LMFDB - L-function 8-3e8-1.1-c23e4-0-0

Dirichlet series

L(s) = 1

− 7.22e6·4^-s + 8.56e9·7^-s − 1.35e13·13^-s − 7.14e12·16^-s + 8.23e14·19^-s − 1.17e16·25^-s − 6.18e16·28^-s + 1.99e17·31^-s + 6.11e18·37^-s + 8.38e18·43^-s − 2.57e19·49^-s + 9.77e19·52^-s − 1.87e20·61^-s − 2.82e19·64^-s + 2.21e21·67^-s + 5.55e21·73^-s − 5.94e21·76^-s − 1.56e21·79^-s − 1.15e23·91^-s + 6.16e22·97^-s + 8.45e22·100^-s + 5.53e23·103^-s + 7.48e23·109^-s − 6.11e22·112^-s − 2.24e24·121^-s − 1.43e24·124^-s + 127^-s + ⋯

L(s) = 1

− 0.860·4^-s + 1.63·7^-s − 2.09·13^-s − 0.101·16^-s + 1.62·19^-s − 0.982·25^-s − 1.40·28^-s + 1.40·31^-s + 5.64·37^-s + 1.37·43^-s − 0.940·49^-s + 1.80·52^-s − 0.552·61^-s − 0.0479·64^-s + 2.21·67^-s + 2.07·73^-s − 1.39·76^-s − 0.235·79^-s − 3.42·91^-s + 0.875·97^-s + 0.845·100^-s + 3.94·103^-s + 2.77·109^-s − 0.166·112^-s − 2.50·121^-s − 1.21·124^-s + 2.65·133^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+23/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$6561$ = $3^{8}$
Sign:	$1$
Analytic conductor:	$828336.$
Root analytic conductor:	$5.49257$
Motivic weight:	$23$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 6561,\ (\ :23/2, 23/2, 23/2, 23/2),\ 1)$

Particular Values

$L(12)$	$\approx$	$7.320149987$
$L(\frac12)$	$\approx$	$7.320149987$
$L(\frac{25}{2})$		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 + 902779 p^{3} T^{2} + 3619815351 p^{14} T^{4} + 902779 p^{49} T^{6} + p^{92} T^{8} $
	5	$C_2^2 \wr C_2$	$ 1 + 468527477472212 p^{2} T^{2} + $$10\!\cdots\!74$$ p^{5} T^{4} + 468527477472212 p^{48} T^{6} + p^{92} T^{8} $
	7	$D_{4}$	$ ( 1 - 611531320 p T + 117661831498634802 p^{3} T^{2} - 611531320 p^{24} T^{3} + p^{46} T^{4} )^{2} $
	11	$C_2^2 \wr C_2$	$ 1 + $$20\!\cdots\!84$$ p T^{2} + $$21\!\cdots\!86$$ p^{3} T^{4} + $$20\!\cdots\!84$$ p^{47} T^{6} + p^{92} T^{8} $
	13	$D_{4}$	$ ( 1 + 6767284999340 T + $$36\!\cdots\!38$$ p T^{2} + 6767284999340 p^{23} T^{3} + p^{46} T^{4} )^{2} $
	17	$C_2^2 \wr C_2$	$ 1 - $$14\!\cdots\!48$$ T^{2} + $$27\!\cdots\!26$$ p^{2} T^{4} - $$14\!\cdots\!48$$ p^{46} T^{6} + p^{92} T^{8} $
	19	$D_{4}$	$ ( 1 - 21660892300048 p T + $$12\!\cdots\!14$$ p^{2} T^{2} - 21660892300048 p^{24} T^{3} + p^{46} T^{4} )^{2} $
	23	$C_2^2 \wr C_2$	$ 1 + $$70\!\cdots\!68$$ T^{2} + $$20\!\cdots\!34$$ T^{4} + $$70\!\cdots\!68$$ p^{46} T^{6} + p^{92} T^{8} $
	29	$C_2^2 \wr C_2$	$ 1 + $$34\!\cdots\!56$$ T^{2} - $$64\!\cdots\!74$$ T^{4} + $$34\!\cdots\!56$$ p^{46} T^{6} + p^{92} T^{8} $
	31	$D_{4}$	$ ( 1 - 99582759296767816 T - $$22\!\cdots\!54$$ T^{2} - 99582759296767816 p^{23} T^{3} + p^{46} T^{4} )^{2} $
	37	$D_{4}$	$ ( 1 - 3057156256712709820 T + $$46\!\cdots\!06$$ T^{2} - 3057156256712709820 p^{23} T^{3} + p^{46} T^{4} )^{2} $
	41	$C_2^2 \wr C_2$	$ 1 + $$24\!\cdots\!84$$ T^{2} + $$33\!\cdots\!46$$ T^{4} + $$24\!\cdots\!84$$ p^{46} T^{6} + p^{92} T^{8} $
	43	$D_{4}$	$ ( 1 - 4192218259444995760 T + $$45\!\cdots\!14$$ T^{2} - 4192218259444995760 p^{23} T^{3} + p^{46} T^{4} )^{2} $
	47	$C_2^2 \wr C_2$	$ 1 + $$86\!\cdots\!92$$ T^{2} + $$34\!\cdots\!74$$ T^{4} + $$86\!\cdots\!92$$ p^{46} T^{6} + p^{92} T^{8} $
	53	$C_2^2 \wr C_2$	$ 1 + $$13\!\cdots\!08$$ T^{2} + $$81\!\cdots\!74$$ T^{4} + $$13\!\cdots\!08$$ p^{46} T^{6} + p^{92} T^{8} $
	59	$C_2^2 \wr C_2$	$ 1 + $$69\!\cdots\!16$$ T^{2} + $$17\!\cdots\!46$$ T^{4} + $$69\!\cdots\!16$$ p^{46} T^{6} + p^{92} T^{8} $
	61	$D_{4}$	$ ( 1 + 93817948640972526836 T + $$19\!\cdots\!86$$ T^{2} + 93817948640972526836 p^{23} T^{3} + p^{46} T^{4} )^{2} $
	67	$D_{4}$	$ ( 1 - $$11\!\cdots\!40$$ T + $$22\!\cdots\!26$$ T^{2} - $$11\!\cdots\!40$$ p^{23} T^{3} + p^{46} T^{4} )^{2} $
	71	$C_2^2 \wr C_2$	$ 1 + $$82\!\cdots\!44$$ T^{2} + $$45\!\cdots\!26$$ T^{4} + $$82\!\cdots\!44$$ p^{46} T^{6} + p^{92} T^{8} $
	73	$D_{4}$	$ ( 1 - $$27\!\cdots\!80$$ T + $$10\!\cdots\!34$$ T^{2} - $$27\!\cdots\!80$$ p^{23} T^{3} + p^{46} T^{4} )^{2} $
	79	$D_{4}$	$ ( 1 + $$78\!\cdots\!32$$ T + $$74\!\cdots\!34$$ T^{2} + $$78\!\cdots\!32$$ p^{23} T^{3} + p^{46} T^{4} )^{2} $
	83	$C_2^2 \wr C_2$	$ 1 - $$11\!\cdots\!52$$ T^{2} + $$37\!\cdots\!14$$ T^{4} - $$11\!\cdots\!52$$ p^{46} T^{6} + p^{92} T^{8} $
	89	$C_2^2 \wr C_2$	$ 1 + $$27\!\cdots\!76$$ T^{2} + $$28\!\cdots\!66$$ T^{4} + $$27\!\cdots\!76$$ p^{46} T^{6} + p^{92} T^{8} $
	97	$D_{4}$	$ ( 1 - $$30\!\cdots\!40$$ T + $$99\!\cdots\!46$$ T^{2} - $$30\!\cdots\!40$$ p^{23} T^{3} + p^{46} 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

−10.62391160345168032842104275791, −9.933370612220743052987728880528, −9.578566846220428344672283432447, −9.481517744903021661234203495737, −9.336338677270473833491085338775, −8.313996480442497466888002669445, −8.016192216748621590815183854125, −7.959271551682075216578262723234, −7.43680692351053601789210555530, −7.21904631428505311504623563798, −6.31662009895272285792141227640, −6.05525723098526086554416400246, −5.50426009732325169538010563041, −5.00209455176749215026207004799, −4.67782008855789247372353534494, −4.45205268515798882569664567042, −4.25884370550113664966237895017, −3.38121399112284757230647235434, −2.90822122051428103170727562707, −2.44485877300628009411234574490, −2.11163952357875927157202795494, −1.68995568615001132665999226100, −0.78015644226417169727662273490, −0.73714582339332141868808836949, −0.58020662713304802923910780443, 0.58020662713304802923910780443, 0.73714582339332141868808836949, 0.78015644226417169727662273490, 1.68995568615001132665999226100, 2.11163952357875927157202795494, 2.44485877300628009411234574490, 2.90822122051428103170727562707, 3.38121399112284757230647235434, 4.25884370550113664966237895017, 4.45205268515798882569664567042, 4.67782008855789247372353534494, 5.00209455176749215026207004799, 5.50426009732325169538010563041, 6.05525723098526086554416400246, 6.31662009895272285792141227640, 7.21904631428505311504623563798, 7.43680692351053601789210555530, 7.959271551682075216578262723234, 8.016192216748621590815183854125, 8.313996480442497466888002669445, 9.336338677270473833491085338775, 9.481517744903021661234203495737, 9.578566846220428344672283432447, 9.933370612220743052987728880528, 10.62391160345168032842104275791

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 $\F_{q}$

\(L(12)\)	\(\approx\)	\(7.320149987\)
\(L(\frac12)\)	\(\approx\)	\(7.320149987\)
\(L(\frac{25}{2})\)		not available
\(L(1)\)		not available

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