LMFDB - L-function 16-6e8-1.1-c22e8-0-0

Dirichlet series

L(s) = 1

− 6.97e4·3^-s − 8.38e6·4^-s + 5.64e9·7^-s − 2.43e10·9^-s + 5.84e11·12^-s − 2.87e12·13^-s + 4.39e13·16^-s − 3.14e14·19^-s − 3.93e14·21^-s + 6.40e15·25^-s + 5.73e15·27^-s − 4.73e16·28^-s − 1.10e17·31^-s + 2.04e17·36^-s − 7.52e17·37^-s + 2.00e17·39^-s − 1.30e18·43^-s − 3.06e18·48^-s + 6.31e18·49^-s + 2.41e19·52^-s + 2.19e19·57^-s − 1.24e20·61^-s − 1.37e20·63^-s − 1.84e20·64^-s − 3.49e20·67^-s − 6.55e19·73^-s − 4.46e20·75^-s + ⋯

L(s) = 1

− 0.393·3^-s − 2·4^-s + 2.85·7^-s − 0.776·9^-s + 0.787·12^-s − 1.60·13^-s + 5/2·16^-s − 2.70·19^-s − 1.12·21^-s + 2.68·25^-s + 1.03·27^-s − 5.71·28^-s − 4.34·31^-s + 1.55·36^-s − 4.22·37^-s + 0.631·39^-s − 1.40·43^-s − 0.983·48^-s + 1.61·49^-s + 3.20·52^-s + 1.06·57^-s − 2.85·61^-s − 2.21·63^-s − 5/2·64^-s − 2.86·67^-s − 0.209·73^-s − 1.05·75^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(23-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+11)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$1679616$ = $2^{8} \cdot 3^{8}$
Sign:	$1$
Analytic conductor:	$1.31523\times 10^{10}$
Root analytic conductor:	$4.28980$
Motivic weight:	$22$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 1679616,\ (\ :[11]^{8}),\ 1)$

Particular Values

$L(\frac{23}{2})$	$\approx$	$0.006966289427$
$L(\frac12)$	$\approx$	$0.006966289427$
$L(12)$		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^{21} T^{2} )^{4} $
	3	$ 1 + 23240 p T + 360822268 p^{4} T^{2} - 33772184600 p^{10} T^{3} + 242900943238 p^{20} T^{4} - 33772184600 p^{32} T^{5} + 360822268 p^{48} T^{6} + 23240 p^{67} T^{7} + p^{88} T^{8} $
good	5	$ 1 - 1280906811524008 p T^{2} + $$90\!\cdots\!16$$ p^{2} T^{4} - $$94\!\cdots\!56$$ p^{7} T^{6} + $$83\!\cdots\!54$$ p^{12} T^{8} - $$94\!\cdots\!56$$ p^{51} T^{10} + $$90\!\cdots\!16$$ p^{90} T^{12} - 1280906811524008 p^{133} T^{14} + p^{176} T^{16} $
	7	$ ( 1 - 2822790920 T + 1256624077538293444 p T^{2} - $$86\!\cdots\!00$$ p^{4} T^{3} + $$22\!\cdots\!58$$ p^{4} T^{4} - $$86\!\cdots\!00$$ p^{26} T^{5} + 1256624077538293444 p^{45} T^{6} - 2822790920 p^{66} T^{7} + p^{88} T^{8} )^{2} $
	11	$ 1 - $$17\!\cdots\!80$$ p T^{2} + $$58\!\cdots\!84$$ p^{2} T^{4} + $$19\!\cdots\!60$$ p^{3} T^{6} - $$14\!\cdots\!54$$ p^{4} T^{8} + $$19\!\cdots\!60$$ p^{47} T^{10} + $$58\!\cdots\!84$$ p^{90} T^{12} - $$17\!\cdots\!80$$ p^{133} T^{14} + p^{176} T^{16} $
	13	$ ( 1 + 1437841190680 T + $$68\!\cdots\!76$$ T^{2} + $$11\!\cdots\!20$$ p T^{3} + $$14\!\cdots\!14$$ p^{2} T^{4} + $$11\!\cdots\!20$$ p^{23} T^{5} + $$68\!\cdots\!76$$ p^{44} T^{6} + 1437841190680 p^{66} T^{7} + p^{88} T^{8} )^{2} $
	17	$ 1 - $$68\!\cdots\!20$$ T^{2} + $$77\!\cdots\!36$$ p^{2} T^{4} - $$55\!\cdots\!60$$ p^{4} T^{6} + $$27\!\cdots\!94$$ p^{6} T^{8} - $$55\!\cdots\!60$$ p^{48} T^{10} + $$77\!\cdots\!36$$ p^{90} T^{12} - $$68\!\cdots\!20$$ p^{132} T^{14} + p^{176} T^{16} $
	19	$ ( 1 + 157401978775768 T + $$12\!\cdots\!92$$ p T^{2} + $$42\!\cdots\!16$$ p^{2} T^{3} + $$20\!\cdots\!10$$ p^{3} T^{4} + $$42\!\cdots\!16$$ p^{24} T^{5} + $$12\!\cdots\!92$$ p^{45} T^{6} + 157401978775768 p^{66} T^{7} + p^{88} T^{8} )^{2} $
	23	$ 1 - $$26\!\cdots\!60$$ T^{2} + $$29\!\cdots\!84$$ T^{4} - $$31\!\cdots\!80$$ T^{6} + $$33\!\cdots\!26$$ T^{8} - $$31\!\cdots\!80$$ p^{44} T^{10} + $$29\!\cdots\!84$$ p^{88} T^{12} - $$26\!\cdots\!60$$ p^{132} T^{14} + p^{176} T^{16} $
	29	$ 1 - $$10\!\cdots\!08$$ T^{2} + $$47\!\cdots\!48$$ T^{4} - $$13\!\cdots\!76$$ T^{6} + $$23\!\cdots\!70$$ T^{8} - $$13\!\cdots\!76$$ p^{44} T^{10} + $$47\!\cdots\!48$$ p^{88} T^{12} - $$10\!\cdots\!08$$ p^{132} T^{14} + p^{176} T^{16} $
	31	$ ( 1 + 55146614124607288 T + $$28\!\cdots\!48$$ T^{2} + $$10\!\cdots\!96$$ T^{3} + $$28\!\cdots\!70$$ T^{4} + $$10\!\cdots\!96$$ p^{22} T^{5} + $$28\!\cdots\!48$$ p^{44} T^{6} + 55146614124607288 p^{66} T^{7} + p^{88} T^{8} )^{2} $
	37	$ ( 1 + 376245183281642200 T + $$13\!\cdots\!88$$ T^{2} + $$27\!\cdots\!40$$ T^{3} + $$59\!\cdots\!98$$ T^{4} + $$27\!\cdots\!40$$ p^{22} T^{5} + $$13\!\cdots\!88$$ p^{44} T^{6} + 376245183281642200 p^{66} T^{7} + p^{88} T^{8} )^{2} $
	41	$ 1 - $$15\!\cdots\!40$$ T^{2} + $$11\!\cdots\!24$$ T^{4} - $$59\!\cdots\!20$$ T^{6} + $$21\!\cdots\!86$$ T^{8} - $$59\!\cdots\!20$$ p^{44} T^{10} + $$11\!\cdots\!24$$ p^{88} T^{12} - $$15\!\cdots\!40$$ p^{132} T^{14} + p^{176} T^{16} $
	43	$ ( 1 + 650507781721037080 T + $$13\!\cdots\!04$$ T^{2} + $$10\!\cdots\!60$$ T^{3} + $$14\!\cdots\!46$$ T^{4} + $$10\!\cdots\!60$$ p^{22} T^{5} + $$13\!\cdots\!04$$ p^{44} T^{6} + 650507781721037080 p^{66} T^{7} + p^{88} T^{8} )^{2} $
	47	$ 1 - $$20\!\cdots\!72$$ T^{2} + $$19\!\cdots\!68$$ T^{4} - $$14\!\cdots\!24$$ T^{6} + $$10\!\cdots\!70$$ T^{8} - $$14\!\cdots\!24$$ p^{44} T^{10} + $$19\!\cdots\!68$$ p^{88} T^{12} - $$20\!\cdots\!72$$ p^{132} T^{14} + p^{176} T^{16} $
	53	$ 1 - $$40\!\cdots\!40$$ T^{2} + $$84\!\cdots\!44$$ T^{4} - $$11\!\cdots\!20$$ T^{6} + $$11\!\cdots\!06$$ T^{8} - $$11\!\cdots\!20$$ p^{44} T^{10} + $$84\!\cdots\!44$$ p^{88} T^{12} - $$40\!\cdots\!40$$ p^{132} T^{14} + p^{176} T^{16} $
	59	$ 1 - $$35\!\cdots\!60$$ T^{2} + $$74\!\cdots\!64$$ T^{4} - $$10\!\cdots\!80$$ T^{6} + $$11\!\cdots\!66$$ T^{8} - $$10\!\cdots\!80$$ p^{44} T^{10} + $$74\!\cdots\!64$$ p^{88} T^{12} - $$35\!\cdots\!60$$ p^{132} T^{14} + p^{176} T^{16} $
	61	$ ( 1 + 62207640705914143384 T + $$70\!\cdots\!60$$ T^{2} + $$31\!\cdots\!96$$ T^{3} + $$20\!\cdots\!74$$ T^{4} + $$31\!\cdots\!96$$ p^{22} T^{5} + $$70\!\cdots\!60$$ p^{44} T^{6} + 62207640705914143384 p^{66} T^{7} + p^{88} T^{8} )^{2} $
	67	$ ( 1 + $$17\!\cdots\!40$$ T + $$39\!\cdots\!08$$ T^{2} + $$32\!\cdots\!80$$ T^{3} + $$58\!\cdots\!18$$ T^{4} + $$32\!\cdots\!80$$ p^{22} T^{5} + $$39\!\cdots\!08$$ p^{44} T^{6} + $$17\!\cdots\!40$$ p^{66} T^{7} + p^{88} T^{8} )^{2} $
	71	$ 1 - $$93\!\cdots\!08$$ T^{2} + $$13\!\cdots\!48$$ T^{4} - $$79\!\cdots\!76$$ T^{6} + $$60\!\cdots\!70$$ T^{8} - $$79\!\cdots\!76$$ p^{44} T^{10} + $$13\!\cdots\!48$$ p^{88} T^{12} - $$93\!\cdots\!08$$ p^{132} T^{14} + p^{176} T^{16} $
	73	$ ( 1 + 32788189594240031800 T + $$37\!\cdots\!96$$ T^{2} + $$91\!\cdots\!00$$ T^{3} + $$54\!\cdots\!86$$ T^{4} + $$91\!\cdots\!00$$ p^{22} T^{5} + $$37\!\cdots\!96$$ p^{44} T^{6} + 32788189594240031800 p^{66} T^{7} + p^{88} T^{8} )^{2} $
	79	$ ( 1 + $$16\!\cdots\!72$$ T + $$16\!\cdots\!28$$ T^{2} + $$26\!\cdots\!04$$ T^{3} + $$11\!\cdots\!90$$ T^{4} + $$26\!\cdots\!04$$ p^{22} T^{5} + $$16\!\cdots\!28$$ p^{44} T^{6} + $$16\!\cdots\!72$$ p^{66} T^{7} + p^{88} T^{8} )^{2} $
	83	$ 1 - $$98\!\cdots\!24$$ T^{2} + $$44\!\cdots\!80$$ T^{4} - $$12\!\cdots\!36$$ T^{6} + $$24\!\cdots\!54$$ T^{8} - $$12\!\cdots\!36$$ p^{44} T^{10} + $$44\!\cdots\!80$$ p^{88} T^{12} - $$98\!\cdots\!24$$ p^{132} T^{14} + p^{176} T^{16} $
	89	$ 1 - $$31\!\cdots\!80$$ T^{2} + $$65\!\cdots\!56$$ p T^{4} - $$68\!\cdots\!40$$ T^{6} + $$15\!\cdots\!26$$ T^{8} - $$68\!\cdots\!40$$ p^{44} T^{10} + $$65\!\cdots\!56$$ p^{89} T^{12} - $$31\!\cdots\!80$$ p^{132} T^{14} + p^{176} T^{16} $
	97	$ ( 1 + $$11\!\cdots\!60$$ T + $$25\!\cdots\!24$$ T^{2} + $$19\!\cdots\!80$$ T^{3} + $$20\!\cdots\!66$$ T^{4} + $$19\!\cdots\!80$$ p^{22} T^{5} + $$25\!\cdots\!24$$ p^{44} T^{6} + $$11\!\cdots\!60$$ p^{66} T^{7} + p^{88} 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

−6.39435692315290611878426193626, −6.09495039423803243113102479723, −5.45658853996681905254703520336, −5.40189581110705210435723816819, −5.26004700493996373502867673102, −5.14407975236591586907266250326, −5.01628889945769371015073619751, −4.55436833473363802863138654614, −4.47412004923420377580163612198, −4.45648588002448971467640395862, −3.90710419240821869368418732219, −3.69332664456683175345029923095, −3.50849191734711728377163142319, −3.06803317682154199827231342508, −2.78652067886083854168466816341, −2.53470554810822639501796941672, −2.27437853850720479353592392927, −1.73250506297629130202421026732, −1.58694709829130444936881517984, −1.42058027988970993848839809919, −1.36193409838823260870900480250, −1.23420671005004624428010884420, −0.23890983665366691288940407413, −0.096661174551647009121265830908, −0.07868669097901799890719335962, 0.07868669097901799890719335962, 0.096661174551647009121265830908, 0.23890983665366691288940407413, 1.23420671005004624428010884420, 1.36193409838823260870900480250, 1.42058027988970993848839809919, 1.58694709829130444936881517984, 1.73250506297629130202421026732, 2.27437853850720479353592392927, 2.53470554810822639501796941672, 2.78652067886083854168466816341, 3.06803317682154199827231342508, 3.50849191734711728377163142319, 3.69332664456683175345029923095, 3.90710419240821869368418732219, 4.45648588002448971467640395862, 4.47412004923420377580163612198, 4.55436833473363802863138654614, 5.01628889945769371015073619751, 5.14407975236591586907266250326, 5.26004700493996373502867673102, 5.40189581110705210435723816819, 5.45658853996681905254703520336, 6.09495039423803243113102479723, 6.39435692315290611878426193626

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 $\F_{q}$

\(L(\frac{23}{2})\)	\(\approx\)	\(0.006966289427\)
\(L(\frac12)\)	\(\approx\)	\(0.006966289427\)
\(L(12)\)		not available
\(L(1)\)		not available

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