} - 362392294979896560 p^{35} T^{13} - 351575602143 p^{42} T^{14} + 569264 p^{49} T^{15} + p^{56} T^{16} $
	47	$ ( 1 + 574200 T + 1181775847100 T^{2} + 840326241178871192 T^{3} + $$77\!\cdots\!90$$ T^{4} + 840326241178871192 p^{7} T^{5} + 1181775847100 p^{14} T^{6} + 574200 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	53	$ ( 1 - 1235350 T + 2684236572461 T^{2} - 3308983772395621830 T^{3} + $$45\!\cdots\!08$$ T^{4} - 3308983772395621830 p^{7} T^{5} + 2684236572461 p^{14} T^{6} - 1235350 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	59	$ 1 - 231504 T - 2356814530631 T^{2} + 792722790467378240 T^{3} - $$33\!\cdots\!19$$ T^{4} + $$84\!\cdots\!52$$ T^{5} + $$82\!\cdots\!82$$ T^{6} - $$49\!\cdots\!08$$ T^{7} + $$67\!\cdots\!78$$ T^{8} - $$49\!\cdots\!08$$ p^{7} T^{9} + $$82\!\cdots\!82$$ p^{14} T^{10} + $$84\!\cdots\!52$$ p^{21} T^{11} - $$33\!\cdots\!19$$ p^{28} T^{12} + 792722790467378240 p^{35} T^{13} - 2356814530631 p^{42} T^{14} - 231504 p^{49} T^{15} + p^{56} T^{16} $
	61	$ 1 - 685684 T + 3433715225862 T^{2} + 3646641375018060600 T^{3} - $$13\!\cdots\!07$$ T^{4} + $$29\!\cdots\!28$$ T^{5} + $$50\!\cdots\!06$$ T^{6} - $$33\!\cdots\!84$$ T^{7} + $$31\!\cdots\!12$$ T^{8} - $$33\!\cdots\!84$$ p^{7} T^{9} + $$50\!\cdots\!06$$ p^{14} T^{10} + $$29\!\cdots\!28$$ p^{21} T^{11} - $$13\!\cdots\!07$$ p^{28} T^{12} + 3646641375018060600 p^{35} T^{13} + 3433715225862 p^{42} T^{14} - 685684 p^{49} T^{15} + p^{56} T^{16} $
	67	$ 1 - 3271056 T - 14689835850767 T^{2} + 35254481255509351008 T^{3} + $$21\!\cdots\!65$$ T^{4} - $$32\!\cdots\!96$$ T^{5} - $$17\!\cdots\!74$$ T^{6} + $$57\!\cdots\!12$$ T^{7} + $$13\!\cdots\!66$$ T^{8} + $$57\!\cdots\!12$$ p^{7} T^{9} - $$17\!\cdots\!74$$ p^{14} T^{10} - $$32\!\cdots\!96$$ p^{21} T^{11} + $$21\!\cdots\!65$$ p^{28} T^{12} + 35254481255509351008 p^{35} T^{13} - 14689835850767 p^{42} T^{14} - 3271056 p^{49} T^{15} + p^{56} T^{16} $
	71	$ 1 + 175012 T - 22406789187435 T^{2} - 11058578082007940412 T^{3} + $$23\!\cdots\!77$$ T^{4} + $$14\!\cdots\!04$$ T^{5} - $$22\!\cdots\!78$$ T^{6} - $$60\!\cdots\!80$$ T^{7} + $$22\!\cdots\!82$$ T^{8} - $$60\!\cdots\!80$$ p^{7} T^{9} - $$22\!\cdots\!78$$ p^{14} T^{10} + $$14\!\cdots\!04$$ p^{21} T^{11} + $$23\!\cdots\!77$$ p^{28} T^{12} - 11058578082007940412 p^{35} T^{13} - 22406789187435 p^{42} T^{14} + 175012 p^{49} T^{15} + p^{56} T^{16} $
	73	$ ( 1 - 7137890 T + 48989573892505 T^{2} - $$21\!\cdots\!78$$ T^{3} + $$81\!\cdots\!12$$ T^{4} - $$21\!\cdots\!78$$ p^{7} T^{5} + 48989573892505 p^{14} T^{6} - 7137890 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	79	$ ( 1 + 7053952 T + 797675870260 p T^{2} + $$36\!\cdots\!40$$ T^{3} + $$17\!\cdots\!66$$ T^{4} + $$36\!\cdots\!40$$ p^{7} T^{5} + 797675870260 p^{15} T^{6} + 7053952 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	83	$ ( 1 - 657288 T + 707565190612 p T^{2} + 83704316584548117240 T^{3} + $$16\!\cdots\!26$$ T^{4} + 83704316584548117240 p^{7} T^{5} + 707565190612 p^{15} T^{6} - 657288 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	89	$ 1 + 11452234 T + 45324518002715 T^{2} + $$11\!\cdots\!06$$ T^{3} + $$99\!\cdots\!45$$ T^{4} + $$24\!\cdots\!60$$ T^{5} + $$52\!\cdots\!74$$ T^{6} + $$37\!\cdots\!08$$ T^{7} + $$66\!\cdots\!22$$ T^{8} + $$37\!\cdots\!08$$ p^{7} T^{9} + $$52\!\cdots\!74$$ p^{14} T^{10} + $$24\!\cdots\!60$$ p^{21} T^{11} + $$99\!\cdots\!45$$ p^{28} T^{12} + $$11\!\cdots\!06$$ p^{35} T^{13} + 45324518002715 p^{42} T^{14} + 11452234 p^{49} T^{15} + p^{56} T^{16} $
	97	$ 1 + 428002 T - 262783180278813 T^{2} - $$24\!\cdots\!50$$ T^{3} + $$39\!\cdots\!73$$ T^{4} + $$35\!\cdots\!32$$ T^{5} - $$42\!\cdots\!26$$ T^{6} - $$13\!\cdots\!80$$ T^{7} + $$37\!\cdots\!66$$ T^{8} - $$13\!\cdots\!80$$ p^{7} T^{9} - $$42\!\cdots\!26$$ p^{14} T^{10} + $$35\!\cdots\!32$$ p^{21} T^{11} + $$39\!\cdots\!73$$ p^{28} T^{12} - $$24\!\cdots\!50$$ p^{35} T^{13} - 262783180278813 p^{42} T^{14} + 428002 p^{49} T^{15} + p^{56} 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

−6.81168357888632453050945433734, −6.41202421035417755654813853444, −6.34734425840558225407562277420, −6.01761739580206786286193250518, −5.78443546173782474486460611623, −5.61392653010655724868642819762, −5.53261499213347158230359298570, −5.46484886820953479772836985539, −4.79274759649410613750469996864, −4.68103046049712956632384351648, −4.67975183157922891483601731651, −4.52559608014640163369828050410, −4.36270168228763664302675118250, −3.83752993980241522668129601175, −3.55158826630784322412726203703, −3.39338269765668062466063672318, −2.78085853788222646183678354315, −2.71018029187727279853891285228, −2.38291324254866173764632379237, −2.20771799778827641035146128298, −2.10824556057425073623084577750, −1.74495065676387056838552167449, −0.77624088404093377737204112699, −0.56010130648833225088231556862, −0.28637075526289471709435495819, 0.28637075526289471709435495819, 0.56010130648833225088231556862, 0.77624088404093377737204112699, 1.74495065676387056838552167449, 2.10824556057425073623084577750, 2.20771799778827641035146128298, 2.38291324254866173764632379237, 2.71018029187727279853891285228, 2.78085853788222646183678354315, 3.39338269765668062466063672318, 3.55158826630784322412726203703, 3.83752993980241522668129601175, 4.36270168228763664302675118250, 4.52559608014640163369828050410, 4.67975183157922891483601731651, 4.68103046049712956632384351648, 4.79274759649410613750469996864, 5.46484886820953479772836985539, 5.53261499213347158230359298570, 5.61392653010655724868642819762, 5.78443546173782474486460611623, 6.01761739580206786286193250518, 6.34734425840558225407562277420, 6.41202421035417755654813853444, 6.81168357888632453050945433734

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(12.74656108\)
\(L(\frac12)\)	\(\approx\)	\(12.74656108\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more