LMFDB - L-function 16-20e8-1.1-c8e8-0-0

Dirichlet series

L(s) = 1

− 70·3^-s + 894·5^-s − 2.03e3·7^-s + 2.45e3·9^-s − 420·11^-s + 3.31e4·13^-s − 6.25e4·15^-s + 4.36e4·17^-s + 1.42e5·21^-s − 6.63e5·23^-s + 4.81e5·25^-s + 3.15e5·27^-s − 3.17e6·31^-s + 2.94e4·33^-s − 1.81e6·35^-s + 5.34e6·37^-s − 2.32e6·39^-s − 1.01e7·41^-s − 1.03e7·43^-s + 2.19e6·45^-s + 1.92e7·47^-s + 2.06e6·49^-s − 3.05e6·51^-s − 2.43e7·53^-s − 3.75e5·55^-s − 8.25e7·61^-s − 4.97e6·63^-s + ⋯

L(s) = 1

− 0.864·3^-s + 1.43·5^-s − 0.845·7^-s + 0.373·9^-s − 0.0286·11^-s + 1.16·13^-s − 1.23·15^-s + 0.522·17^-s + 0.730·21^-s − 2.37·23^-s + 1.23·25^-s + 0.592·27^-s − 3.44·31^-s + 0.0247·33^-s − 1.20·35^-s + 2.85·37^-s − 1.00·39^-s − 3.60·41^-s − 3.02·43^-s + 0.534·45^-s + 3.94·47^-s + 0.357·49^-s − 0.451·51^-s − 3.08·53^-s − 0.0410·55^-s − 5.95·61^-s − 0.315·63^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$2^{16} \cdot 5^{8}$
Sign:	$1$
Analytic conductor:	$1.94189\times 10^{7}$
Root analytic conductor:	$2.85439$
Motivic weight:	$8$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{16} \cdot 5^{8} ,\ ( \ : [4]^{8} ),\ 1 )$

Particular Values

$L(\frac{9}{2})$	$\approx$	$0.01853279274$
$L(\frac12)$	$\approx$	$0.01853279274$
$L(5)$		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 $
	5	$ 1 - 894 T + 63584 p T^{2} + 155646 p^{4} T^{3} - 2923338 p^{7} T^{4} + 155646 p^{12} T^{5} + 63584 p^{17} T^{6} - 894 p^{24} T^{7} + p^{32} T^{8} $
good	3	$ 1 + 70 T + 2450 T^{2} - 3890 p^{4} T^{3} - 2127488 p^{2} T^{4} - 129108590 p^{2} T^{5} + 1690394750 p^{2} T^{6} - 21854802310 p^{6} T^{7} - 13724994098 p^{10} T^{8} - 21854802310 p^{14} T^{9} + 1690394750 p^{18} T^{10} - 129108590 p^{26} T^{11} - 2127488 p^{34} T^{12} - 3890 p^{44} T^{13} + 2450 p^{48} T^{14} + 70 p^{56} T^{15} + p^{64} T^{16} $
	7	$ 1 + 290 p T + 42050 p^{2} T^{2} + 290506630 p^{2} T^{3} + 687872758304 p^{2} T^{4} - 13377173131190 p^{4} T^{5} - 118009162050 p^{10} T^{6} - 30732720771943710 p^{8} T^{7} - $$76\!\cdots\!06$$ p^{6} T^{8} - 30732720771943710 p^{16} T^{9} - 118009162050 p^{26} T^{10} - 13377173131190 p^{28} T^{11} + 687872758304 p^{34} T^{12} + 290506630 p^{42} T^{13} + 42050 p^{50} T^{14} + 290 p^{57} T^{15} + p^{64} T^{16} $
	11	$ ( 1 + 210 T + 276969824 T^{2} - 4317403025970 T^{3} + 22655350934029566 T^{4} - 4317403025970 p^{8} T^{5} + 276969824 p^{16} T^{6} + 210 p^{24} T^{7} + p^{32} T^{8} )^{2} $
	13	$ 1 - 33180 T + 550456200 T^{2} + 3382789736460 T^{3} + 185791033656199948 T^{4} + $$55\!\cdots\!40$$ T^{5} - $$68\!\cdots\!00$$ p^{2} T^{6} + $$19\!\cdots\!20$$ p T^{7} - $$92\!\cdots\!62$$ T^{8} + $$19\!\cdots\!20$$ p^{9} T^{9} - $$68\!\cdots\!00$$ p^{18} T^{10} + $$55\!\cdots\!40$$ p^{24} T^{11} + 185791033656199948 p^{32} T^{12} + 3382789736460 p^{40} T^{13} + 550456200 p^{48} T^{14} - 33180 p^{56} T^{15} + p^{64} T^{16} $
	17	$ 1 - 43620 T + 951352200 T^{2} - 106908559974060 T^{3} - 27247709503236830452 T^{4} + $$73\!\cdots\!60$$ T^{5} - $$40\!\cdots\!00$$ T^{6} - $$35\!\cdots\!60$$ T^{7} + $$44\!\cdots\!98$$ T^{8} - $$35\!\cdots\!60$$ p^{8} T^{9} - $$40\!\cdots\!00$$ p^{16} T^{10} + $$73\!\cdots\!60$$ p^{24} T^{11} - 27247709503236830452 p^{32} T^{12} - 106908559974060 p^{40} T^{13} + 951352200 p^{48} T^{14} - 43620 p^{56} T^{15} + p^{64} T^{16} $
	19	$ 1 - 1479904168 p T^{2} + $$48\!\cdots\!48$$ T^{4} - $$44\!\cdots\!24$$ T^{6} + $$67\!\cdots\!70$$ T^{8} - $$44\!\cdots\!24$$ p^{16} T^{10} + $$48\!\cdots\!48$$ p^{32} T^{12} - 1479904168 p^{49} T^{14} + p^{64} T^{16} $
	23	$ 1 + 663270 T + 219963546450 T^{2} + 50684927414310030 T^{3} + $$11\!\cdots\!56$$ T^{4} + $$25\!\cdots\!90$$ T^{5} + $$41\!\cdots\!50$$ T^{6} - $$25\!\cdots\!10$$ p^{2} T^{7} - $$29\!\cdots\!14$$ p^{4} T^{8} - $$25\!\cdots\!10$$ p^{10} T^{9} + $$41\!\cdots\!50$$ p^{16} T^{10} + $$25\!\cdots\!90$$ p^{24} T^{11} + $$11\!\cdots\!56$$ p^{32} T^{12} + 50684927414310030 p^{40} T^{13} + 219963546450 p^{48} T^{14} + 663270 p^{56} T^{15} + p^{64} T^{16} $
	29	$ 1 - 2743503231112 T^{2} + $$36\!\cdots\!88$$ T^{4} - $$30\!\cdots\!64$$ T^{6} + $$17\!\cdots\!70$$ T^{8} - $$30\!\cdots\!64$$ p^{16} T^{10} + $$36\!\cdots\!88$$ p^{32} T^{12} - 2743503231112 p^{48} T^{14} + p^{64} T^{16} $
	31	$ ( 1 + 51266 p T + 3863416314520 T^{2} + 3956117308235249954 T^{3} + $$16\!\cdots\!34$$ p T^{4} + 3956117308235249954 p^{8} T^{5} + 3863416314520 p^{16} T^{6} + 51266 p^{25} T^{7} + p^{32} T^{8} )^{2} $
	37	$ 1 - 5344080 T + 14279595523200 T^{2} - 29203723934142519120 T^{3} + $$43\!\cdots\!36$$ T^{4} - $$47\!\cdots\!60$$ T^{5} + $$62\!\cdots\!00$$ T^{6} - $$94\!\cdots\!40$$ T^{7} + $$15\!\cdots\!86$$ T^{8} - $$94\!\cdots\!40$$ p^{8} T^{9} + $$62\!\cdots\!00$$ p^{16} T^{10} - $$47\!\cdots\!60$$ p^{24} T^{11} + $$43\!\cdots\!36$$ p^{32} T^{12} - 29203723934142519120 p^{40} T^{13} + 14279595523200 p^{48} T^{14} - 5344080 p^{56} T^{15} + p^{64} T^{16} $
	41	$ ( 1 + 5092626 T + 19188772875200 T^{2} + 33007106306449428174 T^{3} + $$95\!\cdots\!14$$ T^{4} + 33007106306449428174 p^{8} T^{5} + 19188772875200 p^{16} T^{6} + 5092626 p^{24} T^{7} + p^{32} T^{8} )^{2} $
	43	$ 1 + 10342710 T + 53485825072050 T^{2} + $$23\!\cdots\!10$$ T^{3} + $$70\!\cdots\!04$$ T^{4} + $$90\!\cdots\!30$$ T^{5} - $$15\!\cdots\!50$$ T^{6} - $$19\!\cdots\!70$$ T^{7} - $$97\!\cdots\!94$$ T^{8} - $$19\!\cdots\!70$$ p^{8} T^{9} - $$15\!\cdots\!50$$ p^{16} T^{10} + $$90\!\cdots\!30$$ p^{24} T^{11} + $$70\!\cdots\!04$$ p^{32} T^{12} + $$23\!\cdots\!10$$ p^{40} T^{13} + 53485825072050 p^{48} T^{14} + 10342710 p^{56} T^{15} + p^{64} T^{16} $
	47	$ 1 - 19232250 T + 184939720031250 T^{2} - $$12\!\cdots\!50$$ T^{3} + $$71\!\cdots\!48$$ T^{4} - $$32\!\cdots\!50$$ T^{5} + $$12\!\cdots\!50$$ T^{6} - $$42\!\cdots\!50$$ T^{7} + $$15\!\cdots\!58$$ T^{8} - $$42\!\cdots\!50$$ p^{8} T^{9} + $$12\!\cdots\!50$$ p^{16} T^{10} - $$32\!\cdots\!50$$ p^{24} T^{11} + $$71\!\cdots\!48$$ p^{32} T^{12} - $$12\!\cdots\!50$$ p^{40} T^{13} + 184939720031250 p^{48} T^{14} - 19232250 p^{56} T^{15} + p^{64} T^{16} $
	53	$ 1 + 458880 p T + 105285427200 p^{2} T^{2} + $$22\!\cdots\!60$$ T^{3} + $$10\!\cdots\!16$$ T^{4} + $$48\!\cdots\!80$$ T^{5} + $$41\!\cdots\!00$$ T^{6} + $$42\!\cdots\!20$$ T^{7} + $$37\!\cdots\!46$$ T^{8} + $$42\!\cdots\!20$$ p^{8} T^{9} + $$41\!\cdots\!00$$ p^{16} T^{10} + $$48\!\cdots\!80$$ p^{24} T^{11} + $$10\!\cdots\!16$$ p^{32} T^{12} + $$22\!\cdots\!60$$ p^{40} T^{13} + 105285427200 p^{50} T^{14} + 458880 p^{57} T^{15} + p^{64} T^{16} $
	59	$ 1 - 736741399423864 T^{2} + $$26\!\cdots\!00$$ T^{4} - $$60\!\cdots\!56$$ T^{6} + $$10\!\cdots\!74$$ T^{8} - $$60\!\cdots\!56$$ p^{16} T^{10} + $$26\!\cdots\!00$$ p^{32} T^{12} - 736741399423864 p^{48} T^{14} + p^{64} T^{16} $
	61	$ ( 1 + 41257842 T + 1301244818102848 T^{2} + $$26\!\cdots\!74$$ T^{3} + $$42\!\cdots\!70$$ T^{4} + $$26\!\cdots\!74$$ p^{8} T^{5} + 1301244818102848 p^{16} T^{6} + 41257842 p^{24} T^{7} + p^{32} T^{8} )^{2} $
	67	$ 1 - 100675930 T + 5067821440682450 T^{2} - $$17\!\cdots\!70$$ T^{3} + $$51\!\cdots\!76$$ T^{4} - $$13\!\cdots\!10$$ T^{5} + $$30\!\cdots\!50$$ T^{6} - $$65\!\cdots\!90$$ T^{7} + $$13\!\cdots\!66$$ T^{8} - $$65\!\cdots\!90$$ p^{8} T^{9} + $$30\!\cdots\!50$$ p^{16} T^{10} - $$13\!\cdots\!10$$ p^{24} T^{11} + $$51\!\cdots\!76$$ p^{32} T^{12} - $$17\!\cdots\!70$$ p^{40} T^{13} + 5067821440682450 p^{48} T^{14} - 100675930 p^{56} T^{15} + p^{64} T^{16} $
	71	$ ( 1 + 49645038 T + 1836961055683448 T^{2} + $$55\!\cdots\!46$$ T^{3} + $$17\!\cdots\!70$$ T^{4} + $$55\!\cdots\!46$$ p^{8} T^{5} + 1836961055683448 p^{16} T^{6} + 49645038 p^{24} T^{7} + p^{32} T^{8} )^{2} $
	73	$ 1 + 93528520 T + 4373792026695200 T^{2} + $$17\!\cdots\!60$$ T^{3} + $$66\!\cdots\!68$$ T^{4} + $$21\!\cdots\!40$$ T^{5} + $$64\!\cdots\!00$$ T^{6} + $$19\!\cdots\!60$$ T^{7} + $$55\!\cdots\!98$$ T^{8} + $$19\!\cdots\!60$$ p^{8} T^{9} + $$64\!\cdots\!00$$ p^{16} T^{10} + $$21\!\cdots\!40$$ p^{24} T^{11} + $$66\!\cdots\!68$$ p^{32} T^{12} + $$17\!\cdots\!60$$ p^{40} T^{13} + 4373792026695200 p^{48} T^{14} + 93528520 p^{56} T^{15} + p^{64} T^{16} $
	79	$ 1 - 8443828027508744 T^{2} + $$33\!\cdots\!60$$ T^{4} - $$81\!\cdots\!96$$ T^{6} + $$14\!\cdots\!54$$ T^{8} - $$81\!\cdots\!96$$ p^{16} T^{10} + $$33\!\cdots\!60$$ p^{32} T^{12} - 8443828027508744 p^{48} T^{14} + p^{64} T^{16} $
	83	$ 1 + 10450350 T + 54604907561250 T^{2} + $$49\!\cdots\!50$$ T^{3} - $$33\!\cdots\!24$$ T^{4} - $$13\!\cdots\!50$$ T^{5} - $$45\!\cdots\!50$$ T^{6} - $$13\!\cdots\!50$$ T^{7} - $$36\!\cdots\!34$$ T^{8} - $$13\!\cdots\!50$$ p^{8} T^{9} - $$45\!\cdots\!50$$ p^{16} T^{10} - $$13\!\cdots\!50$$ p^{24} T^{11} - $$33\!\cdots\!24$$ p^{32} T^{12} + $$49\!\cdots\!50$$ p^{40} T^{13} + 54604907561250 p^{48} T^{14} + 10450350 p^{56} T^{15} + p^{64} T^{16} $
	89	$ 1 - 29031783018178312 T^{2} + $$37\!\cdots\!48$$ T^{4} - $$28\!\cdots\!04$$ T^{6} + $$13\!\cdots\!70$$ T^{8} - $$28\!\cdots\!04$$ p^{16} T^{10} + $$37\!\cdots\!48$$ p^{32} T^{12} - 29031783018178312 p^{48} T^{14} + p^{64} T^{16} $
	97	$ 1 + 179570760 T + 16122828923488800 T^{2} + $$18\!\cdots\!40$$ T^{3} + $$13\!\cdots\!16$$ T^{4} - $$19\!\cdots\!80$$ T^{5} - $$85\!\cdots\!00$$ T^{6} - $$14\!\cdots\!20$$ T^{7} - $$18\!\cdots\!54$$ T^{8} - $$14\!\cdots\!20$$ p^{8} T^{9} - $$85\!\cdots\!00$$ p^{16} T^{10} - $$19\!\cdots\!80$$ p^{24} T^{11} + $$13\!\cdots\!16$$ p^{32} T^{12} + $$18\!\cdots\!40$$ p^{40} T^{13} + 16122828923488800 p^{48} T^{14} + 179570760 p^{56} T^{15} + p^{64} T^{16} $
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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

−7.12010417819623631121098485997, −6.77200290341409534084951023465, −6.33977016625311350513588413138, −6.23750678637450605036127611333, −6.23202946425692838993933299321, −6.04940699757793439287152211671, −5.82168753480275489952204016449, −5.42206231285029482342443156800, −5.38230094842907061035518594469, −5.06645271533106482217520107901, −4.72593194009678371211659790338, −4.44294057197661979432060333111, −4.15715734147188816438990658587, −3.79119871676131944700207681582, −3.59520153066142442388506976854, −3.18953739523807708609057166360, −3.08519779852104617380974202390, −2.58060495656363156910575741379, −2.31964632469469384098061149790, −1.75865394334544675892727432177, −1.49724429664611773817830664444, −1.40176512064521413425336520465, −1.22716749986544897453117398229, −0.24790021959691861086426109611, −0.03432330432509585533609658412, 0.03432330432509585533609658412, 0.24790021959691861086426109611, 1.22716749986544897453117398229, 1.40176512064521413425336520465, 1.49724429664611773817830664444, 1.75865394334544675892727432177, 2.31964632469469384098061149790, 2.58060495656363156910575741379, 3.08519779852104617380974202390, 3.18953739523807708609057166360, 3.59520153066142442388506976854, 3.79119871676131944700207681582, 4.15715734147188816438990658587, 4.44294057197661979432060333111, 4.72593194009678371211659790338, 5.06645271533106482217520107901, 5.38230094842907061035518594469, 5.42206231285029482342443156800, 5.82168753480275489952204016449, 6.04940699757793439287152211671, 6.23202946425692838993933299321, 6.23750678637450605036127611333, 6.33977016625311350513588413138, 6.77200290341409534084951023465, 7.12010417819623631121098485997

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

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

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 $\F_{q}$

\(L(\frac{9}{2})\)	\(\approx\)	\(0.01853279274\)
\(L(\frac12)\)	\(\approx\)	\(0.01853279274\)
\(L(5)\)		not available
\(L(1)\)		not available

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