LMFDB - L-function 16-208e8-1.1-c5e8-0-0

Dirichlet series

L(s) = 1

− 8·3^-s − 20·5^-s − 68·7^-s + 503·9^-s + 480·11^-s − 234·13^-s + 160·15^-s + 60·17^-s − 520·19^-s + 544·21^-s − 6.64e3·23^-s − 1.90e4·25^-s − 1.73e3·27^-s − 1.23e3·29^-s − 2.47e4·31^-s − 3.84e3·33^-s + 1.36e3·35^-s − 1.19e4·37^-s + 1.87e3·39^-s + 2.44e4·41^-s − 1.63e4·43^-s − 1.00e4·45^-s + 1.53e3·47^-s + 5.91e4·49^-s − 480·51^-s − 4.99e4·53^-s − 9.60e3·55^-s + ⋯

L(s) = 1

− 0.513·3^-s − 0.357·5^-s − 0.524·7^-s + 2.06·9^-s + 1.19·11^-s − 0.384·13^-s + 0.183·15^-s + 0.0503·17^-s − 0.330·19^-s + 0.269·21^-s − 2.61·23^-s − 6.10·25^-s − 0.458·27^-s − 0.272·29^-s − 4.61·31^-s − 0.613·33^-s + 0.187·35^-s − 1.44·37^-s + 0.197·39^-s + 2.26·41^-s − 1.34·43^-s − 0.740·45^-s + 0.101·47^-s + 3.51·49^-s − 0.0258·51^-s − 2.44·53^-s − 0.427·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$2^{32} \cdot 13^{8}$
Sign:	$1$
Analytic conductor:	$1.53387\times 10^{12}$
Root analytic conductor:	$5.77579$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{32} \cdot 13^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$0.03387810085$
$L(\frac12)$	$\approx$	$0.03387810085$
$L(\frac{7}{2})$		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 $
	13	$ 1 + 18 p T - 19 p^{2} T^{2} + 1458 p^{4} T^{3} - 22428 p^{6} T^{4} + 1458 p^{9} T^{5} - 19 p^{12} T^{6} + 18 p^{16} T^{7} + p^{20} T^{8} $
good	3	$ 1 + 8 T - 439 T^{2} - 5800 T^{3} + 79673 T^{4} + 1519712 T^{5} + 6559990 T^{6} - 63322928 p T^{7} - 498485882 p^{2} T^{8} - 63322928 p^{6} T^{9} + 6559990 p^{10} T^{10} + 1519712 p^{15} T^{11} + 79673 p^{20} T^{12} - 5800 p^{25} T^{13} - 439 p^{30} T^{14} + 8 p^{35} T^{15} + p^{40} T^{16} $
	5	$ ( 1 + 2 p T + 9693 T^{2} + 20498 p T^{3} + 41407156 T^{4} + 20498 p^{6} T^{5} + 9693 p^{10} T^{6} + 2 p^{16} T^{7} + p^{20} T^{8} )^{2} $
	7	$ 1 + 68 T - 54491 T^{2} - 2684524 T^{3} + 262283799 p T^{4} + 59935347288 T^{5} - 43769822971458 T^{6} - 406957713276624 T^{7} + 834875075015940646 T^{8} - 406957713276624 p^{5} T^{9} - 43769822971458 p^{10} T^{10} + 59935347288 p^{15} T^{11} + 262283799 p^{21} T^{12} - 2684524 p^{25} T^{13} - 54491 p^{30} T^{14} + 68 p^{35} T^{15} + p^{40} T^{16} $
	11	$ 1 - 480 T - 155183 T^{2} - 52804624 T^{3} + 64836420601 T^{4} + 13855041138608 T^{5} - 822194046799594 T^{6} - 2036702904199155376 T^{7} - $$59\!\cdots\!50$$ T^{8} - 2036702904199155376 p^{5} T^{9} - 822194046799594 p^{10} T^{10} + 13855041138608 p^{15} T^{11} + 64836420601 p^{20} T^{12} - 52804624 p^{25} T^{13} - 155183 p^{30} T^{14} - 480 p^{35} T^{15} + p^{40} T^{16} $
	17	$ 1 - 60 T - 2681298 T^{2} - 3078551288 T^{3} + 3581836705817 T^{4} + 6433447655806760 T^{5} + 182643961132014062 p T^{6} - $$62\!\cdots\!80$$ T^{7} - $$78\!\cdots\!80$$ T^{8} - $$62\!\cdots\!80$$ p^{5} T^{9} + 182643961132014062 p^{11} T^{10} + 6433447655806760 p^{15} T^{11} + 3581836705817 p^{20} T^{12} - 3078551288 p^{25} T^{13} - 2681298 p^{30} T^{14} - 60 p^{35} T^{15} + p^{40} T^{16} $
	19	$ 1 + 520 T - 1790095 T^{2} - 3343893384 T^{3} - 7100741632487 T^{4} - 31105196806528 p T^{5} + 6283621747489888854 T^{6} + $$14\!\cdots\!84$$ T^{7} + $$36\!\cdots\!22$$ T^{8} + $$14\!\cdots\!84$$ p^{5} T^{9} + 6283621747489888854 p^{10} T^{10} - 31105196806528 p^{16} T^{11} - 7100741632487 p^{20} T^{12} - 3343893384 p^{25} T^{13} - 1790095 p^{30} T^{14} + 520 p^{35} T^{15} + p^{40} T^{16} $
	23	$ 1 + 6644 T + 18433381 T^{2} + 47524608372 T^{3} + 104759109500033 T^{4} + 32937162193373384 T^{5} - $$47\!\cdots\!22$$ T^{6} - $$25\!\cdots\!68$$ T^{7} - $$85\!\cdots\!94$$ T^{8} - $$25\!\cdots\!68$$ p^{5} T^{9} - $$47\!\cdots\!22$$ p^{10} T^{10} + 32937162193373384 p^{15} T^{11} + 104759109500033 p^{20} T^{12} + 47524608372 p^{25} T^{13} + 18433381 p^{30} T^{14} + 6644 p^{35} T^{15} + p^{40} T^{16} $
	29	$ 1 + 1236 T - 48485666 T^{2} - 144076283320 T^{3} + 1168123498321753 T^{4} + 4279083840378309176 T^{5} - $$10\!\cdots\!74$$ T^{6} - $$50\!\cdots\!00$$ T^{7} + $$34\!\cdots\!64$$ T^{8} - $$50\!\cdots\!00$$ p^{5} T^{9} - $$10\!\cdots\!74$$ p^{10} T^{10} + 4279083840378309176 p^{15} T^{11} + 1168123498321753 p^{20} T^{12} - 144076283320 p^{25} T^{13} - 48485666 p^{30} T^{14} + 1236 p^{35} T^{15} + p^{40} T^{16} $
	31	$ ( 1 + 12352 T + 142064300 T^{2} + 1081685581376 T^{3} + 6499797864770598 T^{4} + 1081685581376 p^{5} T^{5} + 142064300 p^{10} T^{6} + 12352 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	37	$ 1 + 11996 T - 52149418 T^{2} - 1788004257832 T^{3} - 5667588928811983 T^{4} + 86940773978730557016 T^{5} + $$66\!\cdots\!26$$ T^{6} - $$10\!\cdots\!96$$ T^{7} - $$37\!\cdots\!36$$ T^{8} - $$10\!\cdots\!96$$ p^{5} T^{9} + $$66\!\cdots\!26$$ p^{10} T^{10} + 86940773978730557016 p^{15} T^{11} - 5667588928811983 p^{20} T^{12} - 1788004257832 p^{25} T^{13} - 52149418 p^{30} T^{14} + 11996 p^{35} T^{15} + p^{40} T^{16} $
	41	$ 1 - 24428 T - 17592594 T^{2} + 2508016072168 T^{3} + 47968143670454681 T^{4} - $$75\!\cdots\!04$$ T^{5} - $$62\!\cdots\!06$$ p T^{6} - $$42\!\cdots\!56$$ T^{7} + $$10\!\cdots\!32$$ T^{8} - $$42\!\cdots\!56$$ p^{5} T^{9} - $$62\!\cdots\!06$$ p^{11} T^{10} - $$75\!\cdots\!04$$ p^{15} T^{11} + 47968143670454681 p^{20} T^{12} + 2508016072168 p^{25} T^{13} - 17592594 p^{30} T^{14} - 24428 p^{35} T^{15} + p^{40} T^{16} $
	43	$ 1 + 16304 T - 257986527 T^{2} - 4411561510544 T^{3} + 55485145162633065 T^{4} + $$64\!\cdots\!72$$ T^{5} - $$10\!\cdots\!34$$ T^{6} - $$27\!\cdots\!12$$ T^{7} + $$19\!\cdots\!74$$ T^{8} - $$27\!\cdots\!12$$ p^{5} T^{9} - $$10\!\cdots\!34$$ p^{10} T^{10} + $$64\!\cdots\!72$$ p^{15} T^{11} + 55485145162633065 p^{20} T^{12} - 4411561510544 p^{25} T^{13} - 257986527 p^{30} T^{14} + 16304 p^{35} T^{15} + p^{40} T^{16} $
	47	$ ( 1 - 768 T + 676741308 T^{2} + 457484535552 T^{3} + 207667315209668806 T^{4} + 457484535552 p^{5} T^{5} + 676741308 p^{10} T^{6} - 768 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	53	$ ( 1 + 24970 T + 1271604813 T^{2} + 27718849264554 T^{3} + 709897028413024084 T^{4} + 27718849264554 p^{5} T^{5} + 1271604813 p^{10} T^{6} + 24970 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	59	$ 1 + 52688 T - 90885591 T^{2} - 24234331261120 T^{3} + 669542442327808409 T^{4} + $$49\!\cdots\!84$$ T^{5} - $$10\!\cdots\!90$$ T^{6} - $$58\!\cdots\!08$$ T^{7} + $$67\!\cdots\!42$$ T^{8} - $$58\!\cdots\!08$$ p^{5} T^{9} - $$10\!\cdots\!90$$ p^{10} T^{10} + $$49\!\cdots\!84$$ p^{15} T^{11} + 669542442327808409 p^{20} T^{12} - 24234331261120 p^{25} T^{13} - 90885591 p^{30} T^{14} + 52688 p^{35} T^{15} + p^{40} T^{16} $
	61	$ 1 - 6380 T - 1132424482 T^{2} + 84365791028680 T^{3} + 412033398892526681 T^{4} - $$84\!\cdots\!80$$ T^{5} + $$27\!\cdots\!38$$ T^{6} + $$83\!\cdots\!80$$ p T^{7} - $$74\!\cdots\!56$$ p^{2} T^{8} + $$83\!\cdots\!80$$ p^{6} T^{9} + $$27\!\cdots\!38$$ p^{10} T^{10} - $$84\!\cdots\!80$$ p^{15} T^{11} + 412033398892526681 p^{20} T^{12} + 84365791028680 p^{25} T^{13} - 1132424482 p^{30} T^{14} - 6380 p^{35} T^{15} + p^{40} T^{16} $
	67	$ 1 + 75256 T - 1735816975 T^{2} - 96571542486920 T^{3} + 13745295558106324073 T^{4} + $$38\!\cdots\!44$$ T^{5} - $$20\!\cdots\!94$$ T^{6} + $$36\!\cdots\!40$$ T^{7} + $$50\!\cdots\!98$$ T^{8} + $$36\!\cdots\!40$$ p^{5} T^{9} - $$20\!\cdots\!94$$ p^{10} T^{10} + $$38\!\cdots\!44$$ p^{15} T^{11} + 13745295558106324073 p^{20} T^{12} - 96571542486920 p^{25} T^{13} - 1735816975 p^{30} T^{14} + 75256 p^{35} T^{15} + p^{40} T^{16} $
	71	$ 1 - 31300 T - 2403720995 T^{2} + 112727922460332 T^{3} + 2545377984862033553 T^{4} - $$22\!\cdots\!36$$ T^{5} + $$92\!\cdots\!06$$ T^{6} + $$20\!\cdots\!88$$ T^{7} - $$27\!\cdots\!98$$ T^{8} + $$20\!\cdots\!88$$ p^{5} T^{9} + $$92\!\cdots\!06$$ p^{10} T^{10} - $$22\!\cdots\!36$$ p^{15} T^{11} + 2545377984862033553 p^{20} T^{12} + 112727922460332 p^{25} T^{13} - 2403720995 p^{30} T^{14} - 31300 p^{35} T^{15} + p^{40} T^{16} $
	73	$ ( 1 + 38558 T + 6780938697 T^{2} + 174194191192502 T^{3} + 19170268327511808596 T^{4} + 174194191192502 p^{5} T^{5} + 6780938697 p^{10} T^{6} + 38558 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	79	$ ( 1 + 6016 T + 6186112908 T^{2} + 214360458134272 T^{3} + 19571963124397282982 T^{4} + 214360458134272 p^{5} T^{5} + 6186112908 p^{10} T^{6} + 6016 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	83	$ ( 1 + 38376 T + 11359977564 T^{2} + 203457832843304 T^{3} + 55401291106510623766 T^{4} + 203457832843304 p^{5} T^{5} + 11359977564 p^{10} T^{6} + 38376 p^{15} T^{7} + p^{20} T^{8} )^{2} $
	89	$ 1 + 154 T - 13852990229 T^{2} + 7194074891366 p T^{3} + $$10\!\cdots\!61$$ T^{4} - $$62\!\cdots\!84$$ T^{5} - $$27\!\cdots\!54$$ T^{6} + $$20\!\cdots\!52$$ T^{7} + $$45\!\cdots\!30$$ T^{8} + $$20\!\cdots\!52$$ p^{5} T^{9} - $$27\!\cdots\!54$$ p^{10} T^{10} - $$62\!\cdots\!84$$ p^{15} T^{11} + $$10\!\cdots\!61$$ p^{20} T^{12} + 7194074891366 p^{26} T^{13} - 13852990229 p^{30} T^{14} + 154 p^{35} T^{15} + p^{40} T^{16} $
	97	$ 1 - 137182 T - 6838283293 T^{2} + 747541410948254 T^{3} + $$10\!\cdots\!85$$ T^{4} - $$19\!\cdots\!72$$ T^{5} - $$82\!\cdots\!46$$ T^{6} + $$40\!\cdots\!16$$ T^{7} - $$20\!\cdots\!34$$ T^{8} + $$40\!\cdots\!16$$ p^{5} T^{9} - $$82\!\cdots\!46$$ p^{10} T^{10} - $$19\!\cdots\!72$$ p^{15} T^{11} + $$10\!\cdots\!85$$ p^{20} T^{12} + 747541410948254 p^{25} T^{13} - 6838283293 p^{30} T^{14} - 137182 p^{35} T^{15} + p^{40} 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

−4.43247703583396093043944438080, −4.39079379052464381366449394825, −4.34541584687578633313898781763, −4.06462241977973544805254972052, −3.98594933794478679194914707251, −3.81540931823821355913894489644, −3.72340838221735893511546110280, −3.68888546610727271339017976560, −3.50326104920358843617273694039, −3.29155554181308122107283872220, −3.04891768068493915013530683212, −2.79324149991217264949437970328, −2.30583621856759928836226031439, −2.22168407851190670596894428048, −2.16139360996461235786665025826, −2.02534891484779734379533951947, −1.88512455544293546569650833338, −1.45549432459558163830457960080, −1.45301522251093415573082829141, −1.35233494372223008594643851724, −1.25062993356073804291341272341, −0.72356674294390913453414190695, −0.21242277059015400962498168584, −0.11471940563981231764395322354, −0.097882809477725642211076763336, 0.097882809477725642211076763336, 0.11471940563981231764395322354, 0.21242277059015400962498168584, 0.72356674294390913453414190695, 1.25062993356073804291341272341, 1.35233494372223008594643851724, 1.45301522251093415573082829141, 1.45549432459558163830457960080, 1.88512455544293546569650833338, 2.02534891484779734379533951947, 2.16139360996461235786665025826, 2.22168407851190670596894428048, 2.30583621856759928836226031439, 2.79324149991217264949437970328, 3.04891768068493915013530683212, 3.29155554181308122107283872220, 3.50326104920358843617273694039, 3.68888546610727271339017976560, 3.72340838221735893511546110280, 3.81540931823821355913894489644, 3.98594933794478679194914707251, 4.06462241977973544805254972052, 4.34541584687578633313898781763, 4.39079379052464381366449394825, 4.43247703583396093043944438080

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(0.03387810085\)
\(L(\frac12)\)	\(\approx\)	\(0.03387810085\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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