LMFDB - L-function 8-48e4-1.1-c25e4-0-0

Dirichlet series

L(s) = 1

+ 2.12e6·3^-s + 2.66e8·5^-s − 4.84e8·7^-s + 2.82e12·9^-s − 4.05e12·11^-s + 8.26e13·13^-s + 5.66e14·15^-s + 1.09e15·17^-s + 5.29e15·19^-s − 1.03e15·21^-s − 2.29e16·23^-s − 3.95e17·25^-s + 3.00e18·27^-s − 2.25e18·29^-s + 6.42e18·31^-s − 8.61e18·33^-s − 1.29e17·35^-s − 8.45e19·37^-s + 1.75e20·39^-s − 1.91e20·41^-s + 3.93e20·43^-s + 7.52e20·45^-s + 4.26e20·47^-s − 1.83e21·49^-s + 2.32e21·51^-s + 4.16e21·53^-s − 1.07e21·55^-s + ⋯

L(s) = 1

+ 2.30·3^-s + 0.488·5^-s − 0.0132·7^-s + 10/3·9^-s − 0.389·11^-s + 0.984·13^-s + 1.12·15^-s + 0.454·17^-s + 0.548·19^-s − 0.0305·21^-s − 0.218·23^-s − 1.32·25^-s + 3.84·27^-s − 1.18·29^-s + 1.46·31^-s − 0.899·33^-s − 0.00645·35^-s − 2.11·37^-s + 2.27·39^-s − 1.32·41^-s + 1.50·43^-s + 1.62·45^-s + 0.534·47^-s − 1.36·49^-s + 1.04·51^-s + 1.16·53^-s − 0.190·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+25/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$5308416$ = $2^{16} \cdot 3^{4}$
Sign:	$1$
Analytic conductor:	$1.30536\times 10^{9}$
Root analytic conductor:	$13.7868$
Motivic weight:	$25$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 5308416,\ (\ :25/2, 25/2, 25/2, 25/2),\ 1)$

Particular Values

$L(13)$	$\approx$	$44.21302500$
$L(\frac12)$	$\approx$	$44.21302500$
$L(\frac{27}{2})$		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	2		$ 1 $
	3	$C_1$	$ ( 1 - p^{12} T )^{4} $
good	5	$C_2 \wr S_4$	$ 1 - 266436088 T + 18670153127544716 p^{2} T^{2} - $$50\!\cdots\!48$$ p^{4} T^{3} + $$64\!\cdots\!78$$ p^{6} T^{4} - $$50\!\cdots\!48$$ p^{29} T^{5} + 18670153127544716 p^{52} T^{6} - 266436088 p^{75} T^{7} + p^{100} T^{8} $
	7	$C_2 \wr S_4$	$ 1 + 69226368 p T + $$26\!\cdots\!04$$ p T^{2} + $$27\!\cdots\!72$$ p^{3} T^{3} + $$76\!\cdots\!06$$ p^{6} T^{4} + $$27\!\cdots\!72$$ p^{28} T^{5} + $$26\!\cdots\!04$$ p^{51} T^{6} + 69226368 p^{76} T^{7} + p^{100} T^{8} $
	11	$C_2 \wr S_4$	$ 1 + 4052955557392 T + $$49\!\cdots\!12$$ p T^{2} + $$15\!\cdots\!40$$ p^{3} T^{3} + $$84\!\cdots\!02$$ p^{5} T^{4} + $$15\!\cdots\!40$$ p^{28} T^{5} + $$49\!\cdots\!12$$ p^{51} T^{6} + 4052955557392 p^{75} T^{7} + p^{100} T^{8} $
	13	$C_2 \wr S_4$	$ 1 - 82663381481080 T + $$23\!\cdots\!68$$ T^{2} - $$94\!\cdots\!88$$ p T^{3} + $$12\!\cdots\!98$$ p^{2} T^{4} - $$94\!\cdots\!88$$ p^{26} T^{5} + $$23\!\cdots\!68$$ p^{50} T^{6} - 82663381481080 p^{75} T^{7} + p^{100} T^{8} $
	17	$C_2 \wr S_4$	$ 1 - 1091839531710728 T + $$18\!\cdots\!16$$ p T^{2} + $$47\!\cdots\!60$$ p^{2} T^{3} - $$17\!\cdots\!78$$ p^{3} T^{4} + $$47\!\cdots\!60$$ p^{27} T^{5} + $$18\!\cdots\!16$$ p^{51} T^{6} - 1091839531710728 p^{75} T^{7} + p^{100} T^{8} $
	19	$C_2 \wr S_4$	$ 1 - 5291818285650448 T + $$89\!\cdots\!44$$ p T^{2} - $$49\!\cdots\!48$$ p^{2} T^{3} + $$22\!\cdots\!10$$ p^{3} T^{4} - $$49\!\cdots\!48$$ p^{27} T^{5} + $$89\!\cdots\!44$$ p^{51} T^{6} - 5291818285650448 p^{75} T^{7} + p^{100} T^{8} $
	23	$C_2 \wr S_4$	$ 1 + 998154584418848 p T + $$44\!\cdots\!48$$ p^{2} T^{2} + $$50\!\cdots\!40$$ p^{3} T^{3} + $$10\!\cdots\!58$$ p^{4} T^{4} + $$50\!\cdots\!40$$ p^{28} T^{5} + $$44\!\cdots\!48$$ p^{52} T^{6} + 998154584418848 p^{76} T^{7} + p^{100} T^{8} $
	29	$C_2 \wr S_4$	$ 1 + 77792955158658312 p T + $$27\!\cdots\!36$$ T^{2} - $$29\!\cdots\!72$$ T^{3} - $$88\!\cdots\!10$$ T^{4} - $$29\!\cdots\!72$$ p^{25} T^{5} + $$27\!\cdots\!36$$ p^{50} T^{6} + 77792955158658312 p^{76} T^{7} + p^{100} T^{8} $
	31	$C_2 \wr S_4$	$ 1 - 6422134828063294816 T + $$25\!\cdots\!96$$ T^{2} - $$16\!\cdots\!56$$ T^{3} + $$96\!\cdots\!50$$ T^{4} - $$16\!\cdots\!56$$ p^{25} T^{5} + $$25\!\cdots\!96$$ p^{50} T^{6} - 6422134828063294816 p^{75} T^{7} + p^{100} T^{8} $
	37	$C_2 \wr S_4$	$ 1 + 84572383178406307176 T + $$77\!\cdots\!44$$ T^{2} + $$10\!\cdots\!36$$ p T^{3} + $$19\!\cdots\!14$$ T^{4} + $$10\!\cdots\!36$$ p^{26} T^{5} + $$77\!\cdots\!44$$ p^{50} T^{6} + 84572383178406307176 p^{75} T^{7} + p^{100} T^{8} $
	41	$C_2 \wr S_4$	$ 1 + $$19\!\cdots\!28$$ T + $$35\!\cdots\!72$$ T^{2} - $$11\!\cdots\!60$$ T^{3} - $$14\!\cdots\!18$$ T^{4} - $$11\!\cdots\!60$$ p^{25} T^{5} + $$35\!\cdots\!72$$ p^{50} T^{6} + $$19\!\cdots\!28$$ p^{75} T^{7} + p^{100} T^{8} $
	43	$C_2 \wr S_4$	$ 1 - $$39\!\cdots\!04$$ T + $$24\!\cdots\!28$$ T^{2} - $$58\!\cdots\!20$$ T^{3} + $$23\!\cdots\!86$$ T^{4} - $$58\!\cdots\!20$$ p^{25} T^{5} + $$24\!\cdots\!28$$ p^{50} T^{6} - $$39\!\cdots\!04$$ p^{75} T^{7} + p^{100} T^{8} $
	47	$C_2 \wr S_4$	$ 1 - $$42\!\cdots\!48$$ T + $$17\!\cdots\!28$$ T^{2} - $$80\!\cdots\!08$$ T^{3} + $$14\!\cdots\!94$$ T^{4} - $$80\!\cdots\!08$$ p^{25} T^{5} + $$17\!\cdots\!28$$ p^{50} T^{6} - $$42\!\cdots\!48$$ p^{75} T^{7} + p^{100} T^{8} $
	53	$C_2 \wr S_4$	$ 1 - $$41\!\cdots\!44$$ T + $$45\!\cdots\!24$$ T^{2} - $$11\!\cdots\!08$$ T^{3} + $$79\!\cdots\!26$$ T^{4} - $$11\!\cdots\!08$$ p^{25} T^{5} + $$45\!\cdots\!24$$ p^{50} T^{6} - $$41\!\cdots\!44$$ p^{75} T^{7} + p^{100} T^{8} $
	59	$C_2 \wr S_4$	$ 1 + $$22\!\cdots\!12$$ T + $$24\!\cdots\!04$$ T^{2} - $$97\!\cdots\!92$$ T^{3} - $$29\!\cdots\!50$$ T^{4} - $$97\!\cdots\!92$$ p^{25} T^{5} + $$24\!\cdots\!04$$ p^{50} T^{6} + $$22\!\cdots\!12$$ p^{75} T^{7} + p^{100} T^{8} $
	61	$C_2 \wr S_4$	$ 1 - $$31\!\cdots\!60$$ T + $$19\!\cdots\!80$$ T^{2} - $$40\!\cdots\!24$$ T^{3} + $$13\!\cdots\!70$$ T^{4} - $$40\!\cdots\!24$$ p^{25} T^{5} + $$19\!\cdots\!80$$ p^{50} T^{6} - $$31\!\cdots\!60$$ p^{75} T^{7} + p^{100} T^{8} $
	67	$C_2 \wr S_4$	$ 1 - $$10\!\cdots\!48$$ T + $$18\!\cdots\!08$$ T^{2} - $$12\!\cdots\!00$$ T^{3} + $$12\!\cdots\!18$$ T^{4} - $$12\!\cdots\!00$$ p^{25} T^{5} + $$18\!\cdots\!08$$ p^{50} T^{6} - $$10\!\cdots\!48$$ p^{75} T^{7} + p^{100} T^{8} $
	71	$C_2 \wr S_4$	$ 1 - $$26\!\cdots\!36$$ T + $$74\!\cdots\!76$$ T^{2} - $$10\!\cdots\!96$$ T^{3} + $$18\!\cdots\!10$$ T^{4} - $$10\!\cdots\!96$$ p^{25} T^{5} + $$74\!\cdots\!76$$ p^{50} T^{6} - $$26\!\cdots\!36$$ p^{75} T^{7} + p^{100} T^{8} $
	73	$C_2 \wr S_4$	$ 1 - $$96\!\cdots\!24$$ T + $$39\!\cdots\!12$$ T^{2} - $$52\!\cdots\!96$$ T^{3} + $$20\!\cdots\!34$$ T^{4} - $$52\!\cdots\!96$$ p^{25} T^{5} + $$39\!\cdots\!12$$ p^{50} T^{6} - $$96\!\cdots\!24$$ p^{75} T^{7} + p^{100} T^{8} $
	79	$C_2 \wr S_4$	$ 1 - $$22\!\cdots\!28$$ T + $$10\!\cdots\!04$$ T^{2} - $$17\!\cdots\!16$$ T^{3} + $$41\!\cdots\!18$$ T^{4} - $$17\!\cdots\!16$$ p^{25} T^{5} + $$10\!\cdots\!04$$ p^{50} T^{6} - $$22\!\cdots\!28$$ p^{75} T^{7} + p^{100} T^{8} $
	83	$C_2 \wr S_4$	$ 1 + $$50\!\cdots\!36$$ T + $$26\!\cdots\!44$$ T^{2} + $$14\!\cdots\!84$$ T^{3} + $$33\!\cdots\!14$$ T^{4} + $$14\!\cdots\!84$$ p^{25} T^{5} + $$26\!\cdots\!44$$ p^{50} T^{6} + $$50\!\cdots\!36$$ p^{75} T^{7} + p^{100} T^{8} $
	89	$C_2 \wr S_4$	$ 1 - $$93\!\cdots\!84$$ T + $$49\!\cdots\!28$$ T^{2} - $$17\!\cdots\!28$$ T^{3} + $$46\!\cdots\!70$$ T^{4} - $$17\!\cdots\!28$$ p^{25} T^{5} + $$49\!\cdots\!28$$ p^{50} T^{6} - $$93\!\cdots\!84$$ p^{75} T^{7} + p^{100} T^{8} $
	97	$C_2 \wr S_4$	$ 1 - $$17\!\cdots\!84$$ T + $$23\!\cdots\!68$$ T^{2} - $$22\!\cdots\!48$$ T^{3} + $$17\!\cdots\!18$$ T^{4} - $$22\!\cdots\!48$$ p^{25} T^{5} + $$23\!\cdots\!68$$ p^{50} T^{6} - $$17\!\cdots\!84$$ 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

−7.60111523255754678186891631242, −6.81774250907727344696150844231, −6.64819060083929881437443580360, −6.47639941654634326169756808833, −6.25817748289197178321049141222, −5.51804234032833282296454347405, −5.50620648720302064102827153653, −5.14376264981108529391410331115, −5.01372833510224481008735630989, −4.35335847708913753367135720109, −4.16262228776376066491782153026, −3.79815625953382801411029409052, −3.69470890143113387081148246603, −3.38843550612508737514040111106, −3.20778050350021977845812787701, −2.73754567272669509369242514065, −2.65956584551765634353443287004, −2.15529195210881763437653182460, −1.96690909546178480521192645363, −1.70340678770109061230423241319, −1.68875284423134276618370847208, −1.18891002909417336974421825005, −0.71813340214964862987116786123, −0.54412497720960984440852325402, −0.45336426276406373070844167847, 0.45336426276406373070844167847, 0.54412497720960984440852325402, 0.71813340214964862987116786123, 1.18891002909417336974421825005, 1.68875284423134276618370847208, 1.70340678770109061230423241319, 1.96690909546178480521192645363, 2.15529195210881763437653182460, 2.65956584551765634353443287004, 2.73754567272669509369242514065, 3.20778050350021977845812787701, 3.38843550612508737514040111106, 3.69470890143113387081148246603, 3.79815625953382801411029409052, 4.16262228776376066491782153026, 4.35335847708913753367135720109, 5.01372833510224481008735630989, 5.14376264981108529391410331115, 5.50620648720302064102827153653, 5.51804234032833282296454347405, 6.25817748289197178321049141222, 6.47639941654634326169756808833, 6.64819060083929881437443580360, 6.81774250907727344696150844231, 7.60111523255754678186891631242

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 $\F_{q}$

\(L(13)\)	\(\approx\)	\(44.21302500\)
\(L(\frac12)\)	\(\approx\)	\(44.21302500\)
\(L(\frac{27}{2})\)		not available
\(L(1)\)		not available

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