LMFDB - L-function 8-3e8-1.1-c33e4-0-0

Dirichlet series

L(s) = 1

− 1.35e8·4^-s − 1.28e14·7^-s − 4.46e18·13^-s − 1.47e20·16^-s − 5.36e21·19^-s + 7.08e22·25^-s + 1.74e22·28^-s − 8.79e24·31^-s + 2.63e26·37^-s − 2.61e27·43^-s + 6.63e27·49^-s + 6.04e26·52^-s − 3.26e29·61^-s + 2.99e28·64^-s − 2.76e30·67^-s − 1.18e31·73^-s + 7.26e29·76^-s − 3.27e31·79^-s + 5.75e32·91^-s + 2.28e32·97^-s − 9.59e30·100^-s + 1.57e33·103^-s + 1.48e34·109^-s + 1.90e34·112^-s − 7.25e34·121^-s + 1.19e33·124^-s + 127^-s + ⋯

L(s) = 1

− 0.0157·4^-s − 1.46·7^-s − 1.85·13^-s − 1.99·16^-s − 4.26·19^-s + 0.608·25^-s + 0.0231·28^-s − 2.17·31^-s + 3.51·37^-s − 2.92·43^-s + 0.857·49^-s + 0.0293·52^-s − 1.13·61^-s + 0.0472·64^-s − 2.04·67^-s − 2.12·73^-s + 0.0673·76^-s − 1.60·79^-s + 2.72·91^-s + 0.377·97^-s − 0.00959·100^-s + 0.966·103^-s + 3.59·109^-s + 2.93·112^-s − 3.12·121^-s + 0.0342·124^-s + 6.26·133^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+33/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$6561$ = $3^{8}$
Sign:	$1$
Analytic conductor:	$1.48570\times 10^{7}$
Root analytic conductor:	$7.87937$
Motivic weight:	$33$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$4$
Selberg data:	$(8,\ 6561,\ (\ :33/2, 33/2, 33/2, 33/2),\ 1)$

Particular Values

$L(17)$	$=$	$0$
$L(\frac12)$	$=$	$0$
$L(\frac{35}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$\Gal(F_p)$	$F_p(T)$
bad	3		$ 1 $
good	2	$C_2^2 \wr C_2$	$ 1 + 4233619 p^{5} T^{2} + 281369109913113 p^{19} T^{4} + 4233619 p^{71} T^{6} + p^{132} T^{8} $
	5	$C_2^2 \wr C_2$	$ 1 - $$56\!\cdots\!84$$ p^{3} T^{2} + $$10\!\cdots\!62$$ p^{10} T^{4} - $$56\!\cdots\!84$$ p^{69} T^{6} + p^{132} T^{8} $
	7	$D_{4}$	$ ( 1 + 9211874143160 p T + $$12\!\cdots\!14$$ p^{4} T^{2} + 9211874143160 p^{34} T^{3} + p^{66} T^{4} )^{2} $
	11	$C_2^2 \wr C_2$	$ 1 + $$65\!\cdots\!84$$ p T^{2} + $$17\!\cdots\!86$$ p^{3} T^{4} + $$65\!\cdots\!84$$ p^{67} T^{6} + p^{132} T^{8} $
	13	$D_{4}$	$ ( 1 + 13199077108980380 p^{2} T + $$47\!\cdots\!98$$ p^{3} T^{2} + 13199077108980380 p^{35} T^{3} + p^{66} T^{4} )^{2} $
	17	$C_2^2 \wr C_2$	$ 1 + $$13\!\cdots\!08$$ T^{2} + $$26\!\cdots\!86$$ p^{2} T^{4} + $$13\!\cdots\!08$$ p^{66} T^{6} + p^{132} T^{8} $
	19	$D_{4}$	$ ( 1 + $$26\!\cdots\!52$$ T + $$23\!\cdots\!26$$ p T^{2} + $$26\!\cdots\!52$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	23	$C_2^2 \wr C_2$	$ 1 + $$29\!\cdots\!72$$ T^{2} + $$68\!\cdots\!06$$ p^{2} T^{4} + $$29\!\cdots\!72$$ p^{66} T^{6} + p^{132} T^{8} $
	29	$C_2^2 \wr C_2$	$ 1 + $$19\!\cdots\!16$$ p^{2} T^{2} + $$29\!\cdots\!46$$ p^{4} T^{4} + $$19\!\cdots\!16$$ p^{68} T^{6} + p^{132} T^{8} $
	31	$D_{4}$	$ ( 1 + $$14\!\cdots\!84$$ p T + $$14\!\cdots\!26$$ p^{2} T^{2} + $$14\!\cdots\!84$$ p^{34} T^{3} + p^{66} T^{4} )^{2} $
	37	$D_{4}$	$ ( 1 - $$13\!\cdots\!40$$ T + $$15\!\cdots\!94$$ T^{2} - $$13\!\cdots\!40$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	41	$C_2^2 \wr C_2$	$ 1 + $$31\!\cdots\!84$$ T^{2} + $$64\!\cdots\!46$$ T^{4} + $$31\!\cdots\!84$$ p^{66} T^{6} + p^{132} T^{8} $
	43	$D_{4}$	$ ( 1 + $$13\!\cdots\!20$$ T + $$20\!\cdots\!86$$ T^{2} + $$13\!\cdots\!20$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	47	$C_2^2 \wr C_2$	$ 1 + $$27\!\cdots\!48$$ T^{2} + $$38\!\cdots\!34$$ T^{4} + $$27\!\cdots\!48$$ p^{66} T^{6} + p^{132} T^{8} $
	53	$C_2^2 \wr C_2$	$ 1 + $$81\!\cdots\!52$$ T^{2} + $$14\!\cdots\!34$$ T^{4} + $$81\!\cdots\!52$$ p^{66} T^{6} + p^{132} T^{8} $
	59	$C_2^2 \wr C_2$	$ 1 + $$27\!\cdots\!16$$ T^{2} + $$32\!\cdots\!46$$ T^{4} + $$27\!\cdots\!16$$ p^{66} T^{6} + p^{132} T^{8} $
	61	$D_{4}$	$ ( 1 + $$16\!\cdots\!56$$ T - $$40\!\cdots\!54$$ T^{2} + $$16\!\cdots\!56$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	67	$D_{4}$	$ ( 1 + $$13\!\cdots\!20$$ T + $$74\!\cdots\!74$$ T^{2} + $$13\!\cdots\!20$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	71	$C_2^2 \wr C_2$	$ 1 + $$24\!\cdots\!44$$ T^{2} + $$38\!\cdots\!26$$ T^{4} + $$24\!\cdots\!44$$ p^{66} T^{6} + p^{132} T^{8} $
	73	$D_{4}$	$ ( 1 + $$59\!\cdots\!60$$ T + $$65\!\cdots\!66$$ T^{2} + $$59\!\cdots\!60$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	79	$D_{4}$	$ ( 1 + $$16\!\cdots\!28$$ T + $$65\!\cdots\!74$$ T^{2} + $$16\!\cdots\!28$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
	83	$C_2^2 \wr C_2$	$ 1 + $$57\!\cdots\!12$$ T^{2} + $$15\!\cdots\!74$$ T^{4} + $$57\!\cdots\!12$$ p^{66} T^{6} + p^{132} T^{8} $
	89	$C_2^2 \wr C_2$	$ 1 + $$56\!\cdots\!76$$ T^{2} + $$14\!\cdots\!66$$ T^{4} + $$56\!\cdots\!76$$ p^{66} T^{6} + p^{132} T^{8} $
	97	$D_{4}$	$ ( 1 - $$11\!\cdots\!80$$ T + $$68\!\cdots\!54$$ T^{2} - $$11\!\cdots\!80$$ p^{33} T^{3} + p^{66} T^{4} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−10.04153719546873848783866044477, −9.350870869149623534195643683669, −8.956283450880080548424849581725, −8.930716553045993118793441606994, −8.663142961690271467687070138919, −7.921671808095048584394351994062, −7.64401219241868272281651022471, −7.22355409695059130795967375694, −6.77422879319909121148383826708, −6.59584217119164729742109454332, −6.34775997206784202620697152396, −5.94401091686598080131160557360, −5.63380289281512340050766112642, −4.74071634771013428895914220469, −4.70949590653784026576957934979, −4.34917852639489657393858188843, −4.21673055968675756209845574761, −3.61461452219564571347141895945, −2.96877065665955686043938947203, −2.94635997388534091725912282642, −2.27595698912439494364910107973, −2.25525550205861878086315311277, −1.91130348888562443203650959162, −1.47270280921919547369973333303, −0.792469089790200931181645712899, 0, 0, 0, 0, 0.792469089790200931181645712899, 1.47270280921919547369973333303, 1.91130348888562443203650959162, 2.25525550205861878086315311277, 2.27595698912439494364910107973, 2.94635997388534091725912282642, 2.96877065665955686043938947203, 3.61461452219564571347141895945, 4.21673055968675756209845574761, 4.34917852639489657393858188843, 4.70949590653784026576957934979, 4.74071634771013428895914220469, 5.63380289281512340050766112642, 5.94401091686598080131160557360, 6.34775997206784202620697152396, 6.59584217119164729742109454332, 6.77422879319909121148383826708, 7.22355409695059130795967375694, 7.64401219241868272281651022471, 7.921671808095048584394351994062, 8.663142961690271467687070138919, 8.930716553045993118793441606994, 8.956283450880080548424849581725, 9.350870869149623534195643683669, 10.04153719546873848783866044477

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 $\F_{q}$

\(L(17)\)	\(=\)	\(0\)
\(L(\frac12)\)	\(=\)	\(0\)
\(L(\frac{35}{2})\)		not available
\(L(1)\)		not available

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