LMFDB - L-function 28-74e14-1.1-c5e14-0-1

Dirichlet series

L(s) = 1

+ 14·3^-s − 112·4^-s − 312·7^-s − 1.21e3·9^-s + 262·11^-s − 1.56e3·12^-s + 7.16e3·16^-s − 4.36e3·21^-s + 2.50e4·25^-s − 2.39e4·27^-s + 3.49e4·28^-s + 3.66e3·33^-s + 1.36e5·36^-s + 1.32e4·37^-s + 5.17e3·41^-s − 2.93e4·44^-s + 3.69e4·47^-s + 1.00e5·48^-s − 4.39e4·49^-s + 9.52e3·53^-s + 3.79e5·63^-s − 3.44e5·64^-s − 5.51e4·67^-s + 2.77e4·71^-s + 6.57e4·73^-s + 3.50e5·75^-s − 8.17e4·77^-s + ⋯

L(s) = 1

+ 0.898·3^-s − 7/2·4^-s − 2.40·7^-s − 5.00·9^-s + 0.652·11^-s − 3.14·12^-s + 7·16^-s − 2.16·21^-s + 8.00·25^-s − 6.32·27^-s + 8.42·28^-s + 0.586·33^-s + 17.5·36^-s + 1.59·37^-s + 0.481·41^-s − 2.28·44^-s + 2.44·47^-s + 6.28·48^-s − 2.61·49^-s + 0.465·53^-s + 12.0·63^-s − 10.5·64^-s − 1.50·67^-s + 0.653·71^-s + 1.44·73^-s + 7.18·75^-s − 1.57·77^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$28$
Conductor:	$2^{14} \cdot 37^{14}$
Sign:	$1$
Analytic conductor:	$1.10025\times 10^{15}$
Root analytic conductor:	$3.44505$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(28,\ 2^{14} \cdot 37^{14} ,\ ( \ : [5/2]^{14} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$1.280801125$
$L(\frac12)$	$\approx$	$1.280801125$
$L(\frac{7}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ ( 1 + p^{4} T^{2} )^{7} $
	37	$ 1 - 13286 T + 89471 p T^{2} + 1019000828 p^{2} T^{3} - 269723149815 p^{3} T^{4} + 419225773662 p^{5} T^{5} + 6635688599875 p^{7} T^{6} - 69037006446872 p^{9} T^{7} + 6635688599875 p^{12} T^{8} + 419225773662 p^{15} T^{9} - 269723149815 p^{18} T^{10} + 1019000828 p^{22} T^{11} + 89471 p^{26} T^{12} - 13286 p^{30} T^{13} + p^{35} T^{14} $
good	3	$ ( 1 - 7 T + 682 T^{2} - 1660 T^{3} + 267451 T^{4} + 107192 T^{5} + 24316951 p T^{6} + 18806066 p^{2} T^{7} + 24316951 p^{6} T^{8} + 107192 p^{10} T^{9} + 267451 p^{15} T^{10} - 1660 p^{20} T^{11} + 682 p^{25} T^{12} - 7 p^{30} T^{13} + p^{35} T^{14} )^{2} $
	5	$ 1 - 5003 p T^{2} + 315634264 T^{4} - 528698476352 p T^{6} + 3287745450302522 p T^{8} - 80779845666855298849 T^{10} + $$65\!\cdots\!53$$ p T^{12} - $$11\!\cdots\!36$$ T^{14} + $$65\!\cdots\!53$$ p^{11} T^{16} - 80779845666855298849 p^{20} T^{18} + 3287745450302522 p^{31} T^{20} - 528698476352 p^{41} T^{22} + 315634264 p^{50} T^{24} - 5003 p^{61} T^{26} + p^{70} T^{28} $
	7	$ ( 1 + 156 T + 58475 T^{2} + 10448856 T^{3} + 290500412 p T^{4} + 329158839848 T^{5} + 48814413503906 T^{6} + 6727309849441000 T^{7} + 48814413503906 p^{5} T^{8} + 329158839848 p^{10} T^{9} + 290500412 p^{16} T^{10} + 10448856 p^{20} T^{11} + 58475 p^{25} T^{12} + 156 p^{30} T^{13} + p^{35} T^{14} )^{2} $
	11	$ ( 1 - 131 T + 677236 T^{2} - 141640908 T^{3} + 227283594221 T^{4} - 57409991230728 T^{5} + 4550616830594277 p T^{6} - 12354949387287810566 T^{7} + 4550616830594277 p^{6} T^{8} - 57409991230728 p^{10} T^{9} + 227283594221 p^{15} T^{10} - 141640908 p^{20} T^{11} + 677236 p^{25} T^{12} - 131 p^{30} T^{13} + p^{35} T^{14} )^{2} $
	13	$ 1 - 2591143 T^{2} + 2975978435048 T^{4} - 2033148978338586544 T^{6} + $$74\!\cdots\!46$$ p T^{8} - $$38\!\cdots\!33$$ T^{10} + $$14\!\cdots\!61$$ T^{12} - $$55\!\cdots\!00$$ T^{14} + $$14\!\cdots\!61$$ p^{10} T^{16} - $$38\!\cdots\!33$$ p^{20} T^{18} + $$74\!\cdots\!46$$ p^{31} T^{20} - 2033148978338586544 p^{40} T^{22} + 2975978435048 p^{50} T^{24} - 2591143 p^{60} T^{26} + p^{70} T^{28} $
	17	$ 1 - 14877010 T^{2} + 6317017886251 p T^{4} - $$49\!\cdots\!12$$ T^{6} + $$16\!\cdots\!33$$ T^{8} - $$24\!\cdots\!50$$ p T^{10} + $$84\!\cdots\!71$$ T^{12} - $$13\!\cdots\!36$$ T^{14} + $$84\!\cdots\!71$$ p^{10} T^{16} - $$24\!\cdots\!50$$ p^{21} T^{18} + $$16\!\cdots\!33$$ p^{30} T^{20} - $$49\!\cdots\!12$$ p^{40} T^{22} + 6317017886251 p^{51} T^{24} - 14877010 p^{60} T^{26} + p^{70} T^{28} $
	19	$ 1 - 26374282 T^{2} + 334775213456147 T^{4} - $$27\!\cdots\!24$$ T^{6} + $$15\!\cdots\!89$$ T^{8} - $$69\!\cdots\!46$$ T^{10} + $$24\!\cdots\!87$$ T^{12} - $$67\!\cdots\!24$$ T^{14} + $$24\!\cdots\!87$$ p^{10} T^{16} - $$69\!\cdots\!46$$ p^{20} T^{18} + $$15\!\cdots\!89$$ p^{30} T^{20} - $$27\!\cdots\!24$$ p^{40} T^{22} + 334775213456147 p^{50} T^{24} - 26374282 p^{60} T^{26} + p^{70} T^{28} $
	23	$ 1 - 1284417 p T^{2} + 531385800239636 T^{4} - $$71\!\cdots\!00$$ T^{6} + $$77\!\cdots\!46$$ T^{8} - $$71\!\cdots\!97$$ T^{10} + $$56\!\cdots\!05$$ T^{12} - $$39\!\cdots\!24$$ T^{14} + $$56\!\cdots\!05$$ p^{10} T^{16} - $$71\!\cdots\!97$$ p^{20} T^{18} + $$77\!\cdots\!46$$ p^{30} T^{20} - $$71\!\cdots\!00$$ p^{40} T^{22} + 531385800239636 p^{50} T^{24} - 1284417 p^{61} T^{26} + p^{70} T^{28} $
	29	$ 1 - 166645463 T^{2} + 14384149595503208 T^{4} - $$83\!\cdots\!56$$ T^{6} + $$36\!\cdots\!54$$ T^{8} - $$12\!\cdots\!25$$ T^{10} + $$33\!\cdots\!37$$ T^{12} - $$75\!\cdots\!12$$ T^{14} + $$33\!\cdots\!37$$ p^{10} T^{16} - $$12\!\cdots\!25$$ p^{20} T^{18} + $$36\!\cdots\!54$$ p^{30} T^{20} - $$83\!\cdots\!56$$ p^{40} T^{22} + 14384149595503208 p^{50} T^{24} - 166645463 p^{60} T^{26} + p^{70} T^{28} $
	31	$ 1 - 267975923 T^{2} + 35960686619169652 T^{4} - $$31\!\cdots\!36$$ T^{6} + $$20\!\cdots\!42$$ T^{8} - $$10\!\cdots\!01$$ T^{10} + $$40\!\cdots\!85$$ T^{12} - $$12\!\cdots\!40$$ T^{14} + $$40\!\cdots\!85$$ p^{10} T^{16} - $$10\!\cdots\!01$$ p^{20} T^{18} + $$20\!\cdots\!42$$ p^{30} T^{20} - $$31\!\cdots\!36$$ p^{40} T^{22} + 35960686619169652 p^{50} T^{24} - 267975923 p^{60} T^{26} + p^{70} T^{28} $
	41	$ ( 1 - 2589 T + 5038570 p T^{2} + 995114251822 T^{3} + 45209334084411447 T^{4} + 91361243747170725404 T^{5} + $$62\!\cdots\!27$$ T^{6} + $$31\!\cdots\!96$$ T^{7} + $$62\!\cdots\!27$$ p^{5} T^{8} + 91361243747170725404 p^{10} T^{9} + 45209334084411447 p^{15} T^{10} + 995114251822 p^{20} T^{11} + 5038570 p^{26} T^{12} - 2589 p^{30} T^{13} + p^{35} T^{14} )^{2} $
	43	$ 1 - 1002502830 T^{2} + 543062949998784579 T^{4} - $$20\!\cdots\!96$$ T^{6} + $$58\!\cdots\!21$$ T^{8} - $$13\!\cdots\!66$$ T^{10} + $$25\!\cdots\!35$$ T^{12} - $$41\!\cdots\!16$$ T^{14} + $$25\!\cdots\!35$$ p^{10} T^{16} - $$13\!\cdots\!66$$ p^{20} T^{18} + $$58\!\cdots\!21$$ p^{30} T^{20} - $$20\!\cdots\!96$$ p^{40} T^{22} + 543062949998784579 p^{50} T^{24} - 1002502830 p^{60} T^{26} + p^{70} T^{28} $
	47	$ ( 1 - 18490 T + 781482403 T^{2} - 12926549866712 T^{3} + 336397525050954692 T^{4} - $$42\!\cdots\!80$$ T^{5} + $$92\!\cdots\!38$$ T^{6} - $$11\!\cdots\!16$$ T^{7} + $$92\!\cdots\!38$$ p^{5} T^{8} - $$42\!\cdots\!80$$ p^{10} T^{9} + 336397525050954692 p^{15} T^{10} - 12926549866712 p^{20} T^{11} + 781482403 p^{25} T^{12} - 18490 p^{30} T^{13} + p^{35} T^{14} )^{2} $
	53	$ ( 1 - 4760 T + 1143043049 T^{2} - 15031175619356 T^{3} + 829097320876747596 T^{4} - $$11\!\cdots\!48$$ T^{5} + $$48\!\cdots\!42$$ T^{6} - $$50\!\cdots\!08$$ T^{7} + $$48\!\cdots\!42$$ p^{5} T^{8} - $$11\!\cdots\!48$$ p^{10} T^{9} + 829097320876747596 p^{15} T^{10} - 15031175619356 p^{20} T^{11} + 1143043049 p^{25} T^{12} - 4760 p^{30} T^{13} + p^{35} T^{14} )^{2} $
	59	$ 1 - 5829127162 T^{2} + 16442038905549764483 T^{4} - $$30\!\cdots\!28$$ T^{6} + $$41\!\cdots\!81$$ T^{8} - $$45\!\cdots\!62$$ T^{10} + $$41\!\cdots\!91$$ T^{12} - $$31\!\cdots\!28$$ T^{14} + $$41\!\cdots\!91$$ p^{10} T^{16} - $$45\!\cdots\!62$$ p^{20} T^{18} + $$41\!\cdots\!81$$ p^{30} T^{20} - $$30\!\cdots\!28$$ p^{40} T^{22} + 16442038905549764483 p^{50} T^{24} - 5829127162 p^{60} T^{26} + p^{70} T^{28} $
	61	$ 1 - 6774754047 T^{2} + 22898985107563975152 T^{4} - $$51\!\cdots\!68$$ T^{6} + $$87\!\cdots\!94$$ T^{8} - $$11\!\cdots\!61$$ T^{10} + $$13\!\cdots\!73$$ T^{12} - $$12\!\cdots\!88$$ T^{14} + $$13\!\cdots\!73$$ p^{10} T^{16} - $$11\!\cdots\!61$$ p^{20} T^{18} + $$87\!\cdots\!94$$ p^{30} T^{20} - $$51\!\cdots\!68$$ p^{40} T^{22} + 22898985107563975152 p^{50} T^{24} - 6774754047 p^{60} T^{26} + p^{70} T^{28} $
	67	$ ( 1 + 27569 T + 5402726938 T^{2} + 144640052679966 T^{3} + 16211082020762155464 T^{4} + $$37\!\cdots\!99$$ T^{5} + $$31\!\cdots\!07$$ T^{6} + $$62\!\cdots\!28$$ T^{7} + $$31\!\cdots\!07$$ p^{5} T^{8} + $$37\!\cdots\!99$$ p^{10} T^{9} + 16211082020762155464 p^{15} T^{10} + 144640052679966 p^{20} T^{11} + 5402726938 p^{25} T^{12} + 27569 p^{30} T^{13} + p^{35} T^{14} )^{2} $
	71	$ ( 1 - 13880 T + 9454682355 T^{2} - 97130919360056 T^{3} + 41223493020268668844 T^{4} - $$44\!\cdots\!96$$ p T^{5} + $$11\!\cdots\!70$$ T^{6} - $$66\!\cdots\!36$$ T^{7} + $$11\!\cdots\!70$$ p^{5} T^{8} - $$44\!\cdots\!96$$ p^{11} T^{9} + 41223493020268668844 p^{15} T^{10} - 97130919360056 p^{20} T^{11} + 9454682355 p^{25} T^{12} - 13880 p^{30} T^{13} + p^{35} T^{14} )^{2} $
	73	$ ( 1 - 32891 T + 11141026514 T^{2} - 278490824940290 T^{3} + 56633295238936297223 T^{4} - $$11\!\cdots\!60$$ T^{5} + $$17\!\cdots\!03$$ T^{6} - $$28\!\cdots\!80$$ T^{7} + $$17\!\cdots\!03$$ p^{5} T^{8} - $$11\!\cdots\!60$$ p^{10} T^{9} + 56633295238936297223 p^{15} T^{10} - 278490824940290 p^{20} T^{11} + 11141026514 p^{25} T^{12} - 32891 p^{30} T^{13} + p^{35} T^{14} )^{2} $
	79	$ 1 - 8935558655 T^{2} + 78491154884316596620 T^{4} - $$45\!\cdots\!32$$ T^{6} + $$24\!\cdots\!06$$ T^{8} - $$10\!\cdots\!93$$ T^{10} + $$38\!\cdots\!33$$ T^{12} - $$12\!\cdots\!20$$ T^{14} + $$38\!\cdots\!33$$ p^{10} T^{16} - $$10\!\cdots\!93$$ p^{20} T^{18} + $$24\!\cdots\!06$$ p^{30} T^{20} - $$45\!\cdots\!32$$ p^{40} T^{22} + 78491154884316596620 p^{50} T^{24} - 8935558655 p^{60} T^{26} + p^{70} T^{28} $
	83	$ ( 1 - 89408 T + 22692602067 T^{2} - 1750951627662320 T^{3} + $$24\!\cdots\!12$$ T^{4} - $$15\!\cdots\!68$$ T^{5} + $$15\!\cdots\!66$$ T^{6} - $$77\!\cdots\!16$$ T^{7} + $$15\!\cdots\!66$$ p^{5} T^{8} - $$15\!\cdots\!68$$ p^{10} T^{9} + $$24\!\cdots\!12$$ p^{15} T^{10} - 1750951627662320 p^{20} T^{11} + 22692602067 p^{25} T^{12} - 89408 p^{30} T^{13} + p^{35} T^{14} )^{2} $
	89	$ 1 - 16970815606 T^{2} + $$18\!\cdots\!59$$ T^{4} - $$17\!\cdots\!00$$ T^{6} + $$13\!\cdots\!93$$ T^{8} - $$10\!\cdots\!54$$ T^{10} + $$66\!\cdots\!71$$ T^{12} - $$38\!\cdots\!28$$ T^{14} + $$66\!\cdots\!71$$ p^{10} T^{16} - $$10\!\cdots\!54$$ p^{20} T^{18} + $$13\!\cdots\!93$$ p^{30} T^{20} - $$17\!\cdots\!00$$ p^{40} T^{22} + $$18\!\cdots\!59$$ p^{50} T^{24} - 16970815606 p^{60} T^{26} + p^{70} T^{28} $
	97	$ 1 - 69104852194 T^{2} + $$25\!\cdots\!55$$ T^{4} - $$61\!\cdots\!36$$ T^{6} + $$11\!\cdots\!33$$ T^{8} - $$16\!\cdots\!62$$ T^{10} + $$18\!\cdots\!03$$ T^{12} - $$17\!\cdots\!76$$ T^{14} + $$18\!\cdots\!03$$ p^{10} T^{16} - $$16\!\cdots\!62$$ p^{20} T^{18} + $$11\!\cdots\!33$$ p^{30} T^{20} - $$61\!\cdots\!36$$ p^{40} T^{22} + $$25\!\cdots\!55$$ p^{50} T^{24} - 69104852194 p^{60} T^{26} + p^{70} T^{28} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−3.66017184947855287782315377250, −3.55984704874271522640875638059, −3.36088643998451048491284834297, −3.28272973058366769176384027954, −3.12149530042241503604731268194, −3.07289300561508334296803405137, −2.97038389580453024358400496991, −2.94253867829644199704502512268, −2.84113855163506261203655910413, −2.69383947575536388786363280139, −2.68329402153793475348969444217, −2.41577380123181826896803661841, −2.35792150930712213245092769186, −2.05141094246881212904902874311, −1.72222664149376088526848960030, −1.42900084793544598624512149956, −1.37281323911000312399031656055, −1.29078793737017239405086539845, −0.892959167174251741465231483453, −0.74315725196364482808494821030, −0.68948875359544126853068770805, −0.52167034733356770914519754726, −0.39424802993740217019554598953, −0.27311304701170001477374862875, −0.18283295186490108220956385261, 0.18283295186490108220956385261, 0.27311304701170001477374862875, 0.39424802993740217019554598953, 0.52167034733356770914519754726, 0.68948875359544126853068770805, 0.74315725196364482808494821030, 0.892959167174251741465231483453, 1.29078793737017239405086539845, 1.37281323911000312399031656055, 1.42900084793544598624512149956, 1.72222664149376088526848960030, 2.05141094246881212904902874311, 2.35792150930712213245092769186, 2.41577380123181826896803661841, 2.68329402153793475348969444217, 2.69383947575536388786363280139, 2.84113855163506261203655910413, 2.94253867829644199704502512268, 2.97038389580453024358400496991, 3.07289300561508334296803405137, 3.12149530042241503604731268194, 3.28272973058366769176384027954, 3.36088643998451048491284834297, 3.55984704874271522640875638059, 3.66017184947855287782315377250

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1.280801125\)
\(L(\frac12)\)	\(\approx\)	\(1.280801125\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more