LMFDB - L-function 40-20e40-1.1-c5e20-0-0

Dirichlet series

L(s) = 1

− 804·13^-s + 2.23e3·17^-s − 4.42e4·37^-s − 6.76e3·41^-s − 1.82e5·53^-s − 4.10e4·61^-s − 2.64e5·73^-s − 4.91e3·81^-s − 3.74e5·97^-s − 5.14e5·101^-s − 1.26e6·113^-s + 1.26e6·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 3.23e5·169^-s + 173^-s + 179^-s + 181^-s + 191^-s + 193^-s + ⋯

L(s) = 1

− 1.31·13^-s + 1.87·17^-s − 5.31·37^-s − 0.628·41^-s − 8.92·53^-s − 1.41·61^-s − 5.80·73^-s − 0.0831·81^-s − 4.04·97^-s − 5.01·101^-s − 9.34·113^-s + 7.82·121^-s + 5.50e−6·127^-s + 5.09e−6·131^-s + 4.55e−6·137^-s + 4.38e−6·139^-s + 3.69e−6·149^-s + 3.56e−6·151^-s + 3.23e−6·157^-s + 2.94e−6·163^-s + 2.77e−6·167^-s + 0.870·169^-s + 2.54e−6·173^-s + 2.33e−6·179^-s + 2.26e−6·181^-s + 1.98e−6·191^-s + 1.93e−6·193^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 5^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 5^{40}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$40$
Conductor:	$2^{80} \cdot 5^{40}$
Sign:	$1$
Analytic conductor:	$1.39447\times 10^{36}$
Root analytic conductor:	$8.00958$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(40,\ 2^{80} \cdot 5^{40} ,\ ( \ : [5/2]^{20} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$9.560077077\times10^{-7}$
$L(\frac12)$	$\approx$	$9.560077077\times10^{-7}$
$L(\frac{7}{2})$		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 $
	5	$ 1 $
good	3	$ 1 + 4910 T^{4} + 4404762445 T^{8} + 15335701222280 p^{2} T^{12} - 84567556689796190 p^{4} T^{16} + 14403811239667020148 p^{10} T^{20} - 84567556689796190 p^{24} T^{24} + 15335701222280 p^{42} T^{28} + 4404762445 p^{60} T^{32} + 4910 p^{80} T^{36} + p^{100} T^{40} $
	7	$ 1 - 50878370 T^{4} + 88119764714863965 T^{8} + $$17\!\cdots\!80$$ T^{12} + $$80\!\cdots\!70$$ T^{16} + $$16\!\cdots\!76$$ T^{20} + $$80\!\cdots\!70$$ p^{20} T^{24} + $$17\!\cdots\!80$$ p^{40} T^{28} + 88119764714863965 p^{60} T^{32} - 50878370 p^{80} T^{36} + p^{100} T^{40} $
	11	$ ( 1 - 630250 T^{2} + 230184799605 T^{4} - 63113737617167000 T^{6} + $$13\!\cdots\!10$$ T^{8} - $$24\!\cdots\!00$$ T^{10} + $$13\!\cdots\!10$$ p^{10} T^{12} - 63113737617167000 p^{20} T^{14} + 230184799605 p^{30} T^{16} - 630250 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	13	$ ( 1 + 402 T + 80802 T^{2} + 412195466 T^{3} + 274076716325 T^{4} - 98802411059208 T^{5} + 23088035017184312 T^{6} + 33522742681396790296 T^{7} - $$32\!\cdots\!50$$ p^{2} T^{8} - $$26\!\cdots\!36$$ p T^{9} + $$30\!\cdots\!32$$ T^{10} - $$26\!\cdots\!36$$ p^{6} T^{11} - $$32\!\cdots\!50$$ p^{12} T^{12} + 33522742681396790296 p^{15} T^{13} + 23088035017184312 p^{20} T^{14} - 98802411059208 p^{25} T^{15} + 274076716325 p^{30} T^{16} + 412195466 p^{35} T^{17} + 80802 p^{40} T^{18} + 402 p^{45} T^{19} + p^{50} T^{20} )^{2} $
	17	$ ( 1 - 1118 T + 624962 T^{2} - 69923678 p T^{3} - 1739300659939 T^{4} + 3459207301051384 T^{5} - 2073890095879259656 T^{6} + $$50\!\cdots\!88$$ T^{7} + $$54\!\cdots\!26$$ T^{8} - $$82\!\cdots\!84$$ T^{9} + $$52\!\cdots\!56$$ T^{10} - $$82\!\cdots\!84$$ p^{5} T^{11} + $$54\!\cdots\!26$$ p^{10} T^{12} + $$50\!\cdots\!88$$ p^{15} T^{13} - 2073890095879259656 p^{20} T^{14} + 3459207301051384 p^{25} T^{15} - 1739300659939 p^{30} T^{16} - 69923678 p^{36} T^{17} + 624962 p^{40} T^{18} - 1118 p^{45} T^{19} + p^{50} T^{20} )^{2} $
	19	$ ( 1 + 8249550 T^{2} + 36376777568405 T^{4} + $$13\!\cdots\!00$$ T^{6} + $$42\!\cdots\!10$$ T^{8} + $$11\!\cdots\!00$$ T^{10} + $$42\!\cdots\!10$$ p^{10} T^{12} + $$13\!\cdots\!00$$ p^{20} T^{14} + 36376777568405 p^{30} T^{16} + 8249550 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	23	$ 1 - 207859623483490 T^{4} + $$17\!\cdots\!45$$ T^{8} - $$53\!\cdots\!80$$ T^{12} - $$11\!\cdots\!90$$ T^{16} + $$13\!\cdots\!52$$ T^{20} - $$11\!\cdots\!90$$ p^{20} T^{24} - $$53\!\cdots\!80$$ p^{40} T^{28} + $$17\!\cdots\!45$$ p^{60} T^{32} - 207859623483490 p^{80} T^{36} + p^{100} T^{40} $
	29	$ ( 1 - 99799410 T^{2} + 5177263888867445 T^{4} - $$18\!\cdots\!20$$ T^{6} + $$49\!\cdots\!10$$ T^{8} - $$10\!\cdots\!52$$ T^{10} + $$49\!\cdots\!10$$ p^{10} T^{12} - $$18\!\cdots\!20$$ p^{20} T^{14} + 5177263888867445 p^{30} T^{16} - 99799410 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	31	$ ( 1 - 101180050 T^{2} + 5842226806223805 T^{4} - $$25\!\cdots\!00$$ T^{6} + $$95\!\cdots\!10$$ T^{8} - $$29\!\cdots\!00$$ T^{10} + $$95\!\cdots\!10$$ p^{10} T^{12} - $$25\!\cdots\!00$$ p^{20} T^{14} + 5842226806223805 p^{30} T^{16} - 101180050 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	37	$ ( 1 + 22130 T + 244868450 T^{2} + 1308308732410 T^{3} + 13809729158765845 T^{4} + $$24\!\cdots\!80$$ T^{5} + $$28\!\cdots\!00$$ T^{6} + $$15\!\cdots\!60$$ T^{7} + $$10\!\cdots\!10$$ T^{8} + $$15\!\cdots\!80$$ T^{9} + $$18\!\cdots\!00$$ T^{10} + $$15\!\cdots\!80$$ p^{5} T^{11} + $$10\!\cdots\!10$$ p^{10} T^{12} + $$15\!\cdots\!60$$ p^{15} T^{13} + $$28\!\cdots\!00$$ p^{20} T^{14} + $$24\!\cdots\!80$$ p^{25} T^{15} + 13809729158765845 p^{30} T^{16} + 1308308732410 p^{35} T^{17} + 244868450 p^{40} T^{18} + 22130 p^{45} T^{19} + p^{50} T^{20} )^{2} $
	41	$ ( 1 + 1690 T + 164119545 T^{2} - 1028151329120 T^{3} + 25291870422416110 T^{4} - 93379183076463398452 T^{5} + 25291870422416110 p^{5} T^{6} - 1028151329120 p^{10} T^{7} + 164119545 p^{15} T^{8} + 1690 p^{20} T^{9} + p^{25} T^{10} )^{4} $
	43	$ 1 + 22292551011039310 T^{4} + $$40\!\cdots\!45$$ T^{8} + $$23\!\cdots\!20$$ T^{12} + $$67\!\cdots\!10$$ T^{16} + $$21\!\cdots\!52$$ T^{20} + $$67\!\cdots\!10$$ p^{20} T^{24} + $$23\!\cdots\!20$$ p^{40} T^{28} + $$40\!\cdots\!45$$ p^{60} T^{32} + 22292551011039310 p^{80} T^{36} + p^{100} T^{40} $
	47	$ 1 + 74279974827999230 T^{4} + $$48\!\cdots\!65$$ T^{8} + $$19\!\cdots\!80$$ T^{12} + $$14\!\cdots\!70$$ T^{16} - $$25\!\cdots\!24$$ T^{20} + $$14\!\cdots\!70$$ p^{20} T^{24} + $$19\!\cdots\!80$$ p^{40} T^{28} + $$48\!\cdots\!65$$ p^{60} T^{32} + 74279974827999230 p^{80} T^{36} + p^{100} T^{40} $
	53	$ ( 1 + 91226 T + 4161091538 T^{2} + 141622957033378 T^{3} + 3893612180777232821 T^{4} + $$81\!\cdots\!12$$ T^{5} + $$12\!\cdots\!56$$ T^{6} + $$10\!\cdots\!76$$ T^{7} - $$19\!\cdots\!54$$ T^{8} - $$10\!\cdots\!72$$ T^{9} - $$26\!\cdots\!96$$ T^{10} - $$10\!\cdots\!72$$ p^{5} T^{11} - $$19\!\cdots\!54$$ p^{10} T^{12} + $$10\!\cdots\!76$$ p^{15} T^{13} + $$12\!\cdots\!56$$ p^{20} T^{14} + $$81\!\cdots\!12$$ p^{25} T^{15} + 3893612180777232821 p^{30} T^{16} + 141622957033378 p^{35} T^{17} + 4161091538 p^{40} T^{18} + 91226 p^{45} T^{19} + p^{50} T^{20} )^{2} $
	59	$ ( 1 + 4265422750 T^{2} + 8278744468566020805 T^{4} + $$98\!\cdots\!00$$ T^{6} + $$85\!\cdots\!10$$ T^{8} + $$63\!\cdots\!00$$ T^{10} + $$85\!\cdots\!10$$ p^{10} T^{12} + $$98\!\cdots\!00$$ p^{20} T^{14} + 8278744468566020805 p^{30} T^{16} + 4265422750 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	61	$ ( 1 + 10270 T + 3306972765 T^{2} + 29698468018720 T^{3} + 4981733552475787870 T^{4} + $$35\!\cdots\!24$$ T^{5} + 4981733552475787870 p^{5} T^{6} + 29698468018720 p^{10} T^{7} + 3306972765 p^{15} T^{8} + 10270 p^{20} T^{9} + p^{25} T^{10} )^{4} $
	67	$ 1 - 2038648142805486290 T^{4} - $$25\!\cdots\!55$$ T^{8} + $$14\!\cdots\!20$$ T^{12} + $$73\!\cdots\!10$$ T^{16} - $$57\!\cdots\!48$$ T^{20} + $$73\!\cdots\!10$$ p^{20} T^{24} + $$14\!\cdots\!20$$ p^{40} T^{28} - $$25\!\cdots\!55$$ p^{60} T^{32} - 2038648142805486290 p^{80} T^{36} + p^{100} T^{40} $
	71	$ ( 1 - 14421599170 T^{2} + 96751044929093627565 T^{4} - $$40\!\cdots\!20$$ T^{6} + $$11\!\cdots\!70$$ T^{8} - $$24\!\cdots\!24$$ T^{10} + $$11\!\cdots\!70$$ p^{10} T^{12} - $$40\!\cdots\!20$$ p^{20} T^{14} + 96751044929093627565 p^{30} T^{16} - 14421599170 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	73	$ ( 1 + 132186 T + 8736569298 T^{2} + 431249337673738 T^{3} + 10750357541220477581 T^{4} - $$12\!\cdots\!88$$ T^{5} - $$17\!\cdots\!84$$ T^{6} - $$58\!\cdots\!44$$ T^{7} + $$47\!\cdots\!06$$ T^{8} + $$60\!\cdots\!08$$ T^{9} + $$32\!\cdots\!04$$ T^{10} + $$60\!\cdots\!08$$ p^{5} T^{11} + $$47\!\cdots\!06$$ p^{10} T^{12} - $$58\!\cdots\!44$$ p^{15} T^{13} - $$17\!\cdots\!84$$ p^{20} T^{14} - $$12\!\cdots\!88$$ p^{25} T^{15} + 10750357541220477581 p^{30} T^{16} + 431249337673738 p^{35} T^{17} + 8736569298 p^{40} T^{18} + 132186 p^{45} T^{19} + p^{50} T^{20} )^{2} $
	79	$ ( 1 + 18614864790 T^{2} + $$16\!\cdots\!45$$ T^{4} + $$10\!\cdots\!80$$ T^{6} + $$44\!\cdots\!10$$ T^{8} + $$15\!\cdots\!48$$ T^{10} + $$44\!\cdots\!10$$ p^{10} T^{12} + $$10\!\cdots\!80$$ p^{20} T^{14} + $$16\!\cdots\!45$$ p^{30} T^{16} + 18614864790 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	83	$ 1 - 18609684090616711570 T^{4} - $$46\!\cdots\!35$$ T^{8} + $$34\!\cdots\!80$$ T^{12} + $$25\!\cdots\!70$$ T^{16} - $$17\!\cdots\!24$$ T^{20} + $$25\!\cdots\!70$$ p^{20} T^{24} + $$34\!\cdots\!80$$ p^{40} T^{28} - $$46\!\cdots\!35$$ p^{60} T^{32} - 18609684090616711570 p^{80} T^{36} + p^{100} T^{40} $
	89	$ ( 1 - 33120686970 T^{2} + $$41\!\cdots\!65$$ T^{4} - $$20\!\cdots\!20$$ T^{6} - $$32\!\cdots\!30$$ T^{8} + $$73\!\cdots\!76$$ T^{10} - $$32\!\cdots\!30$$ p^{10} T^{12} - $$20\!\cdots\!20$$ p^{20} T^{14} + $$41\!\cdots\!65$$ p^{30} T^{16} - 33120686970 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	97	$ ( 1 + 187386 T + 17556756498 T^{2} + 497237634071002 T^{3} - $$23\!\cdots\!35$$ T^{4} - $$42\!\cdots\!24$$ T^{5} - $$37\!\cdots\!32$$ T^{6} - $$18\!\cdots\!68$$ T^{7} + $$11\!\cdots\!50$$ T^{8} + $$33\!\cdots\!76$$ T^{9} + $$36\!\cdots\!68$$ T^{10} + $$33\!\cdots\!76$$ p^{5} T^{11} + $$11\!\cdots\!50$$ p^{10} T^{12} - $$18\!\cdots\!68$$ p^{15} T^{13} - $$37\!\cdots\!32$$ p^{20} T^{14} - $$42\!\cdots\!24$$ p^{25} T^{15} - $$23\!\cdots\!35$$ p^{30} T^{16} + 497237634071002 p^{35} T^{17} + 17556756498 p^{40} T^{18} + 187386 p^{45} T^{19} + p^{50} T^{20} )^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−1.74663653428394800200385680805, −1.74244284991654380808247478001, −1.74013426726122584695134011877, −1.60971863199526786138454127384, −1.52232250054447050691835868233, −1.35595837749519278103062343107, −1.31604730590091623918576145527, −1.30229910416081825439820396567, −1.27722535194451609479649153366, −1.25737456047239116134134240870, −1.17217311609079102971519755452, −1.11082329599322566100158986854, −1.06264290682430618726279382730, −0.893142287553534978706838897200, −0.848320237546306944606134934816, −0.73577286294649735242541365981, −0.51760015209701835320862208670, −0.43841097996065122671556964886, −0.41585492723339501593748943006, −0.34848590167076951132319722845, −0.29424390781487581980526424777, −0.07822291184859071298741930727, −0.05924374916914969134820585636, −0.01270343749286886269451444490, −0.00422404471715352567082216762, 0.00422404471715352567082216762, 0.01270343749286886269451444490, 0.05924374916914969134820585636, 0.07822291184859071298741930727, 0.29424390781487581980526424777, 0.34848590167076951132319722845, 0.41585492723339501593748943006, 0.43841097996065122671556964886, 0.51760015209701835320862208670, 0.73577286294649735242541365981, 0.848320237546306944606134934816, 0.893142287553534978706838897200, 1.06264290682430618726279382730, 1.11082329599322566100158986854, 1.17217311609079102971519755452, 1.25737456047239116134134240870, 1.27722535194451609479649153366, 1.30229910416081825439820396567, 1.31604730590091623918576145527, 1.35595837749519278103062343107, 1.52232250054447050691835868233, 1.60971863199526786138454127384, 1.74013426726122584695134011877, 1.74244284991654380808247478001, 1.74663653428394800200385680805

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(9.560077077\times10^{-7}\)
\(L(\frac12)\)	\(\approx\)	\(9.560077077\times10^{-7}\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more