LMFDB - L-function 40-30e20-1.1-c5e20-0-0

Dirichlet series

L(s) = 1

− 4·3^-s + 76·7^-s + 8·9^-s + 2.64e3·13^-s − 1.28e3·16^-s − 304·21^-s − 7.88e3·25^-s + 5.50e3·27^-s + 2.39e4·31^-s − 6.06e4·37^-s − 1.05e4·39^-s + 4.80e4·43^-s + 5.12e3·48^-s + 2.88e3·49^-s + 1.60e5·61^-s + 608·63^-s − 1.18e5·67^-s + 3.79e5·73^-s + 3.15e4·75^-s − 9.26e4·81^-s + 2.00e5·91^-s − 9.58e4·93^-s − 2.88e5·97^-s − 2.58e5·103^-s + 2.42e5·111^-s − 9.72e4·112^-s + 2.11e4·117^-s + ⋯

L(s) = 1

− 0.256·3^-s + 0.586·7^-s + 0.0329·9^-s + 4.33·13^-s − 5/4·16^-s − 0.150·21^-s − 2.52·25^-s + 1.45·27^-s + 4.47·31^-s − 7.28·37^-s − 1.11·39^-s + 3.96·43^-s + 0.320·48^-s + 0.171·49^-s + 5.52·61^-s + 0.0192·63^-s − 3.23·67^-s + 8.32·73^-s + 0.647·75^-s − 1.56·81^-s + 2.53·91^-s − 1.14·93^-s − 3.11·97^-s − 2.40·103^-s + 1.87·111^-s − 0.732·112^-s + 0.142·117^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$40$
Conductor:	$2^{20} \cdot 3^{20} \cdot 5^{20}$
Sign:	$1$
Analytic conductor:	$4.42217\times 10^{13}$
Root analytic conductor:	$2.19351$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(40,\ 2^{20} \cdot 3^{20} \cdot 5^{20} ,\ ( \ : [5/2]^{20} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$5.079886084$
$L(\frac12)$	$\approx$	$5.079886084$
$L(\frac{7}{2})$		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^{8} T^{4} )^{5} $
	3	$ 1 + 4 T + 8 T^{2} - 68 p^{4} T^{3} + 5389 p^{2} T^{4} + 197632 p^{2} T^{5} + 2432864 p^{2} T^{6} - 1159072 p^{5} T^{7} - 936974 p^{8} T^{8} + 3751384 p^{9} T^{9} + 40911536 p^{10} T^{10} + 3751384 p^{14} T^{11} - 936974 p^{18} T^{12} - 1159072 p^{20} T^{13} + 2432864 p^{22} T^{14} + 197632 p^{27} T^{15} + 5389 p^{32} T^{16} - 68 p^{39} T^{17} + 8 p^{40} T^{18} + 4 p^{45} T^{19} + p^{50} T^{20} $
	5	$ 1 + 7888 T^{2} + 286449 p T^{4} - 4533485472 p^{2} T^{6} - 15137290374 p^{5} T^{8} + 2803847326368 p^{8} T^{10} - 15137290374 p^{15} T^{12} - 4533485472 p^{22} T^{14} + 286449 p^{31} T^{16} + 7888 p^{40} T^{18} + p^{50} T^{20} $
good	7	$ ( 1 - 38 T + 722 T^{2} - 373306 T^{3} - 404492099 T^{4} + 8762278984 T^{5} + 4107911272 p T^{6} - 14662282378136 p T^{7} + 48056166685046066 T^{8} + 354879382256814708 p T^{9} - 831831674726644756 p^{2} T^{10} + 354879382256814708 p^{6} T^{11} + 48056166685046066 p^{10} T^{12} - 14662282378136 p^{16} T^{13} + 4107911272 p^{21} T^{14} + 8762278984 p^{25} T^{15} - 404492099 p^{30} T^{16} - 373306 p^{35} T^{17} + 722 p^{40} T^{18} - 38 p^{45} T^{19} + p^{50} T^{20} )^{2} $
	11	$ ( 1 - 534700 T^{2} + 217993474005 T^{4} - 58449397396978800 T^{6} + $$13\!\cdots\!10$$ T^{8} - $$22\!\cdots\!00$$ T^{10} + $$13\!\cdots\!10$$ p^{10} T^{12} - 58449397396978800 p^{20} T^{14} + 217993474005 p^{30} T^{16} - 534700 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	13	$ ( 1 - 1320 T + 871200 T^{2} - 46790520 p T^{3} + 498052172645 T^{4} - 26876511655440 p T^{5} + 212298195577075200 T^{6} - $$14\!\cdots\!60$$ T^{7} + $$11\!\cdots\!10$$ T^{8} - $$79\!\cdots\!20$$ T^{9} + $$45\!\cdots\!00$$ T^{10} - $$79\!\cdots\!20$$ p^{5} T^{11} + $$11\!\cdots\!10$$ p^{10} T^{12} - $$14\!\cdots\!60$$ p^{15} T^{13} + 212298195577075200 p^{20} T^{14} - 26876511655440 p^{26} T^{15} + 498052172645 p^{30} T^{16} - 46790520 p^{36} T^{17} + 871200 p^{40} T^{18} - 1320 p^{45} T^{19} + p^{50} T^{20} )^{2} $
	17	$ 1 + 2264383601210 T^{4} - $$91\!\cdots\!55$$ T^{8} - $$32\!\cdots\!80$$ T^{12} + $$21\!\cdots\!10$$ T^{16} + $$19\!\cdots\!52$$ T^{20} + $$21\!\cdots\!10$$ p^{20} T^{24} - $$32\!\cdots\!80$$ p^{40} T^{28} - $$91\!\cdots\!55$$ p^{60} T^{32} + 2264383601210 p^{80} T^{36} + p^{100} T^{40} $
	19	$ ( 1 - 8366510 T^{2} + 41451679984245 T^{4} - 7948228875277133880 p T^{6} + $$44\!\cdots\!10$$ T^{8} - $$11\!\cdots\!52$$ T^{10} + $$44\!\cdots\!10$$ p^{10} T^{12} - 7948228875277133880 p^{21} T^{14} + 41451679984245 p^{30} T^{16} - 8366510 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	23	$ 1 - 25947947080390 T^{4} + $$70\!\cdots\!45$$ T^{8} + $$70\!\cdots\!20$$ T^{12} - $$75\!\cdots\!90$$ T^{16} + $$13\!\cdots\!52$$ T^{20} - $$75\!\cdots\!90$$ p^{20} T^{24} + $$70\!\cdots\!20$$ p^{40} T^{28} + $$70\!\cdots\!45$$ p^{60} T^{32} - 25947947080390 p^{80} T^{36} + p^{100} T^{40} $
	29	$ ( 1 + 4212560 p T^{2} + 7588783179910245 T^{4} + $$31\!\cdots\!80$$ T^{6} + $$93\!\cdots\!10$$ T^{8} + $$21\!\cdots\!48$$ T^{10} + $$93\!\cdots\!10$$ p^{10} T^{12} + $$31\!\cdots\!80$$ p^{20} T^{14} + 7588783179910245 p^{30} T^{16} + 4212560 p^{41} T^{18} + p^{50} T^{20} )^{2} $
	31	$ ( 1 - 5990 T + 74009595 T^{2} - 330422803080 T^{3} + 1982306679039210 T^{4} - 9271998012306323748 T^{5} + 1982306679039210 p^{5} T^{6} - 330422803080 p^{10} T^{7} + 74009595 p^{15} T^{8} - 5990 p^{20} T^{9} + p^{25} T^{10} )^{4} $
	37	$ ( 1 + 30348 T + 460500552 T^{2} + 4468409009676 T^{3} + 29609859809383061 T^{4} + $$19\!\cdots\!16$$ T^{5} + $$22\!\cdots\!84$$ T^{6} + $$21\!\cdots\!08$$ p^{2} T^{7} + $$29\!\cdots\!66$$ T^{8} + $$22\!\cdots\!04$$ T^{9} + $$16\!\cdots\!16$$ T^{10} + $$22\!\cdots\!04$$ p^{5} T^{11} + $$29\!\cdots\!66$$ p^{10} T^{12} + $$21\!\cdots\!08$$ p^{17} T^{13} + $$22\!\cdots\!84$$ p^{20} T^{14} + $$19\!\cdots\!16$$ p^{25} T^{15} + 29609859809383061 p^{30} T^{16} + 4468409009676 p^{35} T^{17} + 460500552 p^{40} T^{18} + 30348 p^{45} T^{19} + p^{50} T^{20} )^{2} $
	41	$ ( 1 - 538322650 T^{2} + 140052760154294205 T^{4} - $$23\!\cdots\!00$$ T^{6} + $$29\!\cdots\!10$$ T^{8} - $$34\!\cdots\!00$$ T^{10} + $$29\!\cdots\!10$$ p^{10} T^{12} - $$23\!\cdots\!00$$ p^{20} T^{14} + 140052760154294205 p^{30} T^{16} - 538322650 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	43	$ ( 1 - 24012 T + 288288072 T^{2} - 5119692161076 T^{3} + 67270891536531365 T^{4} - $$27\!\cdots\!72$$ T^{5} + $$29\!\cdots\!72$$ T^{6} + $$32\!\cdots\!24$$ T^{7} - $$18\!\cdots\!90$$ T^{8} + $$26\!\cdots\!28$$ T^{9} - $$25\!\cdots\!68$$ T^{10} + $$26\!\cdots\!28$$ p^{5} T^{11} - $$18\!\cdots\!90$$ p^{10} T^{12} + $$32\!\cdots\!24$$ p^{15} T^{13} + $$29\!\cdots\!72$$ p^{20} T^{14} - $$27\!\cdots\!72$$ p^{25} T^{15} + 67270891536531365 p^{30} T^{16} - 5119692161076 p^{35} T^{17} + 288288072 p^{40} T^{18} - 24012 p^{45} T^{19} + p^{50} T^{20} )^{2} $
	47	$ 1 - 177111979945716710 T^{4} + $$15\!\cdots\!45$$ T^{8} - $$90\!\cdots\!20$$ T^{12} + $$46\!\cdots\!10$$ T^{16} - $$23\!\cdots\!52$$ T^{20} + $$46\!\cdots\!10$$ p^{20} T^{24} - $$90\!\cdots\!20$$ p^{40} T^{28} + $$15\!\cdots\!45$$ p^{60} T^{32} - 177111979945716710 p^{80} T^{36} + p^{100} T^{40} $
	53	$ 1 - 102415973340331990 T^{4} - $$46\!\cdots\!55$$ T^{8} - $$13\!\cdots\!80$$ T^{12} + $$62\!\cdots\!10$$ T^{16} + $$22\!\cdots\!52$$ T^{20} + $$62\!\cdots\!10$$ p^{20} T^{24} - $$13\!\cdots\!80$$ p^{40} T^{28} - $$46\!\cdots\!55$$ p^{60} T^{32} - 102415973340331990 p^{80} T^{36} + p^{100} T^{40} $
	59	$ ( 1 + 1024012300 T^{2} + 623643773443709205 T^{4} + $$29\!\cdots\!00$$ T^{6} - $$11\!\cdots\!90$$ T^{8} - $$18\!\cdots\!00$$ T^{10} - $$11\!\cdots\!90$$ p^{10} T^{12} + $$29\!\cdots\!00$$ p^{20} T^{14} + 623643773443709205 p^{30} T^{16} + 1024012300 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	61	$ ( 1 - 40170 T + 2735635865 T^{2} - 80916443625720 T^{3} + 3964575852612866770 T^{4} - $$95\!\cdots\!24$$ T^{5} + 3964575852612866770 p^{5} T^{6} - 80916443625720 p^{10} T^{7} + 2735635865 p^{15} T^{8} - 40170 p^{20} T^{9} + p^{25} T^{10} )^{4} $
	67	$ ( 1 + 59368 T + 1762279712 T^{2} + 95006019983336 T^{3} + 9835273950700969141 T^{4} + $$41\!\cdots\!16$$ T^{5} + $$11\!\cdots\!44$$ T^{6} + $$59\!\cdots\!72$$ T^{7} + $$36\!\cdots\!86$$ T^{8} + $$11\!\cdots\!44$$ T^{9} + $$30\!\cdots\!76$$ T^{10} + $$11\!\cdots\!44$$ p^{5} T^{11} + $$36\!\cdots\!86$$ p^{10} T^{12} + $$59\!\cdots\!72$$ p^{15} T^{13} + $$11\!\cdots\!44$$ p^{20} T^{14} + $$41\!\cdots\!16$$ p^{25} T^{15} + 9835273950700969141 p^{30} T^{16} + 95006019983336 p^{35} T^{17} + 1762279712 p^{40} T^{18} + 59368 p^{45} T^{19} + p^{50} T^{20} )^{2} $
	71	$ ( 1 - 15342678910 T^{2} + $$10\!\cdots\!45$$ T^{4} - $$47\!\cdots\!20$$ T^{6} + $$14\!\cdots\!10$$ T^{8} - $$30\!\cdots\!52$$ T^{10} + $$14\!\cdots\!10$$ p^{10} T^{12} - $$47\!\cdots\!20$$ p^{20} T^{14} + $$10\!\cdots\!45$$ p^{30} T^{16} - 15342678910 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	73	$ ( 1 - 189550 T + 17964601250 T^{2} - 1343145510953150 T^{3} + 96689218656030546445 T^{4} - $$63\!\cdots\!00$$ T^{5} + $$36\!\cdots\!00$$ T^{6} - $$19\!\cdots\!00$$ T^{7} + $$10\!\cdots\!10$$ T^{8} - $$51\!\cdots\!00$$ T^{9} + $$23\!\cdots\!00$$ T^{10} - $$51\!\cdots\!00$$ p^{5} T^{11} + $$10\!\cdots\!10$$ p^{10} T^{12} - $$19\!\cdots\!00$$ p^{15} T^{13} + $$36\!\cdots\!00$$ p^{20} T^{14} - $$63\!\cdots\!00$$ p^{25} T^{15} + 96689218656030546445 p^{30} T^{16} - 1343145510953150 p^{35} T^{17} + 17964601250 p^{40} T^{18} - 189550 p^{45} T^{19} + p^{50} T^{20} )^{2} $
	79	$ ( 1 - 16427550610 T^{2} + $$14\!\cdots\!45$$ T^{4} - $$85\!\cdots\!20$$ T^{6} + $$38\!\cdots\!10$$ T^{8} - $$13\!\cdots\!52$$ T^{10} + $$38\!\cdots\!10$$ p^{10} T^{12} - $$85\!\cdots\!20$$ p^{20} T^{14} + $$14\!\cdots\!45$$ p^{30} T^{16} - 16427550610 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	83	$ 1 + 20495218598689857130 T^{4} + $$77\!\cdots\!65$$ T^{8} + $$78\!\cdots\!80$$ T^{12} + $$11\!\cdots\!70$$ T^{16} - $$11\!\cdots\!24$$ T^{20} + $$11\!\cdots\!70$$ p^{20} T^{24} + $$78\!\cdots\!80$$ p^{40} T^{28} + $$77\!\cdots\!65$$ p^{60} T^{32} + 20495218598689857130 p^{80} T^{36} + p^{100} T^{40} $
	89	$ ( 1 + 28606773250 T^{2} + $$42\!\cdots\!05$$ T^{4} + $$43\!\cdots\!00$$ T^{6} + $$34\!\cdots\!10$$ T^{8} + $$21\!\cdots\!00$$ T^{10} + $$34\!\cdots\!10$$ p^{10} T^{12} + $$43\!\cdots\!00$$ p^{20} T^{14} + $$42\!\cdots\!05$$ p^{30} T^{16} + 28606773250 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	97	$ ( 1 + 144114 T + 10384422498 T^{2} + 1456446247238418 T^{3} + $$28\!\cdots\!45$$ T^{4} + $$24\!\cdots\!84$$ T^{5} + $$16\!\cdots\!28$$ T^{6} + $$22\!\cdots\!48$$ T^{7} + $$36\!\cdots\!50$$ T^{8} + $$27\!\cdots\!24$$ T^{9} + $$18\!\cdots\!68$$ T^{10} + $$27\!\cdots\!24$$ p^{5} T^{11} + $$36\!\cdots\!50$$ p^{10} T^{12} + $$22\!\cdots\!48$$ p^{15} T^{13} + $$16\!\cdots\!28$$ p^{20} T^{14} + $$24\!\cdots\!84$$ p^{25} T^{15} + $$28\!\cdots\!45$$ p^{30} T^{16} + 1456446247238418 p^{35} T^{17} + 10384422498 p^{40} T^{18} + 144114 p^{45} T^{19} + p^{50} T^{20} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−3.80859091074757487850974936714, −3.72070512903641488314487143015, −3.63016556075728750582024353448, −3.42565588239157183584378026620, −3.30950020945997667447814007407, −3.21745763464534729500663300125, −3.06897558330073925493596810496, −2.89964463019255216295340710628, −2.70144085047970014138880036675, −2.67860239351804847991661297165, −2.50945097706297401115357708282, −2.21092941734293549700417657345, −2.13316325416151499761908298066, −2.03350019070634838243582521838, −1.92806029674340752063049904311, −1.69173723188088551616174609535, −1.56928736150139327533630206695, −1.49226839165426329983135249661, −1.19570835322245555588201040101, −0.996777642771152236137434202564, −0.74753169377280859737822659431, −0.71428018020541956705176682232, −0.67238192984367575080138654452, −0.53910298134700280199767692296, −0.083271461048344415205597559061, 0.083271461048344415205597559061, 0.53910298134700280199767692296, 0.67238192984367575080138654452, 0.71428018020541956705176682232, 0.74753169377280859737822659431, 0.996777642771152236137434202564, 1.19570835322245555588201040101, 1.49226839165426329983135249661, 1.56928736150139327533630206695, 1.69173723188088551616174609535, 1.92806029674340752063049904311, 2.03350019070634838243582521838, 2.13316325416151499761908298066, 2.21092941734293549700417657345, 2.50945097706297401115357708282, 2.67860239351804847991661297165, 2.70144085047970014138880036675, 2.89964463019255216295340710628, 3.06897558330073925493596810496, 3.21745763464534729500663300125, 3.30950020945997667447814007407, 3.42565588239157183584378026620, 3.63016556075728750582024353448, 3.72070512903641488314487143015, 3.80859091074757487850974936714

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 $\F_{q}$
Belyi maps

\(L(3)\)	\(\approx\)	\(5.079886084\)
\(L(\frac12)\)	\(\approx\)	\(5.079886084\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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