LMFDB - L-function 32-450e16-1.1-c5e16-0-1

Dirichlet series

L(s) = 1

+ 528·7^-s − 192·13^-s − 1.02e3·16^-s + 1.30e4·31^-s + 4.73e4·37^-s + 5.54e4·43^-s + 1.39e5·49^-s + 2.84e4·61^-s − 2.42e5·67^-s + 4.30e5·73^-s − 1.01e5·91^-s − 4.57e5·97^-s + 1.52e6·103^-s − 5.40e5·112^-s + 1.07e6·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 1.84e4·169^-s + 173^-s + 179^-s + ⋯

L(s) = 1

+ 4.07·7^-s − 0.315·13^-s − 16^-s + 2.43·31^-s + 5.68·37^-s + 4.57·43^-s + 8.29·49^-s + 0.977·61^-s − 6.59·67^-s + 9.46·73^-s − 1.28·91^-s − 4.93·97^-s + 14.2·103^-s − 4.07·112^-s + 6.68·121^-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 + 0.0496·169^-s + 2.54e−6·173^-s + 2.33e−6·179^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$32$
Conductor:	$2^{16} \cdot 3^{32} \cdot 5^{32}$
Sign:	$1$
Analytic conductor:	$5.41962\times 10^{29}$
Root analytic conductor:	$8.49545$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(32,\ 2^{16} \cdot 3^{32} \cdot 5^{32} ,\ ( \ : [5/2]^{16} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$1012.577677$
$L(\frac12)$	$\approx$	$1012.577677$
$L(\frac{7}{2})$		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} )^{4} $
	3	$ 1 $
	5	$ 1 $
good	7	$ ( 1 - 264 T + 34848 T^{2} - 779808 T^{3} - 945939794 T^{4} + 161509856616 T^{5} - 191233509120 p^{2} T^{6} - 202683166028712 p T^{7} + 408268569929291859 T^{8} - 202683166028712 p^{6} T^{9} - 191233509120 p^{12} T^{10} + 161509856616 p^{15} T^{11} - 945939794 p^{20} T^{12} - 779808 p^{25} T^{13} + 34848 p^{30} T^{14} - 264 p^{35} T^{15} + p^{40} T^{16} )^{2} $
	11	$ ( 1 - 538096 T^{2} + 13175210060 p T^{4} - 31717938071112784 T^{6} + $$58\!\cdots\!94$$ T^{8} - 31717938071112784 p^{10} T^{10} + 13175210060 p^{21} T^{12} - 538096 p^{30} T^{14} + p^{40} T^{16} )^{2} $
	13	$ ( 1 + 96 T + 4608 T^{2} - 305562432 T^{3} - 225729084242 T^{4} - 58338192184992 T^{5} + 249253805297664 p^{2} T^{6} + 1483945455354192288 p T^{7} + $$42\!\cdots\!43$$ T^{8} + 1483945455354192288 p^{6} T^{9} + 249253805297664 p^{12} T^{10} - 58338192184992 p^{15} T^{11} - 225729084242 p^{20} T^{12} - 305562432 p^{25} T^{13} + 4608 p^{30} T^{14} + 96 p^{35} T^{15} + p^{40} T^{16} )^{2} $
	17	$ 1 + 4453513364024 T^{4} + $$20\!\cdots\!60$$ p T^{8} - $$11\!\cdots\!64$$ T^{12} - $$32\!\cdots\!66$$ T^{16} - $$11\!\cdots\!64$$ p^{20} T^{20} + $$20\!\cdots\!60$$ p^{41} T^{24} + 4453513364024 p^{60} T^{28} + p^{80} T^{32} $
	19	$ ( 1 - 7345156 T^{2} + 26976805996330 T^{4} - 69970525905267916144 T^{6} + $$16\!\cdots\!99$$ T^{8} - 69970525905267916144 p^{10} T^{10} + 26976805996330 p^{20} T^{12} - 7345156 p^{30} T^{14} + p^{40} T^{16} )^{2} $
	23	$ 1 + 136282946866808 T^{4} + $$10\!\cdots\!28$$ T^{8} + $$52\!\cdots\!56$$ T^{12} + $$22\!\cdots\!70$$ T^{16} + $$52\!\cdots\!56$$ p^{20} T^{20} + $$10\!\cdots\!28$$ p^{40} T^{24} + 136282946866808 p^{60} T^{28} + p^{80} T^{32} $
	29	$ ( 1 + 109827904 T^{2} + 204313789713140 p T^{4} + $$20\!\cdots\!16$$ T^{6} + $$49\!\cdots\!94$$ T^{8} + $$20\!\cdots\!16$$ p^{10} T^{10} + 204313789713140 p^{21} T^{12} + 109827904 p^{30} T^{14} + p^{40} T^{16} )^{2} $
	31	$ ( 1 - 3256 T + 107067130 T^{2} - 243362140144 T^{3} + 4458532261264699 T^{4} - 243362140144 p^{5} T^{5} + 107067130 p^{10} T^{6} - 3256 p^{15} T^{7} + p^{20} T^{8} )^{4} $
	37	$ ( 1 - 23664 T + 279992448 T^{2} - 3307606234608 T^{3} + 47316915450833956 T^{4} - $$52\!\cdots\!84$$ T^{5} + $$47\!\cdots\!20$$ T^{6} - $$45\!\cdots\!84$$ T^{7} + $$41\!\cdots\!34$$ T^{8} - $$45\!\cdots\!84$$ p^{5} T^{9} + $$47\!\cdots\!20$$ p^{10} T^{10} - $$52\!\cdots\!84$$ p^{15} T^{11} + 47316915450833956 p^{20} T^{12} - 3307606234608 p^{25} T^{13} + 279992448 p^{30} T^{14} - 23664 p^{35} T^{15} + p^{40} T^{16} )^{2} $
	41	$ ( 1 - 571646536 T^{2} + 169641938060392540 T^{4} - $$32\!\cdots\!24$$ T^{6} + $$44\!\cdots\!14$$ T^{8} - $$32\!\cdots\!24$$ p^{10} T^{10} + 169641938060392540 p^{20} T^{12} - 571646536 p^{30} T^{14} + p^{40} T^{16} )^{2} $
	43	$ ( 1 - 27720 T + 384199200 T^{2} - 7867055856960 T^{3} + 85720027706446846 T^{4} + 99917127454959929160 T^{5} - $$47\!\cdots\!00$$ T^{6} + $$16\!\cdots\!80$$ T^{7} - $$37\!\cdots\!69$$ T^{8} + $$16\!\cdots\!80$$ p^{5} T^{9} - $$47\!\cdots\!00$$ p^{10} T^{10} + 99917127454959929160 p^{15} T^{11} + 85720027706446846 p^{20} T^{12} - 7867055856960 p^{25} T^{13} + 384199200 p^{30} T^{14} - 27720 p^{35} T^{15} + p^{40} T^{16} )^{2} $
	47	$ 1 + 3996859080844936 p T^{4} + $$14\!\cdots\!28$$ T^{8} + $$12\!\cdots\!52$$ p T^{12} + $$21\!\cdots\!70$$ T^{16} + $$12\!\cdots\!52$$ p^{21} T^{20} + $$14\!\cdots\!28$$ p^{40} T^{24} + 3996859080844936 p^{61} T^{28} + p^{80} T^{32} $
	53	$ 1 - 1173284725784529592 T^{4} + $$63\!\cdots\!28$$ T^{8} - $$20\!\cdots\!44$$ T^{12} + $$44\!\cdots\!70$$ T^{16} - $$20\!\cdots\!44$$ p^{20} T^{20} + $$63\!\cdots\!28$$ p^{40} T^{24} - 1173284725784529592 p^{60} T^{28} + p^{80} T^{32} $
	59	$ ( 1 + 2222735920 T^{2} + 2663290134105595204 T^{4} + $$24\!\cdots\!60$$ T^{6} + $$18\!\cdots\!06$$ T^{8} + $$24\!\cdots\!60$$ p^{10} T^{10} + 2663290134105595204 p^{20} T^{12} + 2222735920 p^{30} T^{14} + p^{40} T^{16} )^{2} $
	61	$ ( 1 - 7100 T + 1700268754 T^{2} + 10725361061200 T^{3} + 1428617355565068331 T^{4} + 10725361061200 p^{5} T^{5} + 1700268754 p^{10} T^{6} - 7100 p^{15} T^{7} + p^{20} T^{8} )^{4} $
	67	$ ( 1 + 121128 T + 7335996192 T^{2} + 377443661453376 T^{3} + 14531672059354841758 T^{4} + $$29\!\cdots\!44$$ T^{5} + $$52\!\cdots\!84$$ T^{6} - $$35\!\cdots\!92$$ T^{7} - $$21\!\cdots\!57$$ T^{8} - $$35\!\cdots\!92$$ p^{5} T^{9} + $$52\!\cdots\!84$$ p^{10} T^{10} + $$29\!\cdots\!44$$ p^{15} T^{11} + 14531672059354841758 p^{20} T^{12} + 377443661453376 p^{25} T^{13} + 7335996192 p^{30} T^{14} + 121128 p^{35} T^{15} + p^{40} T^{16} )^{2} $
	71	$ ( 1 - 6873784120 T^{2} + 26078175028590999004 T^{4} - $$71\!\cdots\!60$$ T^{6} + $$14\!\cdots\!06$$ T^{8} - $$71\!\cdots\!60$$ p^{10} T^{10} + 26078175028590999004 p^{20} T^{12} - 6873784120 p^{30} T^{14} + p^{40} T^{16} )^{2} $
	73	$ ( 1 - 215472 T + 23214091392 T^{2} - 1796035546968816 T^{3} + $$11\!\cdots\!76$$ T^{4} - $$58\!\cdots\!92$$ T^{5} + $$27\!\cdots\!60$$ T^{6} - $$11\!\cdots\!48$$ T^{7} + $$52\!\cdots\!74$$ T^{8} - $$11\!\cdots\!48$$ p^{5} T^{9} + $$27\!\cdots\!60$$ p^{10} T^{10} - $$58\!\cdots\!92$$ p^{15} T^{11} + $$11\!\cdots\!76$$ p^{20} T^{12} - 1796035546968816 p^{25} T^{13} + 23214091392 p^{30} T^{14} - 215472 p^{35} T^{15} + p^{40} T^{16} )^{2} $
	79	$ ( 1 - 8840349736 T^{2} + 56636855401030130140 T^{4} - $$25\!\cdots\!24$$ T^{6} + $$89\!\cdots\!14$$ T^{8} - $$25\!\cdots\!24$$ p^{10} T^{10} + 56636855401030130140 p^{20} T^{12} - 8840349736 p^{30} T^{14} + p^{40} T^{16} )^{2} $
	83	$ 1 - 61334510299692769096 T^{4} + $$19\!\cdots\!60$$ T^{8} - $$47\!\cdots\!84$$ T^{12} + $$84\!\cdots\!94$$ T^{16} - $$47\!\cdots\!84$$ p^{20} T^{20} + $$19\!\cdots\!60$$ p^{40} T^{24} - 61334510299692769096 p^{60} T^{28} + p^{80} T^{32} $
	89	$ ( 1 + 9080488360 T^{2} + 61529078298336754204 T^{4} + $$45\!\cdots\!80$$ T^{6} + $$26\!\cdots\!06$$ T^{8} + $$45\!\cdots\!80$$ p^{10} T^{10} + 61529078298336754204 p^{20} T^{12} + 9080488360 p^{30} T^{14} + p^{40} T^{16} )^{2} $
	97	$ ( 1 + 228576 T + 26123493888 T^{2} + 2780148939051072 T^{3} + $$24\!\cdots\!86$$ T^{4} + $$17\!\cdots\!76$$ T^{5} + $$14\!\cdots\!00$$ T^{6} + $$14\!\cdots\!96$$ T^{7} + $$13\!\cdots\!19$$ T^{8} + $$14\!\cdots\!96$$ p^{5} T^{9} + $$14\!\cdots\!00$$ p^{10} T^{10} + $$17\!\cdots\!76$$ p^{15} T^{11} + $$24\!\cdots\!86$$ p^{20} T^{12} + 2780148939051072 p^{25} T^{13} + 26123493888 p^{30} T^{14} + 228576 p^{35} T^{15} + p^{40} T^{16} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−2.19338459952226065204815971342, −2.12367778342954439753567852657, −1.98843174937285472882003415445, −1.94078259036216266452430301106, −1.77694289958487158539846705285, −1.74606800297374680765227328830, −1.72189534625187524362505829306, −1.66532719152106014221194358014, −1.44328799172644842837334907521, −1.25825681409293564650398178518, −1.25731669275470770678071483568, −1.14056479340578642001246083981, −1.10982905973530627148954820748, −1.05915917745943889675923354193, −0.962235880404212810895910161158, −0.940569567668271333858021144027, −0.77128941195205316534657454850, −0.68078773405721461219423784954, −0.67591130455786755124519104060, −0.66019374126202243136758812408, −0.48571818754112702660667151182, −0.35703893432034945140823335646, −0.34513961071640972107927803671, −0.23827907368995574645745774975, −0.22467282501031599211427066684, 0.22467282501031599211427066684, 0.23827907368995574645745774975, 0.34513961071640972107927803671, 0.35703893432034945140823335646, 0.48571818754112702660667151182, 0.66019374126202243136758812408, 0.67591130455786755124519104060, 0.68078773405721461219423784954, 0.77128941195205316534657454850, 0.940569567668271333858021144027, 0.962235880404212810895910161158, 1.05915917745943889675923354193, 1.10982905973530627148954820748, 1.14056479340578642001246083981, 1.25731669275470770678071483568, 1.25825681409293564650398178518, 1.44328799172644842837334907521, 1.66532719152106014221194358014, 1.72189534625187524362505829306, 1.74606800297374680765227328830, 1.77694289958487158539846705285, 1.94078259036216266452430301106, 1.98843174937285472882003415445, 2.12367778342954439753567852657, 2.19338459952226065204815971342

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1012.577677\)
\(L(\frac12)\)	\(\approx\)	\(1012.577677\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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