LMFDB - L-function 8-15e4-1.1-c25e4-0-0

Dirichlet series

L(s) = 1

+ 7.68e3·2^-s + 2.12e6·3^-s − 2.01e7·4^-s − 9.76e8·5^-s + 1.63e10·6^-s + 2.49e10·7^-s − 1.65e11·8^-s + 2.82e12·9^-s − 7.50e12·10^-s + 9.98e12·11^-s − 4.29e13·12^-s − 1.43e14·13^-s + 1.91e14·14^-s − 2.07e15·15^-s + 1.23e15·16^-s + 2.80e15·17^-s + 2.16e16·18^-s + 1.50e16·19^-s + 1.97e16·20^-s + 5.29e16·21^-s + 7.67e16·22^-s + 4.87e16·23^-s − 3.51e17·24^-s + 5.96e17·25^-s − 1.10e18·26^-s + 3.00e18·27^-s − 5.02e17·28^-s + ⋯

L(s) = 1

+ 1.32·2^-s + 2.30·3^-s − 0.601·4^-s − 1.78·5^-s + 3.06·6^-s + 0.680·7^-s − 0.850·8^-s + 10/3·9^-s − 2.37·10^-s + 0.959·11^-s − 1.38·12^-s − 1.70·13^-s + 0.902·14^-s − 4.13·15^-s + 1.09·16^-s + 1.16·17^-s + 4.42·18^-s + 1.55·19^-s + 1.07·20^-s + 1.57·21^-s + 1.27·22^-s + 0.464·23^-s − 1.96·24^-s + 2·25^-s − 2.26·26^-s + 3.84·27^-s − 0.409·28^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+25/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$50625$ = $3^{4} \cdot 5^{4}$
Sign:	$1$
Analytic conductor:	$1.24489\times 10^{7}$
Root analytic conductor:	$7.70710$
Motivic weight:	$25$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 50625,\ (\ :25/2, 25/2, 25/2, 25/2),\ 1)$

Particular Values

$L(13)$	$\approx$	$22.90427774$
$L(\frac12)$	$\approx$	$22.90427774$
$L(\frac{27}{2})$		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	$C_1$	$ ( 1 - p^{12} T )^{4} $
	5	$C_1$	$ ( 1 + p^{12} T )^{4} $
good	2	$C_2 \wr S_4$	$ 1 - 7683 T + 19803275 p^{2} T^{2} - 9350685853 p^{6} T^{3} + 451253267883 p^{13} T^{4} - 9350685853 p^{31} T^{5} + 19803275 p^{52} T^{6} - 7683 p^{75} T^{7} + p^{100} T^{8} $
	7	$C_2 \wr S_4$	$ 1 - 508422464 p^{2} T + 53246328028610032828 p^{2} T^{2} - $$61\!\cdots\!88$$ p^{5} T^{3} + $$94\!\cdots\!30$$ p^{9} T^{4} - $$61\!\cdots\!88$$ p^{30} T^{5} + 53246328028610032828 p^{52} T^{6} - 508422464 p^{77} T^{7} + p^{100} T^{8} $
	11	$C_2 \wr S_4$	$ 1 - 9989114586816 T + $$20\!\cdots\!12$$ p T^{2} - $$67\!\cdots\!60$$ p^{3} T^{3} + $$12\!\cdots\!86$$ p^{5} T^{4} - $$67\!\cdots\!60$$ p^{28} T^{5} + $$20\!\cdots\!12$$ p^{51} T^{6} - 9989114586816 p^{75} T^{7} + p^{100} T^{8} $
	13	$C_2 \wr S_4$	$ 1 + 143366199676168 T + $$19\!\cdots\!96$$ p^{2} T^{2} + $$17\!\cdots\!44$$ p^{2} T^{3} + $$16\!\cdots\!10$$ p^{3} T^{4} + $$17\!\cdots\!44$$ p^{27} T^{5} + $$19\!\cdots\!96$$ p^{52} T^{6} + 143366199676168 p^{75} T^{7} + p^{100} T^{8} $
	17	$C_2 \wr S_4$	$ 1 - 2801218397375112 T + $$11\!\cdots\!32$$ T^{2} - $$16\!\cdots\!80$$ p T^{3} + $$35\!\cdots\!54$$ p^{2} T^{4} - $$16\!\cdots\!80$$ p^{26} T^{5} + $$11\!\cdots\!32$$ p^{50} T^{6} - 2801218397375112 p^{75} T^{7} + p^{100} T^{8} $
	19	$C_2 \wr S_4$	$ 1 - 15008254302340928 T + $$18\!\cdots\!08$$ p T^{2} - $$96\!\cdots\!96$$ p^{2} T^{3} + $$67\!\cdots\!66$$ p^{3} T^{4} - $$96\!\cdots\!96$$ p^{27} T^{5} + $$18\!\cdots\!08$$ p^{51} T^{6} - 15008254302340928 p^{75} T^{7} + p^{100} T^{8} $
	23	$C_2 \wr S_4$	$ 1 - 48788561474090928 T + $$10\!\cdots\!36$$ p T^{2} - $$26\!\cdots\!88$$ p^{2} T^{3} + $$28\!\cdots\!30$$ p^{3} T^{4} - $$26\!\cdots\!88$$ p^{27} T^{5} + $$10\!\cdots\!36$$ p^{51} T^{6} - 48788561474090928 p^{75} T^{7} + p^{100} T^{8} $
	29	$C_2 \wr S_4$	$ 1 + 366855400436538264 T + $$37\!\cdots\!60$$ T^{2} + $$81\!\cdots\!88$$ T^{3} + $$13\!\cdots\!78$$ T^{4} + $$81\!\cdots\!88$$ p^{25} T^{5} + $$37\!\cdots\!60$$ p^{50} T^{6} + 366855400436538264 p^{75} T^{7} + p^{100} T^{8} $
	31	$C_2 \wr S_4$	$ 1 - 5266475512689038864 T + $$25\!\cdots\!28$$ T^{2} - $$36\!\cdots\!92$$ T^{3} + $$33\!\cdots\!34$$ p T^{4} - $$36\!\cdots\!92$$ p^{25} T^{5} + $$25\!\cdots\!28$$ p^{50} T^{6} - 5266475512689038864 p^{75} T^{7} + p^{100} T^{8} $
	37	$C_2 \wr S_4$	$ 1 + 82582977571412504584 T + $$49\!\cdots\!52$$ T^{2} + $$21\!\cdots\!84$$ T^{3} + $$90\!\cdots\!30$$ T^{4} + $$21\!\cdots\!84$$ p^{25} T^{5} + $$49\!\cdots\!52$$ p^{50} T^{6} + 82582977571412504584 p^{75} T^{7} + p^{100} T^{8} $
	41	$C_2 \wr S_4$	$ 1 - $$11\!\cdots\!44$$ T + $$68\!\cdots\!68$$ T^{2} - $$75\!\cdots\!92$$ T^{3} + $$20\!\cdots\!34$$ T^{4} - $$75\!\cdots\!92$$ p^{25} T^{5} + $$68\!\cdots\!68$$ p^{50} T^{6} - $$11\!\cdots\!44$$ p^{75} T^{7} + p^{100} T^{8} $
	43	$C_2 \wr S_4$	$ 1 - $$29\!\cdots\!60$$ T + $$24\!\cdots\!80$$ T^{2} - $$55\!\cdots\!80$$ T^{3} + $$24\!\cdots\!98$$ T^{4} - $$55\!\cdots\!80$$ p^{25} T^{5} + $$24\!\cdots\!80$$ p^{50} T^{6} - $$29\!\cdots\!60$$ p^{75} T^{7} + p^{100} T^{8} $
	47	$C_2 \wr S_4$	$ 1 + $$25\!\cdots\!20$$ T + $$11\!\cdots\!20$$ T^{2} - $$22\!\cdots\!60$$ T^{3} + $$75\!\cdots\!98$$ T^{4} - $$22\!\cdots\!60$$ p^{25} T^{5} + $$11\!\cdots\!20$$ p^{50} T^{6} + $$25\!\cdots\!20$$ p^{75} T^{7} + p^{100} T^{8} $
	53	$C_2 \wr S_4$	$ 1 + $$17\!\cdots\!12$$ T + $$37\!\cdots\!48$$ T^{2} + $$50\!\cdots\!88$$ T^{3} + $$65\!\cdots\!30$$ T^{4} + $$50\!\cdots\!88$$ p^{25} T^{5} + $$37\!\cdots\!48$$ p^{50} T^{6} + $$17\!\cdots\!12$$ p^{75} T^{7} + p^{100} T^{8} $
	59	$C_2 \wr S_4$	$ 1 - $$53\!\cdots\!32$$ T + $$50\!\cdots\!92$$ T^{2} - $$20\!\cdots\!44$$ T^{3} + $$13\!\cdots\!14$$ T^{4} - $$20\!\cdots\!44$$ p^{25} T^{5} + $$50\!\cdots\!92$$ p^{50} T^{6} - $$53\!\cdots\!32$$ p^{75} T^{7} + p^{100} T^{8} $
	61	$C_2 \wr S_4$	$ 1 - $$19\!\cdots\!00$$ T + $$90\!\cdots\!76$$ T^{2} - $$90\!\cdots\!00$$ T^{3} + $$36\!\cdots\!46$$ T^{4} - $$90\!\cdots\!00$$ p^{25} T^{5} + $$90\!\cdots\!76$$ p^{50} T^{6} - $$19\!\cdots\!00$$ p^{75} T^{7} + p^{100} T^{8} $
	67	$C_2 \wr S_4$	$ 1 - $$22\!\cdots\!72$$ T + $$28\!\cdots\!72$$ T^{2} - $$24\!\cdots\!40$$ T^{3} + $$18\!\cdots\!86$$ T^{4} - $$24\!\cdots\!40$$ p^{25} T^{5} + $$28\!\cdots\!72$$ p^{50} T^{6} - $$22\!\cdots\!72$$ p^{75} T^{7} + p^{100} T^{8} $
	71	$C_2 \wr S_4$	$ 1 - $$82\!\cdots\!08$$ T + $$59\!\cdots\!28$$ T^{2} - $$37\!\cdots\!56$$ T^{3} + $$15\!\cdots\!70$$ T^{4} - $$37\!\cdots\!56$$ p^{25} T^{5} + $$59\!\cdots\!28$$ p^{50} T^{6} - $$82\!\cdots\!08$$ p^{75} T^{7} + p^{100} T^{8} $
	73	$C_2 \wr S_4$	$ 1 - $$55\!\cdots\!68$$ T + $$19\!\cdots\!28$$ T^{2} - $$45\!\cdots\!72$$ T^{3} + $$94\!\cdots\!10$$ T^{4} - $$45\!\cdots\!72$$ p^{25} T^{5} + $$19\!\cdots\!28$$ p^{50} T^{6} - $$55\!\cdots\!68$$ p^{75} T^{7} + p^{100} T^{8} $
	79	$C_2 \wr S_4$	$ 1 - $$11\!\cdots\!60$$ T + $$13\!\cdots\!96$$ T^{2} - $$89\!\cdots\!20$$ T^{3} + $$60\!\cdots\!06$$ T^{4} - $$89\!\cdots\!20$$ p^{25} T^{5} + $$13\!\cdots\!96$$ p^{50} T^{6} - $$11\!\cdots\!60$$ p^{75} T^{7} + p^{100} T^{8} $
	83	$C_2 \wr S_4$	$ 1 - $$24\!\cdots\!04$$ T + $$51\!\cdots\!80$$ T^{2} - $$68\!\cdots\!24$$ T^{3} + $$79\!\cdots\!86$$ T^{4} - $$68\!\cdots\!24$$ p^{25} T^{5} + $$51\!\cdots\!80$$ p^{50} T^{6} - $$24\!\cdots\!04$$ p^{75} T^{7} + p^{100} T^{8} $
	89	$C_2 \wr S_4$	$ 1 - $$70\!\cdots\!08$$ T + $$29\!\cdots\!92$$ T^{2} - $$99\!\cdots\!96$$ T^{3} + $$26\!\cdots\!14$$ T^{4} - $$99\!\cdots\!96$$ p^{25} T^{5} + $$29\!\cdots\!92$$ p^{50} T^{6} - $$70\!\cdots\!08$$ p^{75} T^{7} + p^{100} T^{8} $
	97	$C_2 \wr S_4$	$ 1 + $$15\!\cdots\!48$$ T + $$25\!\cdots\!92$$ T^{2} + $$22\!\cdots\!20$$ T^{3} + $$19\!\cdots\!26$$ T^{4} + $$22\!\cdots\!20$$ p^{25} T^{5} + $$25\!\cdots\!92$$ p^{50} T^{6} + $$15\!\cdots\!48$$ p^{75} T^{7} + p^{100} 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

−9.346873645726292149294916488371, −8.325547528233825224424389211631, −8.317259751826826183969629110515, −8.136292522333369605359543349736, −7.64468916716814958706246288650, −7.58826328908224768637822562626, −7.17524084573394015672300056272, −6.64978320328107853398323875115, −6.49399085545852135687838526245, −5.12235171484064995209276883425, −5.07634183991207681680698787477, −5.02556259219720869027533416339, −4.92807955516625055707674161483, −3.94149463711404978625598405558, −3.87431930887480337485871275675, −3.73617421678470521195760291944, −3.73157841179353888640425622708, −3.01043675034396624608411295354, −2.71331543781675627947522952760, −2.37971236324420732853021365012, −1.84497679327000558596779433659, −1.40337375114415808711643740281, −0.893521687496667237505534265448, −0.861197981949507243806230952593, −0.35443486371006702438996976480, 0.35443486371006702438996976480, 0.861197981949507243806230952593, 0.893521687496667237505534265448, 1.40337375114415808711643740281, 1.84497679327000558596779433659, 2.37971236324420732853021365012, 2.71331543781675627947522952760, 3.01043675034396624608411295354, 3.73157841179353888640425622708, 3.73617421678470521195760291944, 3.87431930887480337485871275675, 3.94149463711404978625598405558, 4.92807955516625055707674161483, 5.02556259219720869027533416339, 5.07634183991207681680698787477, 5.12235171484064995209276883425, 6.49399085545852135687838526245, 6.64978320328107853398323875115, 7.17524084573394015672300056272, 7.58826328908224768637822562626, 7.64468916716814958706246288650, 8.136292522333369605359543349736, 8.317259751826826183969629110515, 8.325547528233825224424389211631, 9.346873645726292149294916488371

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 $\F_{q}$

\(L(13)\)	\(\approx\)	\(22.90427774\)
\(L(\frac12)\)	\(\approx\)	\(22.90427774\)
\(L(\frac{27}{2})\)		not available
\(L(1)\)		not available

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