LMFDB - L-function 8-3e8-1.1-c43e4-0-0

Dirichlet series

L(s) = 1

− 1.66e6·2^-s − 1.54e12·4^-s − 1.65e15·5^-s + 1.14e17·7^-s − 1.70e19·8^-s + 2.74e21·10^-s − 8.71e21·11^-s − 1.62e24·13^-s − 1.90e23·14^-s + 3.74e25·16^-s − 2.93e26·17^-s − 2.59e27·19^-s + 2.55e27·20^-s + 1.44e28·22^-s + 1.65e29·23^-s + 1.70e29·25^-s + 2.69e30·26^-s − 1.77e29·28^-s − 5.01e31·29^-s + 1.53e32·31^-s − 1.13e32·32^-s + 4.87e32·34^-s − 1.88e32·35^-s + 1.95e34·37^-s + 4.30e33·38^-s + 2.81e34·40^-s + 9.03e33·41^-s + ⋯

L(s) = 1

− 0.559·2^-s − 0.175·4^-s − 1.54·5^-s + 0.0774·7^-s − 0.652·8^-s + 0.866·10^-s − 0.355·11^-s − 1.82·13^-s − 0.0433·14^-s + 0.484·16^-s − 1.02·17^-s − 0.833·19^-s + 0.272·20^-s + 0.198·22^-s + 0.875·23^-s + 0.149·25^-s + 1.01·26^-s − 0.0136·28^-s − 1.81·29^-s + 1.32·31^-s − 0.493·32^-s + 0.576·34^-s − 0.119·35^-s + 3.76·37^-s + 0.466·38^-s + 1.01·40^-s + 0.191·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+43/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$6561$ = $3^{8}$
Sign:	$1$
Analytic conductor:	$1.23410\times 10^{8}$
Root analytic conductor:	$10.2664$
Motivic weight:	$43$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$4$
Selberg data:	$(8,\ 6561,\ (\ :43/2, 43/2, 43/2, 43/2),\ 1)$

Particular Values

$L(22)$	$=$	$0$
$L(\frac12)$	$=$	$0$
$L(\frac{45}{2})$		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	$C_2 \wr S_4$	$ 1 + 830007 p T + 67236468785 p^{6} T^{2} + 408021684903009 p^{16} T^{3} + 77971628085324573 p^{29} T^{4} + 408021684903009 p^{59} T^{5} + 67236468785 p^{92} T^{6} + 830007 p^{130} T^{7} + p^{172} T^{8} $
	5	$C_2 \wr S_4$	$ 1 + 330130161507384 p T + $$81\!\cdots\!32$$ p^{5} T^{2} + $$39\!\cdots\!08$$ p^{10} T^{3} + $$59\!\cdots\!42$$ p^{17} T^{4} + $$39\!\cdots\!08$$ p^{53} T^{5} + $$81\!\cdots\!32$$ p^{91} T^{6} + 330130161507384 p^{130} T^{7} + p^{172} T^{8} $
	7	$C_2 \wr S_4$	$ 1 - 114481425784209728 T + $$93\!\cdots\!84$$ p T^{2} - $$52\!\cdots\!92$$ p^{4} T^{3} + $$23\!\cdots\!90$$ p^{7} T^{4} - $$52\!\cdots\!92$$ p^{47} T^{5} + $$93\!\cdots\!84$$ p^{87} T^{6} - 114481425784209728 p^{129} T^{7} + p^{172} T^{8} $
	11	$C_2 \wr S_4$	$ 1 + $$79\!\cdots\!36$$ p T + $$10\!\cdots\!52$$ p^{3} T^{2} + $$52\!\cdots\!60$$ p^{3} T^{3} + $$62\!\cdots\!26$$ p^{6} T^{4} + $$52\!\cdots\!60$$ p^{46} T^{5} + $$10\!\cdots\!52$$ p^{89} T^{6} + $$79\!\cdots\!36$$ p^{130} T^{7} + p^{172} T^{8} $
	13	$C_2 \wr S_4$	$ 1 + $$16\!\cdots\!96$$ T + $$21\!\cdots\!72$$ p T^{2} + $$14\!\cdots\!84$$ p^{3} T^{3} + $$68\!\cdots\!50$$ p^{6} T^{4} + $$14\!\cdots\!84$$ p^{46} T^{5} + $$21\!\cdots\!72$$ p^{87} T^{6} + $$16\!\cdots\!96$$ p^{129} T^{7} + p^{172} T^{8} $
	17	$C_2 \wr S_4$	$ 1 + $$29\!\cdots\!76$$ T + $$17\!\cdots\!04$$ p T^{2} + $$13\!\cdots\!00$$ p^{3} T^{3} + $$25\!\cdots\!18$$ p^{5} T^{4} + $$13\!\cdots\!00$$ p^{46} T^{5} + $$17\!\cdots\!04$$ p^{87} T^{6} + $$29\!\cdots\!76$$ p^{129} T^{7} + p^{172} T^{8} $
	19	$C_2 \wr S_4$	$ 1 + $$25\!\cdots\!88$$ T + $$10\!\cdots\!28$$ p T^{2} + $$11\!\cdots\!56$$ p^{2} T^{3} + $$15\!\cdots\!34$$ p^{4} T^{4} + $$11\!\cdots\!56$$ p^{45} T^{5} + $$10\!\cdots\!28$$ p^{87} T^{6} + $$25\!\cdots\!88$$ p^{129} T^{7} + p^{172} T^{8} $
	23	$C_2 \wr S_4$	$ 1 - $$16\!\cdots\!44$$ T + $$11\!\cdots\!12$$ T^{2} - $$67\!\cdots\!68$$ p T^{3} + $$10\!\cdots\!70$$ p^{2} T^{4} - $$67\!\cdots\!68$$ p^{44} T^{5} + $$11\!\cdots\!12$$ p^{86} T^{6} - $$16\!\cdots\!44$$ p^{129} T^{7} + p^{172} T^{8} $
	29	$C_2 \wr S_4$	$ 1 + $$50\!\cdots\!64$$ T + $$11\!\cdots\!00$$ p T^{2} + $$44\!\cdots\!52$$ p^{3} T^{3} + $$15\!\cdots\!22$$ p^{3} T^{4} + $$44\!\cdots\!52$$ p^{46} T^{5} + $$11\!\cdots\!00$$ p^{87} T^{6} + $$50\!\cdots\!64$$ p^{129} T^{7} + p^{172} T^{8} $
	31	$C_2 \wr S_4$	$ 1 - $$15\!\cdots\!24$$ T + $$45\!\cdots\!08$$ T^{2} - $$46\!\cdots\!32$$ p^{2} T^{3} + $$86\!\cdots\!54$$ p^{2} T^{4} - $$46\!\cdots\!32$$ p^{45} T^{5} + $$45\!\cdots\!08$$ p^{86} T^{6} - $$15\!\cdots\!24$$ p^{129} T^{7} + p^{172} T^{8} $
	37	$C_2 \wr S_4$	$ 1 - $$19\!\cdots\!28$$ T + $$61\!\cdots\!04$$ p T^{2} - $$13\!\cdots\!48$$ p^{2} T^{3} + $$21\!\cdots\!90$$ p^{3} T^{4} - $$13\!\cdots\!48$$ p^{45} T^{5} + $$61\!\cdots\!04$$ p^{87} T^{6} - $$19\!\cdots\!28$$ p^{129} T^{7} + p^{172} T^{8} $
	41	$C_2 \wr S_4$	$ 1 - $$22\!\cdots\!16$$ p T - $$22\!\cdots\!72$$ p^{2} T^{2} - $$59\!\cdots\!68$$ p^{3} T^{3} + $$25\!\cdots\!34$$ p^{4} T^{4} - $$59\!\cdots\!68$$ p^{46} T^{5} - $$22\!\cdots\!72$$ p^{88} T^{6} - $$22\!\cdots\!16$$ p^{130} T^{7} + p^{172} T^{8} $
	43	$C_2 \wr S_4$	$ 1 - $$57\!\cdots\!00$$ T + $$28\!\cdots\!40$$ T^{2} - $$91\!\cdots\!00$$ T^{3} + $$40\!\cdots\!98$$ T^{4} - $$91\!\cdots\!00$$ p^{43} T^{5} + $$28\!\cdots\!40$$ p^{86} T^{6} - $$57\!\cdots\!00$$ p^{129} T^{7} + p^{172} T^{8} $
	47	$C_2 \wr S_4$	$ 1 + $$48\!\cdots\!40$$ T + $$19\!\cdots\!80$$ T^{2} + $$82\!\cdots\!20$$ T^{3} + $$20\!\cdots\!58$$ T^{4} + $$82\!\cdots\!20$$ p^{43} T^{5} + $$19\!\cdots\!80$$ p^{86} T^{6} + $$48\!\cdots\!40$$ p^{129} T^{7} + p^{172} T^{8} $
	53	$C_2 \wr S_4$	$ 1 - $$40\!\cdots\!64$$ T + $$11\!\cdots\!12$$ T^{2} - $$19\!\cdots\!44$$ T^{3} + $$27\!\cdots\!90$$ T^{4} - $$19\!\cdots\!44$$ p^{43} T^{5} + $$11\!\cdots\!12$$ p^{86} T^{6} - $$40\!\cdots\!64$$ p^{129} T^{7} + p^{172} T^{8} $
	59	$C_2 \wr S_4$	$ 1 - $$31\!\cdots\!32$$ T + $$32\!\cdots\!72$$ T^{2} - $$69\!\cdots\!84$$ T^{3} + $$57\!\cdots\!94$$ T^{4} - $$69\!\cdots\!84$$ p^{43} T^{5} + $$32\!\cdots\!72$$ p^{86} T^{6} - $$31\!\cdots\!32$$ p^{129} T^{7} + p^{172} T^{8} $
	61	$C_2 \wr S_4$	$ 1 + $$24\!\cdots\!60$$ T + $$18\!\cdots\!76$$ T^{2} - $$57\!\cdots\!60$$ T^{3} - $$20\!\cdots\!34$$ T^{4} - $$57\!\cdots\!60$$ p^{43} T^{5} + $$18\!\cdots\!76$$ p^{86} T^{6} + $$24\!\cdots\!60$$ p^{129} T^{7} + p^{172} T^{8} $
	67	$C_2 \wr S_4$	$ 1 - $$10\!\cdots\!76$$ T + $$13\!\cdots\!68$$ T^{2} - $$10\!\cdots\!00$$ T^{3} + $$65\!\cdots\!26$$ T^{4} - $$10\!\cdots\!00$$ p^{43} T^{5} + $$13\!\cdots\!68$$ p^{86} T^{6} - $$10\!\cdots\!76$$ p^{129} T^{7} + p^{172} T^{8} $
	71	$C_2 \wr S_4$	$ 1 - $$71\!\cdots\!72$$ T + $$94\!\cdots\!88$$ T^{2} - $$62\!\cdots\!04$$ T^{3} + $$46\!\cdots\!70$$ T^{4} - $$62\!\cdots\!04$$ p^{43} T^{5} + $$94\!\cdots\!88$$ p^{86} T^{6} - $$71\!\cdots\!72$$ p^{129} T^{7} + p^{172} T^{8} $
	73	$C_2 \wr S_4$	$ 1 - $$54\!\cdots\!96$$ T + $$34\!\cdots\!32$$ T^{2} - $$18\!\cdots\!76$$ T^{3} + $$60\!\cdots\!10$$ T^{4} - $$18\!\cdots\!76$$ p^{43} T^{5} + $$34\!\cdots\!32$$ p^{86} T^{6} - $$54\!\cdots\!96$$ p^{129} T^{7} + p^{172} T^{8} $
	79	$C_2 \wr S_4$	$ 1 + $$18\!\cdots\!60$$ T - $$12\!\cdots\!44$$ T^{2} + $$59\!\cdots\!20$$ T^{3} + $$25\!\cdots\!26$$ T^{4} + $$59\!\cdots\!20$$ p^{43} T^{5} - $$12\!\cdots\!44$$ p^{86} T^{6} + $$18\!\cdots\!60$$ p^{129} T^{7} + p^{172} T^{8} $
	83	$C_2 \wr S_4$	$ 1 + $$20\!\cdots\!08$$ T + $$10\!\cdots\!80$$ T^{2} + $$13\!\cdots\!52$$ T^{3} + $$43\!\cdots\!46$$ T^{4} + $$13\!\cdots\!52$$ p^{43} T^{5} + $$10\!\cdots\!80$$ p^{86} T^{6} + $$20\!\cdots\!08$$ p^{129} T^{7} + p^{172} T^{8} $
	89	$C_2 \wr S_4$	$ 1 - $$44\!\cdots\!68$$ T + $$24\!\cdots\!92$$ T^{2} - $$88\!\cdots\!16$$ T^{3} + $$23\!\cdots\!74$$ T^{4} - $$88\!\cdots\!16$$ p^{43} T^{5} + $$24\!\cdots\!92$$ p^{86} T^{6} - $$44\!\cdots\!68$$ p^{129} T^{7} + p^{172} T^{8} $
	97	$C_2 \wr S_4$	$ 1 + $$50\!\cdots\!64$$ T + $$60\!\cdots\!28$$ T^{2} + $$93\!\cdots\!00$$ T^{3} + $$13\!\cdots\!66$$ T^{4} + $$93\!\cdots\!00$$ p^{43} T^{5} + $$60\!\cdots\!28$$ p^{86} T^{6} + $$50\!\cdots\!64$$ p^{129} T^{7} + p^{172} T^{8} $
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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

−8.931540656948352853778022520065, −8.223186282229929957687131382840, −8.128827884430048812821009546942, −8.058576614074357235740715306333, −7.60272045614349586943430624039, −7.29560904312258193354322102480, −6.92800794839397141543499735615, −6.69734430681113054944306686898, −6.18403265819907447778514322104, −6.01118244347781824597018823376, −5.36700163616483570130750906925, −5.23114981489662416703672607986, −4.84374782121906602117978132376, −4.32915966400087821445245978989, −4.15941511573661423835645147304, −4.12375565418582483477199449051, −3.46693129450593846832177981771, −3.27625094239747564895977362703, −2.70261357090689046836844828210, −2.39419296734731610415870540065, −2.37440803085602366538592330170, −2.07218531761947109629976809635, −1.17117040845226247451701611640, −1.15392804742021642644888944651, −0.78823924266162235445872667846, 0, 0, 0, 0, 0.78823924266162235445872667846, 1.15392804742021642644888944651, 1.17117040845226247451701611640, 2.07218531761947109629976809635, 2.37440803085602366538592330170, 2.39419296734731610415870540065, 2.70261357090689046836844828210, 3.27625094239747564895977362703, 3.46693129450593846832177981771, 4.12375565418582483477199449051, 4.15941511573661423835645147304, 4.32915966400087821445245978989, 4.84374782121906602117978132376, 5.23114981489662416703672607986, 5.36700163616483570130750906925, 6.01118244347781824597018823376, 6.18403265819907447778514322104, 6.69734430681113054944306686898, 6.92800794839397141543499735615, 7.29560904312258193354322102480, 7.60272045614349586943430624039, 8.058576614074357235740715306333, 8.128827884430048812821009546942, 8.223186282229929957687131382840, 8.931540656948352853778022520065

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 $\F_{q}$

\(L(22)\)	\(=\)	\(0\)
\(L(\frac12)\)	\(=\)	\(0\)
\(L(\frac{45}{2})\)		not available
\(L(1)\)		not available

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