LMFDB - L-function 16-18e8-1.1-c46e8-0-0

Dirichlet series

L(s) = 1

− 1.40e14·4^-s + 5.49e19·7^-s + 1.16e25·13^-s + 1.23e28·16^-s − 6.44e29·19^-s + 3.60e32·25^-s − 7.73e33·28^-s − 8.99e33·31^-s − 3.85e36·37^-s − 4.32e37·43^-s − 1.07e39·49^-s − 1.63e39·52^-s + 3.07e41·61^-s − 8.71e41·64^-s + 6.42e42·67^-s + 2.05e43·73^-s + 9.07e43·76^-s + 5.05e43·79^-s + 6.39e44·91^-s + 6.91e45·97^-s − 5.06e46·100^-s − 2.65e46·103^-s + 4.45e47·109^-s + 6.80e47·112^-s + 3.22e48·121^-s + 1.26e48·124^-s + 127^-s + ⋯

L(s) = 1

− 2·4^-s + 2.00·7^-s + 0.278·13^-s + 5/2·16^-s − 2.50·19^-s + 2.53·25^-s − 4.01·28^-s − 0.449·31^-s − 3.29·37^-s − 1.16·43^-s − 1.43·49^-s − 0.557·52^-s + 2.66·61^-s − 5/2·64^-s + 6.42·67^-s + 2.85·73^-s + 5.00·76^-s + 1.14·79^-s + 0.559·91^-s + 1.39·97^-s − 5.06·100^-s − 1.34·103^-s + 6.13·109^-s + 5.02·112^-s + 4.02·121^-s + 0.899·124^-s − 5.02·133^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(47-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+23)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$2^{8} \cdot 3^{16}$
Sign:	$1$
Analytic conductor:	$1.14681\times 10^{19}$
Root analytic conductor:	$15.5316$
Motivic weight:	$46$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{8} \cdot 3^{16} ,\ ( \ : [23]^{8} ),\ 1 )$

Particular Values

$L(\frac{47}{2})$	$\approx$	$11.28634342$
$L(\frac12)$	$\approx$	$11.28634342$
$L(24)$		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^{45} T^{2} )^{4} $
	3	$ 1 $
good	5	$ 1 - $$36\!\cdots\!92$$ T^{2} + $$14\!\cdots\!36$$ p^{5} T^{4} - $$37\!\cdots\!52$$ p^{10} T^{6} + $$19\!\cdots\!66$$ p^{22} T^{8} - $$37\!\cdots\!52$$ p^{102} T^{10} + $$14\!\cdots\!36$$ p^{189} T^{12} - $$36\!\cdots\!92$$ p^{276} T^{14} + p^{368} T^{16} $
	7	$ ( 1 - 80154419535090608 p^{3} T + $$48\!\cdots\!04$$ p^{3} T^{2} - $$88\!\cdots\!92$$ p^{8} T^{3} + $$98\!\cdots\!70$$ p^{12} T^{4} - $$88\!\cdots\!92$$ p^{54} T^{5} + $$48\!\cdots\!04$$ p^{95} T^{6} - 80154419535090608 p^{141} T^{7} + p^{184} T^{8} )^{2} $
	11	$ 1 - $$32\!\cdots\!16$$ T^{2} + $$40\!\cdots\!80$$ p^{4} T^{4} - $$31\!\cdots\!04$$ p^{9} T^{6} + $$18\!\cdots\!74$$ p^{14} T^{8} - $$31\!\cdots\!04$$ p^{101} T^{10} + $$40\!\cdots\!80$$ p^{188} T^{12} - $$32\!\cdots\!16$$ p^{276} T^{14} + p^{368} T^{16} $
	13	$ ( 1 - $$58\!\cdots\!12$$ T + $$31\!\cdots\!80$$ p T^{2} - $$23\!\cdots\!56$$ p^{3} T^{3} + $$16\!\cdots\!46$$ p^{6} T^{4} - $$23\!\cdots\!56$$ p^{49} T^{5} + $$31\!\cdots\!80$$ p^{93} T^{6} - $$58\!\cdots\!12$$ p^{138} T^{7} + p^{184} T^{8} )^{2} $
	17	$ 1 - $$98\!\cdots\!24$$ T^{2} + $$56\!\cdots\!60$$ T^{4} - $$63\!\cdots\!04$$ p^{2} T^{6} + $$26\!\cdots\!46$$ p^{6} T^{8} - $$63\!\cdots\!04$$ p^{94} T^{10} + $$56\!\cdots\!60$$ p^{184} T^{12} - $$98\!\cdots\!24$$ p^{276} T^{14} + p^{368} T^{16} $
	19	$ ( 1 + $$16\!\cdots\!16$$ p T + $$34\!\cdots\!80$$ p^{2} T^{2} + $$22\!\cdots\!56$$ p^{4} T^{3} + $$23\!\cdots\!94$$ p^{6} T^{4} + $$22\!\cdots\!56$$ p^{50} T^{5} + $$34\!\cdots\!80$$ p^{94} T^{6} + $$16\!\cdots\!16$$ p^{139} T^{7} + p^{184} T^{8} )^{2} $
	23	$ 1 - $$72\!\cdots\!20$$ p T^{2} + $$27\!\cdots\!96$$ p^{2} T^{4} - $$69\!\cdots\!40$$ p^{3} T^{6} + $$14\!\cdots\!06$$ p^{4} T^{8} - $$69\!\cdots\!40$$ p^{95} T^{10} + $$27\!\cdots\!96$$ p^{186} T^{12} - $$72\!\cdots\!20$$ p^{277} T^{14} + p^{368} T^{16} $
	29	$ 1 - $$94\!\cdots\!76$$ T^{2} + $$38\!\cdots\!80$$ T^{4} - $$11\!\cdots\!24$$ p^{2} T^{6} + $$26\!\cdots\!74$$ p^{4} T^{8} - $$11\!\cdots\!24$$ p^{94} T^{10} + $$38\!\cdots\!80$$ p^{184} T^{12} - $$94\!\cdots\!76$$ p^{276} T^{14} + p^{368} T^{16} $
	31	$ ( 1 + $$44\!\cdots\!44$$ T + $$46\!\cdots\!00$$ p T^{2} + $$53\!\cdots\!56$$ p^{2} T^{3} + $$27\!\cdots\!34$$ p^{3} T^{4} + $$53\!\cdots\!56$$ p^{48} T^{5} + $$46\!\cdots\!00$$ p^{93} T^{6} + $$44\!\cdots\!44$$ p^{138} T^{7} + p^{184} T^{8} )^{2} $
	37	$ ( 1 + $$19\!\cdots\!36$$ T + $$94\!\cdots\!88$$ p^{2} T^{2} - $$10\!\cdots\!48$$ p^{2} T^{3} - $$36\!\cdots\!10$$ p^{3} T^{4} - $$10\!\cdots\!48$$ p^{48} T^{5} + $$94\!\cdots\!88$$ p^{94} T^{6} + $$19\!\cdots\!36$$ p^{138} T^{7} + p^{184} T^{8} )^{2} $
	41	$ 1 - $$68\!\cdots\!76$$ T^{2} + $$11\!\cdots\!40$$ p^{2} T^{4} - $$11\!\cdots\!64$$ p^{4} T^{6} + $$10\!\cdots\!94$$ p^{6} T^{8} - $$11\!\cdots\!64$$ p^{96} T^{10} + $$11\!\cdots\!40$$ p^{186} T^{12} - $$68\!\cdots\!76$$ p^{276} T^{14} + p^{368} T^{16} $
	43	$ ( 1 + $$50\!\cdots\!40$$ p T + $$16\!\cdots\!04$$ p^{2} T^{2} + $$80\!\cdots\!20$$ p^{3} T^{3} + $$17\!\cdots\!06$$ p^{4} T^{4} + $$80\!\cdots\!20$$ p^{49} T^{5} + $$16\!\cdots\!04$$ p^{94} T^{6} + $$50\!\cdots\!40$$ p^{139} T^{7} + p^{184} T^{8} )^{2} $
	47	$ 1 - $$33\!\cdots\!32$$ T^{2} + $$12\!\cdots\!84$$ p T^{4} - $$64\!\cdots\!84$$ T^{6} + $$58\!\cdots\!70$$ T^{8} - $$64\!\cdots\!84$$ p^{92} T^{10} + $$12\!\cdots\!84$$ p^{185} T^{12} - $$33\!\cdots\!32$$ p^{276} T^{14} + p^{368} T^{16} $
	53	$ 1 - $$76\!\cdots\!60$$ T^{2} + $$54\!\cdots\!64$$ T^{4} - $$14\!\cdots\!80$$ T^{6} + $$18\!\cdots\!86$$ T^{8} - $$14\!\cdots\!80$$ p^{92} T^{10} + $$54\!\cdots\!64$$ p^{184} T^{12} - $$76\!\cdots\!60$$ p^{276} T^{14} + p^{368} T^{16} $
	59	$ 1 - $$20\!\cdots\!60$$ T^{2} + $$28\!\cdots\!24$$ T^{4} - $$48\!\cdots\!80$$ T^{6} + $$34\!\cdots\!66$$ T^{8} - $$48\!\cdots\!80$$ p^{92} T^{10} + $$28\!\cdots\!24$$ p^{184} T^{12} - $$20\!\cdots\!60$$ p^{276} T^{14} + p^{368} T^{16} $
	61	$ ( 1 - $$15\!\cdots\!60$$ T + $$52\!\cdots\!44$$ T^{2} - $$54\!\cdots\!80$$ T^{3} + $$10\!\cdots\!26$$ T^{4} - $$54\!\cdots\!80$$ p^{46} T^{5} + $$52\!\cdots\!44$$ p^{92} T^{6} - $$15\!\cdots\!60$$ p^{138} T^{7} + p^{184} T^{8} )^{2} $
	67	$ ( 1 - $$32\!\cdots\!52$$ T + $$74\!\cdots\!40$$ T^{2} - $$10\!\cdots\!52$$ T^{3} + $$13\!\cdots\!34$$ T^{4} - $$10\!\cdots\!52$$ p^{46} T^{5} + $$74\!\cdots\!40$$ p^{92} T^{6} - $$32\!\cdots\!52$$ p^{138} T^{7} + p^{184} T^{8} )^{2} $
	71	$ 1 - $$33\!\cdots\!20$$ T^{2} + $$91\!\cdots\!64$$ T^{4} - $$18\!\cdots\!60$$ T^{6} + $$30\!\cdots\!86$$ T^{8} - $$18\!\cdots\!60$$ p^{92} T^{10} + $$91\!\cdots\!64$$ p^{184} T^{12} - $$33\!\cdots\!20$$ p^{276} T^{14} + p^{368} T^{16} $
	73	$ ( 1 - $$10\!\cdots\!16$$ T + $$12\!\cdots\!52$$ T^{2} - $$92\!\cdots\!28$$ T^{3} + $$91\!\cdots\!70$$ T^{4} - $$92\!\cdots\!28$$ p^{46} T^{5} + $$12\!\cdots\!52$$ p^{92} T^{6} - $$10\!\cdots\!16$$ p^{138} T^{7} + p^{184} T^{8} )^{2} $
	79	$ ( 1 - $$25\!\cdots\!76$$ T + $$72\!\cdots\!00$$ T^{2} - $$14\!\cdots\!24$$ T^{3} + $$20\!\cdots\!14$$ T^{4} - $$14\!\cdots\!24$$ p^{46} T^{5} + $$72\!\cdots\!00$$ p^{92} T^{6} - $$25\!\cdots\!76$$ p^{138} T^{7} + p^{184} T^{8} )^{2} $
	83	$ 1 - $$10\!\cdots\!04$$ T^{2} + $$51\!\cdots\!00$$ T^{4} - $$23\!\cdots\!24$$ p^{2} T^{6} + $$36\!\cdots\!34$$ T^{8} - $$23\!\cdots\!24$$ p^{94} T^{10} + $$51\!\cdots\!00$$ p^{184} T^{12} - $$10\!\cdots\!04$$ p^{276} T^{14} + p^{368} T^{16} $
	89	$ 1 - $$83\!\cdots\!40$$ T^{2} + $$55\!\cdots\!84$$ T^{4} - $$27\!\cdots\!20$$ T^{6} + $$15\!\cdots\!46$$ T^{8} - $$27\!\cdots\!20$$ p^{92} T^{10} + $$55\!\cdots\!84$$ p^{184} T^{12} - $$83\!\cdots\!40$$ p^{276} T^{14} + p^{368} T^{16} $
	97	$ ( 1 - $$34\!\cdots\!04$$ T + $$64\!\cdots\!72$$ T^{2} - $$19\!\cdots\!52$$ T^{3} + $$20\!\cdots\!70$$ T^{4} - $$19\!\cdots\!52$$ p^{46} T^{5} + $$64\!\cdots\!72$$ p^{92} T^{6} - $$34\!\cdots\!04$$ p^{138} T^{7} + p^{184} T^{8} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−3.54701079176766549982551855149, −3.41840678189899455313646356464, −3.41246880674351665583976105361, −3.38628420621448629209539163558, −3.21109497162091663384343101060, −2.53377826220150837452319348245, −2.50181098457380270312940387228, −2.36452764388166249289933905074, −2.35343101153502566940172479886, −2.24267230042029865700597521378, −2.09546924906032673939551798380, −1.91136126407243455538636438122, −1.81444346370195904376060117783, −1.66888244296464020819013451125, −1.45539208351605054916523271375, −1.24999027268478797635613147330, −1.22246664512331654100250089523, −1.13450638955377868111881265737, −0.78649929658657519471309278610, −0.72437799735641009235735983478, −0.69181960667538650289652253420, −0.50995018840754142438846779755, −0.37444998173019020209926740041, −0.26309427848446902908801041183, −0.16175185994799297480944083751, 0.16175185994799297480944083751, 0.26309427848446902908801041183, 0.37444998173019020209926740041, 0.50995018840754142438846779755, 0.69181960667538650289652253420, 0.72437799735641009235735983478, 0.78649929658657519471309278610, 1.13450638955377868111881265737, 1.22246664512331654100250089523, 1.24999027268478797635613147330, 1.45539208351605054916523271375, 1.66888244296464020819013451125, 1.81444346370195904376060117783, 1.91136126407243455538636438122, 2.09546924906032673939551798380, 2.24267230042029865700597521378, 2.35343101153502566940172479886, 2.36452764388166249289933905074, 2.50181098457380270312940387228, 2.53377826220150837452319348245, 3.21109497162091663384343101060, 3.38628420621448629209539163558, 3.41246880674351665583976105361, 3.41840678189899455313646356464, 3.54701079176766549982551855149

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{47}{2})\)	\(\approx\)	\(11.28634342\)
\(L(\frac12)\)	\(\approx\)	\(11.28634342\)
\(L(24)\)		not available
\(L(1)\)		not available

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