LMFDB - L-function 16-6e8-1.1-c24e8-0-0

Dirichlet series

L(s) = 1

− 1.31e5·3^-s − 3.35e7·4^-s − 1.01e10·7^-s + 1.56e11·9^-s + 4.42e12·12^-s + 5.05e13·13^-s + 7.03e14·16^-s − 9.78e14·19^-s + 1.34e15·21^-s + 2.42e17·25^-s − 4.87e16·27^-s + 3.40e17·28^-s + 3.09e18·31^-s − 5.24e18·36^-s + 2.25e19·37^-s − 6.66e18·39^-s + 2.26e20·43^-s − 9.28e19·48^-s − 6.62e20·49^-s − 1.69e21·52^-s + 1.28e20·57^-s − 6.50e21·61^-s − 1.58e21·63^-s − 1.18e22·64^-s − 7.04e21·67^-s + 3.62e22·73^-s − 3.19e22·75^-s + ⋯

L(s) = 1

− 0.248·3^-s − 2·4^-s − 0.734·7^-s + 0.553·9^-s + 0.496·12^-s + 2.17·13^-s + 5/2·16^-s − 0.441·19^-s + 0.182·21^-s + 4.06·25^-s − 0.324·27^-s + 1.46·28^-s + 3.92·31^-s − 1.10·36^-s + 3.43·37^-s − 0.538·39^-s + 5.66·43^-s − 0.620·48^-s − 3.45·49^-s − 4.34·52^-s + 0.109·57^-s − 2.45·61^-s − 0.406·63^-s − 5/2·64^-s − 0.860·67^-s + 1.58·73^-s − 1.00·75^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(25-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+12)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$1679616$ = $2^{8} \cdot 3^{8}$
Sign:	$1$
Analytic conductor:	$5.28737\times 10^{10}$
Root analytic conductor:	$4.67953$
Motivic weight:	$24$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 1679616,\ (\ :[12]^{8}),\ 1)$

Particular Values

$L(\frac{25}{2})$	$\approx$	$27.95206882$
$L(\frac12)$	$\approx$	$27.95206882$
$L(13)$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ ( 1 + p^{23} T^{2} )^{4} $
	3	$ 1 + 43960 p T - 190519012 p^{6} T^{2} + 6136572680 p^{13} T^{3} + 2567030625142 p^{22} T^{4} + 6136572680 p^{37} T^{5} - 190519012 p^{54} T^{6} + 43960 p^{73} T^{7} + p^{96} T^{8} $
good	5	$ 1 - 242233888485069512 T^{2} + $$16\!\cdots\!76$$ p^{3} T^{4} - $$42\!\cdots\!12$$ p^{6} T^{6} + $$89\!\cdots\!26$$ p^{14} T^{8} - $$42\!\cdots\!12$$ p^{54} T^{10} + $$16\!\cdots\!76$$ p^{99} T^{12} - 242233888485069512 p^{144} T^{14} + p^{192} T^{16} $
	7	$ ( 1 + 5080397320 T + 52842286586751462436 p T^{2} + $$12\!\cdots\!40$$ p^{3} T^{3} + $$56\!\cdots\!02$$ p^{6} T^{4} + $$12\!\cdots\!40$$ p^{27} T^{5} + 52842286586751462436 p^{49} T^{6} + 5080397320 p^{72} T^{7} + p^{96} T^{8} )^{2} $
	11	$ 1 - $$43\!\cdots\!80$$ T^{2} + $$87\!\cdots\!44$$ p^{2} T^{4} - $$95\!\cdots\!40$$ p^{6} T^{6} + $$74\!\cdots\!66$$ p^{10} T^{8} - $$95\!\cdots\!40$$ p^{54} T^{10} + $$87\!\cdots\!44$$ p^{98} T^{12} - $$43\!\cdots\!80$$ p^{144} T^{14} + p^{192} T^{16} $
	13	$ ( 1 - 25284181839560 T + $$94\!\cdots\!44$$ T^{2} - $$30\!\cdots\!60$$ p T^{3} + $$17\!\cdots\!54$$ p^{2} T^{4} - $$30\!\cdots\!60$$ p^{25} T^{5} + $$94\!\cdots\!44$$ p^{48} T^{6} - 25284181839560 p^{72} T^{7} + p^{96} T^{8} )^{2} $
	17	$ 1 - $$15\!\cdots\!60$$ T^{2} + $$44\!\cdots\!16$$ p^{2} T^{4} - $$85\!\cdots\!80$$ p^{4} T^{6} + $$11\!\cdots\!74$$ p^{6} T^{8} - $$85\!\cdots\!80$$ p^{52} T^{10} + $$44\!\cdots\!16$$ p^{98} T^{12} - $$15\!\cdots\!60$$ p^{144} T^{14} + p^{192} T^{16} $
	19	$ ( 1 + 489041815670632 T + $$87\!\cdots\!12$$ p T^{2} + $$22\!\cdots\!84$$ p^{2} T^{3} + $$16\!\cdots\!70$$ p^{3} T^{4} + $$22\!\cdots\!84$$ p^{26} T^{5} + $$87\!\cdots\!12$$ p^{49} T^{6} + 489041815670632 p^{72} T^{7} + p^{96} T^{8} )^{2} $
	23	$ 1 - $$23\!\cdots\!20$$ p^{2} T^{2} + $$41\!\cdots\!24$$ p^{4} T^{4} - $$56\!\cdots\!60$$ p^{6} T^{6} + $$54\!\cdots\!06$$ p^{8} T^{8} - $$56\!\cdots\!60$$ p^{54} T^{10} + $$41\!\cdots\!24$$ p^{100} T^{12} - $$23\!\cdots\!20$$ p^{146} T^{14} + p^{192} T^{16} $
	29	$ 1 - $$36\!\cdots\!88$$ T^{2} + $$15\!\cdots\!48$$ T^{4} + $$40\!\cdots\!04$$ T^{6} + $$13\!\cdots\!70$$ T^{8} + $$40\!\cdots\!04$$ p^{48} T^{10} + $$15\!\cdots\!48$$ p^{96} T^{12} - $$36\!\cdots\!88$$ p^{144} T^{14} + p^{192} T^{16} $
	31	$ ( 1 - 1546059893411354552 T + $$25\!\cdots\!48$$ T^{2} - $$21\!\cdots\!64$$ T^{3} + $$21\!\cdots\!70$$ T^{4} - $$21\!\cdots\!64$$ p^{24} T^{5} + $$25\!\cdots\!48$$ p^{48} T^{6} - 1546059893411354552 p^{72} T^{7} + p^{96} T^{8} )^{2} $
	37	$ ( 1 - 11295146611996391240 T + $$17\!\cdots\!52$$ T^{2} - $$12\!\cdots\!60$$ T^{3} + $$10\!\cdots\!38$$ T^{4} - $$12\!\cdots\!60$$ p^{24} T^{5} + $$17\!\cdots\!52$$ p^{48} T^{6} - 11295146611996391240 p^{72} T^{7} + p^{96} T^{8} )^{2} $
	41	$ 1 - $$20\!\cdots\!40$$ T^{2} + $$23\!\cdots\!24$$ T^{4} - $$18\!\cdots\!20$$ T^{6} + $$11\!\cdots\!26$$ T^{8} - $$18\!\cdots\!20$$ p^{48} T^{10} + $$23\!\cdots\!24$$ p^{96} T^{12} - $$20\!\cdots\!40$$ p^{144} T^{14} + p^{192} T^{16} $
	43	$ ( 1 - $$11\!\cdots\!40$$ T + $$89\!\cdots\!36$$ T^{2} - $$47\!\cdots\!00$$ T^{3} + $$21\!\cdots\!46$$ T^{4} - $$47\!\cdots\!00$$ p^{24} T^{5} + $$89\!\cdots\!36$$ p^{48} T^{6} - $$11\!\cdots\!40$$ p^{72} T^{7} + p^{96} T^{8} )^{2} $
	47	$ 1 - $$68\!\cdots\!48$$ T^{2} + $$23\!\cdots\!08$$ T^{4} - $$54\!\cdots\!96$$ T^{6} + $$86\!\cdots\!70$$ T^{8} - $$54\!\cdots\!96$$ p^{48} T^{10} + $$23\!\cdots\!08$$ p^{96} T^{12} - $$68\!\cdots\!48$$ p^{144} T^{14} + p^{192} T^{16} $
	53	$ 1 - $$59\!\cdots\!80$$ T^{2} + $$23\!\cdots\!04$$ T^{4} - $$73\!\cdots\!40$$ T^{6} + $$17\!\cdots\!46$$ T^{8} - $$73\!\cdots\!40$$ p^{48} T^{10} + $$23\!\cdots\!04$$ p^{96} T^{12} - $$59\!\cdots\!80$$ p^{144} T^{14} + p^{192} T^{16} $
	59	$ 1 - $$12\!\cdots\!80$$ T^{2} + $$79\!\cdots\!44$$ T^{4} - $$35\!\cdots\!40$$ T^{6} + $$12\!\cdots\!66$$ T^{8} - $$35\!\cdots\!40$$ p^{48} T^{10} + $$79\!\cdots\!44$$ p^{96} T^{12} - $$12\!\cdots\!80$$ p^{144} T^{14} + p^{192} T^{16} $
	61	$ ( 1 + $$32\!\cdots\!24$$ T + $$16\!\cdots\!20$$ T^{2} + $$41\!\cdots\!96$$ T^{3} + $$13\!\cdots\!14$$ T^{4} + $$41\!\cdots\!96$$ p^{24} T^{5} + $$16\!\cdots\!20$$ p^{48} T^{6} + $$32\!\cdots\!24$$ p^{72} T^{7} + p^{96} T^{8} )^{2} $
	67	$ ( 1 + $$35\!\cdots\!60$$ T + $$20\!\cdots\!12$$ T^{2} + $$42\!\cdots\!00$$ T^{3} + $$17\!\cdots\!98$$ T^{4} + $$42\!\cdots\!00$$ p^{24} T^{5} + $$20\!\cdots\!12$$ p^{48} T^{6} + $$35\!\cdots\!60$$ p^{72} T^{7} + p^{96} T^{8} )^{2} $
	71	$ 1 - $$10\!\cdots\!88$$ T^{2} + $$63\!\cdots\!48$$ T^{4} - $$26\!\cdots\!96$$ T^{6} + $$83\!\cdots\!70$$ T^{8} - $$26\!\cdots\!96$$ p^{48} T^{10} + $$63\!\cdots\!48$$ p^{96} T^{12} - $$10\!\cdots\!88$$ p^{144} T^{14} + p^{192} T^{16} $
	73	$ ( 1 - $$18\!\cdots\!60$$ T + $$72\!\cdots\!24$$ T^{2} - $$40\!\cdots\!80$$ T^{3} + $$36\!\cdots\!06$$ T^{4} - $$40\!\cdots\!80$$ p^{24} T^{5} + $$72\!\cdots\!24$$ p^{48} T^{6} - $$18\!\cdots\!60$$ p^{72} T^{7} + p^{96} T^{8} )^{2} $
	79	$ ( 1 - $$15\!\cdots\!12$$ T + $$21\!\cdots\!88$$ T^{2} - $$17\!\cdots\!84$$ T^{3} + $$12\!\cdots\!30$$ T^{4} - $$17\!\cdots\!84$$ p^{24} T^{5} + $$21\!\cdots\!88$$ p^{48} T^{6} - $$15\!\cdots\!12$$ p^{72} T^{7} + p^{96} T^{8} )^{2} $
	83	$ 1 - $$34\!\cdots\!76$$ T^{2} + $$78\!\cdots\!20$$ T^{4} - $$13\!\cdots\!04$$ T^{6} + $$23\!\cdots\!26$$ p^{2} T^{8} - $$13\!\cdots\!04$$ p^{48} T^{10} + $$78\!\cdots\!20$$ p^{96} T^{12} - $$34\!\cdots\!76$$ p^{144} T^{14} + p^{192} T^{16} $
	89	$ 1 - $$15\!\cdots\!60$$ p T^{2} + $$10\!\cdots\!84$$ T^{4} - $$79\!\cdots\!20$$ T^{6} + $$58\!\cdots\!86$$ T^{8} - $$79\!\cdots\!20$$ p^{48} T^{10} + $$10\!\cdots\!84$$ p^{96} T^{12} - $$15\!\cdots\!60$$ p^{145} T^{14} + p^{192} T^{16} $
	97	$ ( 1 - $$40\!\cdots\!40$$ T + $$12\!\cdots\!76$$ T^{2} - $$73\!\cdots\!00$$ T^{3} + $$76\!\cdots\!46$$ T^{4} - $$73\!\cdots\!00$$ p^{24} T^{5} + $$12\!\cdots\!76$$ p^{48} T^{6} - $$40\!\cdots\!40$$ p^{72} T^{7} + p^{96} T^{8} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−6.13435039509716438315737053670, −5.94896674554765800587462846112, −5.58259360166925018181375303861, −5.43916590617675305517866381095, −4.77829198380849522749416059564, −4.77507282208977231312617984814, −4.51683358311718038156666834832, −4.41935538334447090486624688009, −4.39453258195576504780431382451, −4.13828093260468041706972791137, −3.61993267637823713923120818443, −3.32089522526169056143070494622, −3.19496260735458942556410499950, −3.06021780283389970378503335732, −2.70409848570300123722015936925, −2.65544190318909419815519909205, −2.12495071391108041714130029764, −1.82589380788275317168779895448, −1.49601786135068435117804322497, −1.18930130381941536553543523570, −0.76899749529298679323563685936, −0.74175779133109537033970377374, −0.63596541731637220328247265360, −0.61567017855446826661169655473, −0.61232541560928236478977761919, 0.61232541560928236478977761919, 0.61567017855446826661169655473, 0.63596541731637220328247265360, 0.74175779133109537033970377374, 0.76899749529298679323563685936, 1.18930130381941536553543523570, 1.49601786135068435117804322497, 1.82589380788275317168779895448, 2.12495071391108041714130029764, 2.65544190318909419815519909205, 2.70409848570300123722015936925, 3.06021780283389970378503335732, 3.19496260735458942556410499950, 3.32089522526169056143070494622, 3.61993267637823713923120818443, 4.13828093260468041706972791137, 4.39453258195576504780431382451, 4.41935538334447090486624688009, 4.51683358311718038156666834832, 4.77507282208977231312617984814, 4.77829198380849522749416059564, 5.43916590617675305517866381095, 5.58259360166925018181375303861, 5.94896674554765800587462846112, 6.13435039509716438315737053670

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{25}{2})\)	\(\approx\)	\(27.95206882\)
\(L(\frac12)\)	\(\approx\)	\(27.95206882\)
\(L(13)\)		not available
\(L(1)\)		not available

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