LMFDB - L-function 16-336e8-1.1-c6e8-0-7

Dirichlet series

L(s) = 1

+ 108·3^-s − 42·5^-s + 92·7^-s + 6.31e3·9^-s − 126·11^-s − 4.53e3·15^-s − 2.53e3·17^-s − 1.99e4·19^-s + 9.93e3·21^-s + 1.56e4·23^-s − 9.01e3·25^-s + 2.62e5·27^-s + 1.29e5·29^-s + 4.20e4·31^-s − 1.36e4·33^-s − 3.86e3·35^-s + 9.86e3·37^-s + 7.17e4·43^-s − 2.65e5·45^-s − 8.69e4·47^-s + 1.61e5·49^-s − 2.73e5·51^-s + 3.91e5·53^-s + 5.29e3·55^-s − 2.15e6·57^-s + 5.53e5·59^-s − 1.00e6·61^-s + ⋯

L(s) = 1

+ 4·3^-s − 0.335·5^-s + 0.268·7^-s + 26/3·9^-s − 0.0946·11^-s − 1.34·15^-s − 0.515·17^-s − 2.91·19^-s + 1.07·21^-s + 1.28·23^-s − 0.577·25^-s + 40/3·27^-s + 5.30·29^-s + 1.41·31^-s − 0.378·33^-s − 0.0901·35^-s + 0.194·37^-s + 0.902·43^-s − 2.91·45^-s − 0.837·47^-s + 1.37·49^-s − 2.06·51^-s + 2.63·53^-s + 0.0318·55^-s − 11.6·57^-s + 2.69·59^-s − 4.44·61^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$2^{32} \cdot 3^{8} \cdot 7^{8}$
Sign:	$1$
Analytic conductor:	$1.27454\times 10^{15}$
Root analytic conductor:	$8.79193$
Motivic weight:	$6$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{32} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : [3]^{8} ),\ 1 )$

Particular Values

$L(\frac{7}{2})$	$\approx$	$122.0256330$
$L(\frac12)$	$\approx$	$122.0256330$
$L(4)$		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 $
	3	$ ( 1 - p^{3} T + p^{5} T^{2} )^{4} $
	7	$ 1 - 92 T - 21866 p T^{2} + 17296 p^{3} T^{3} + 103315 p^{6} T^{4} + 17296 p^{9} T^{5} - 21866 p^{13} T^{6} - 92 p^{18} T^{7} + p^{24} T^{8} $
good	5	$ 1 + 42 T + 10783 T^{2} + 85638 p T^{3} + 5055781 p^{2} T^{4} + 3342708 p^{5} T^{5} - 6123397118 p^{4} T^{6} - 2172720264 p^{7} T^{7} - 4336120101206 p^{6} T^{8} - 2172720264 p^{13} T^{9} - 6123397118 p^{16} T^{10} + 3342708 p^{23} T^{11} + 5055781 p^{26} T^{12} + 85638 p^{31} T^{13} + 10783 p^{36} T^{14} + 42 p^{42} T^{15} + p^{48} T^{16} $
	11	$ 1 + 126 T - 5257453 T^{2} - 464436702 T^{3} + 14512948661161 T^{4} + 643973588741916 T^{5} - 36441609127203872914 T^{6} - $$29\!\cdots\!68$$ T^{7} + $$77\!\cdots\!90$$ T^{8} - $$29\!\cdots\!68$$ p^{6} T^{9} - 36441609127203872914 p^{12} T^{10} + 643973588741916 p^{18} T^{11} + 14512948661161 p^{24} T^{12} - 464436702 p^{30} T^{13} - 5257453 p^{36} T^{14} + 126 p^{42} T^{15} + p^{48} T^{16} $
	13	$ 1 - 25871282 T^{2} + 24361348154677 p T^{4} - $$24\!\cdots\!54$$ T^{6} + $$13\!\cdots\!40$$ T^{8} - $$24\!\cdots\!54$$ p^{12} T^{10} + 24361348154677 p^{25} T^{12} - 25871282 p^{36} T^{14} + p^{48} T^{16} $
	17	$ 1 + 2532 T + 18867376 T^{2} + 42361291776 T^{3} - 418607858989358 T^{4} + 384384340831876020 T^{5} - $$65\!\cdots\!40$$ T^{6} - $$94\!\cdots\!64$$ T^{7} - $$11\!\cdots\!25$$ T^{8} - $$94\!\cdots\!64$$ p^{6} T^{9} - $$65\!\cdots\!40$$ p^{12} T^{10} + 384384340831876020 p^{18} T^{11} - 418607858989358 p^{24} T^{12} + 42361291776 p^{30} T^{13} + 18867376 p^{36} T^{14} + 2532 p^{42} T^{15} + p^{48} T^{16} $
	19	$ 1 + 19998 T + 294723631 T^{2} + 3228016426074 T^{3} + 29196210121257421 T^{4} + $$25\!\cdots\!96$$ T^{5} + $$20\!\cdots\!02$$ T^{6} + $$83\!\cdots\!24$$ p T^{7} + $$32\!\cdots\!98$$ p^{2} T^{8} + $$83\!\cdots\!24$$ p^{7} T^{9} + $$20\!\cdots\!02$$ p^{12} T^{10} + $$25\!\cdots\!96$$ p^{18} T^{11} + 29196210121257421 p^{24} T^{12} + 3228016426074 p^{30} T^{13} + 294723631 p^{36} T^{14} + 19998 p^{42} T^{15} + p^{48} T^{16} $
	23	$ 1 - 15648 T + 99012092 T^{2} - 5792005505472 T^{3} + 85374505198585162 T^{4} - $$63\!\cdots\!96$$ T^{5} + $$18\!\cdots\!32$$ T^{6} - $$24\!\cdots\!08$$ T^{7} + $$15\!\cdots\!27$$ T^{8} - $$24\!\cdots\!08$$ p^{6} T^{9} + $$18\!\cdots\!32$$ p^{12} T^{10} - $$63\!\cdots\!96$$ p^{18} T^{11} + 85374505198585162 p^{24} T^{12} - 5792005505472 p^{30} T^{13} + 99012092 p^{36} T^{14} - 15648 p^{42} T^{15} + p^{48} T^{16} $
	29	$ ( 1 - 64734 T + 2691471109 T^{2} - 90673931971242 T^{3} + 2582526423094543896 T^{4} - 90673931971242 p^{6} T^{5} + 2691471109 p^{12} T^{6} - 64734 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	31	$ 1 - 42096 T + 2189282974 T^{2} - 67294324706592 T^{3} + 1718498578394878849 T^{4} + $$14\!\cdots\!48$$ T^{5} - $$12\!\cdots\!22$$ T^{6} + $$93\!\cdots\!80$$ T^{7} - $$30\!\cdots\!60$$ T^{8} + $$93\!\cdots\!80$$ p^{6} T^{9} - $$12\!\cdots\!22$$ p^{12} T^{10} + $$14\!\cdots\!48$$ p^{18} T^{11} + 1718498578394878849 p^{24} T^{12} - 67294324706592 p^{30} T^{13} + 2189282974 p^{36} T^{14} - 42096 p^{42} T^{15} + p^{48} T^{16} $
	37	$ 1 - 9866 T - 4908798717 T^{2} - 34845117213094 T^{3} + 8197436045497152377 T^{4} + $$27\!\cdots\!36$$ T^{5} - $$18\!\cdots\!38$$ T^{6} - $$42\!\cdots\!24$$ T^{7} + $$76\!\cdots\!42$$ T^{8} - $$42\!\cdots\!24$$ p^{6} T^{9} - $$18\!\cdots\!38$$ p^{12} T^{10} + $$27\!\cdots\!36$$ p^{18} T^{11} + 8197436045497152377 p^{24} T^{12} - 34845117213094 p^{30} T^{13} - 4908798717 p^{36} T^{14} - 9866 p^{42} T^{15} + p^{48} T^{16} $
	41	$ 1 - 5866747544 T^{2} + 32558293618486887772 T^{4} + $$98\!\cdots\!84$$ T^{6} - $$48\!\cdots\!70$$ T^{8} + $$98\!\cdots\!84$$ p^{12} T^{10} + 32558293618486887772 p^{24} T^{12} - 5866747544 p^{36} T^{14} + p^{48} T^{16} $
	43	$ ( 1 - 35882 T + 8509612825 T^{2} + 179302326632422 T^{3} + 19061746338854663908 T^{4} + 179302326632422 p^{6} T^{5} + 8509612825 p^{12} T^{6} - 35882 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	47	$ 1 + 86988 T + 28747343296 T^{2} + 2281263714105024 T^{3} + $$41\!\cdots\!02$$ T^{4} + $$39\!\cdots\!52$$ T^{5} + $$43\!\cdots\!40$$ T^{6} + $$50\!\cdots\!44$$ T^{7} + $$43\!\cdots\!63$$ T^{8} + $$50\!\cdots\!44$$ p^{6} T^{9} + $$43\!\cdots\!40$$ p^{12} T^{10} + $$39\!\cdots\!52$$ p^{18} T^{11} + $$41\!\cdots\!02$$ p^{24} T^{12} + 2281263714105024 p^{30} T^{13} + 28747343296 p^{36} T^{14} + 86988 p^{42} T^{15} + p^{48} T^{16} $
	53	$ 1 - 391710 T + 26778943247 T^{2} + 2630029820576766 T^{3} + $$12\!\cdots\!53$$ T^{4} - $$28\!\cdots\!16$$ T^{5} - $$27\!\cdots\!98$$ T^{6} - $$21\!\cdots\!16$$ T^{7} + $$12\!\cdots\!10$$ T^{8} - $$21\!\cdots\!16$$ p^{6} T^{9} - $$27\!\cdots\!98$$ p^{12} T^{10} - $$28\!\cdots\!16$$ p^{18} T^{11} + $$12\!\cdots\!53$$ p^{24} T^{12} + 2630029820576766 p^{30} T^{13} + 26778943247 p^{36} T^{14} - 391710 p^{42} T^{15} + p^{48} T^{16} $
	59	$ 1 - 553434 T + 290046846259 T^{2} - 104018168685053238 T^{3} + $$36\!\cdots\!41$$ T^{4} - $$10\!\cdots\!36$$ T^{5} + $$27\!\cdots\!02$$ T^{6} - $$64\!\cdots\!24$$ T^{7} + $$13\!\cdots\!54$$ T^{8} - $$64\!\cdots\!24$$ p^{6} T^{9} + $$27\!\cdots\!02$$ p^{12} T^{10} - $$10\!\cdots\!36$$ p^{18} T^{11} + $$36\!\cdots\!41$$ p^{24} T^{12} - 104018168685053238 p^{30} T^{13} + 290046846259 p^{36} T^{14} - 553434 p^{42} T^{15} + p^{48} T^{16} $
	61	$ 1 + 1009104 T + 617835019972 T^{2} + 280939322322772800 T^{3} + $$10\!\cdots\!34$$ T^{4} + $$34\!\cdots\!56$$ T^{5} + $$98\!\cdots\!80$$ T^{6} + $$25\!\cdots\!28$$ T^{7} + $$60\!\cdots\!31$$ T^{8} + $$25\!\cdots\!28$$ p^{6} T^{9} + $$98\!\cdots\!80$$ p^{12} T^{10} + $$34\!\cdots\!56$$ p^{18} T^{11} + $$10\!\cdots\!34$$ p^{24} T^{12} + 280939322322772800 p^{30} T^{13} + 617835019972 p^{36} T^{14} + 1009104 p^{42} T^{15} + p^{48} T^{16} $
	67	$ 1 + 229762 T - 224949744729 T^{2} - 13964293996954978 T^{3} + $$35\!\cdots\!05$$ T^{4} - $$73\!\cdots\!16$$ T^{5} - $$38\!\cdots\!46$$ T^{6} + $$12\!\cdots\!96$$ T^{7} + $$31\!\cdots\!26$$ T^{8} + $$12\!\cdots\!96$$ p^{6} T^{9} - $$38\!\cdots\!46$$ p^{12} T^{10} - $$73\!\cdots\!16$$ p^{18} T^{11} + $$35\!\cdots\!05$$ p^{24} T^{12} - 13964293996954978 p^{30} T^{13} - 224949744729 p^{36} T^{14} + 229762 p^{42} T^{15} + p^{48} T^{16} $
	71	$ ( 1 + 104244 T + 467419225432 T^{2} + 35722328009904828 T^{3} + $$86\!\cdots\!42$$ T^{4} + 35722328009904828 p^{6} T^{5} + 467419225432 p^{12} T^{6} + 104244 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	73	$ 1 + 1249290 T + 920860106635 T^{2} + 500488400945676150 T^{3} + $$20\!\cdots\!41$$ T^{4} + $$57\!\cdots\!24$$ T^{5} + $$41\!\cdots\!02$$ T^{6} - $$46\!\cdots\!84$$ T^{7} - $$27\!\cdots\!82$$ T^{8} - $$46\!\cdots\!84$$ p^{6} T^{9} + $$41\!\cdots\!02$$ p^{12} T^{10} + $$57\!\cdots\!24$$ p^{18} T^{11} + $$20\!\cdots\!41$$ p^{24} T^{12} + 500488400945676150 p^{30} T^{13} + 920860106635 p^{36} T^{14} + 1249290 p^{42} T^{15} + p^{48} T^{16} $
	79	$ 1 + 693808 T - 64434336306 T^{2} - 176593384856154400 T^{3} - $$85\!\cdots\!03$$ T^{4} - $$46\!\cdots\!28$$ T^{5} - $$95\!\cdots\!10$$ T^{6} + $$11\!\cdots\!88$$ T^{7} + $$10\!\cdots\!96$$ T^{8} + $$11\!\cdots\!88$$ p^{6} T^{9} - $$95\!\cdots\!10$$ p^{12} T^{10} - $$46\!\cdots\!28$$ p^{18} T^{11} - $$85\!\cdots\!03$$ p^{24} T^{12} - 176593384856154400 p^{30} T^{13} - 64434336306 p^{36} T^{14} + 693808 p^{42} T^{15} + p^{48} T^{16} $
	83	$ 1 - 2111537278586 T^{2} + $$20\!\cdots\!09$$ T^{4} - $$12\!\cdots\!74$$ T^{6} + $$49\!\cdots\!36$$ T^{8} - $$12\!\cdots\!74$$ p^{12} T^{10} + $$20\!\cdots\!09$$ p^{24} T^{12} - 2111537278586 p^{36} T^{14} + p^{48} T^{16} $
	89	$ 1 + 1414692 T + 2458060840048 T^{2} + 2533632765339697920 T^{3} + $$27\!\cdots\!02$$ T^{4} + $$22\!\cdots\!52$$ T^{5} + $$20\!\cdots\!56$$ T^{6} + $$14\!\cdots\!88$$ T^{7} + $$11\!\cdots\!51$$ T^{8} + $$14\!\cdots\!88$$ p^{6} T^{9} + $$20\!\cdots\!56$$ p^{12} T^{10} + $$22\!\cdots\!52$$ p^{18} T^{11} + $$27\!\cdots\!02$$ p^{24} T^{12} + 2533632765339697920 p^{30} T^{13} + 2458060840048 p^{36} T^{14} + 1414692 p^{42} T^{15} + p^{48} T^{16} $
	97	$ 1 - 4886369036882 T^{2} + $$11\!\cdots\!73$$ T^{4} - $$16\!\cdots\!34$$ T^{6} + $$16\!\cdots\!20$$ T^{8} - $$16\!\cdots\!34$$ p^{12} T^{10} + $$11\!\cdots\!73$$ p^{24} T^{12} - 4886369036882 p^{36} T^{14} + p^{48} 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

−4.21389574580403756123017196531, −3.94465296532921731932075514677, −3.44305395823054184320622474228, −3.40689461310276343446424631652, −3.33242808122474221995327403571, −3.29007433898478648829891145914, −3.17558487411633074234809689886, −2.94725774710829250650097978635, −2.91077994180661801034459583059, −2.66210759459276165913189991512, −2.39458375571346075597582544937, −2.37363659432115446750852436546, −2.24624557512753211269994683361, −2.18996415811791980342764399688, −2.15424717468343957309587729491, −1.71643712206865327308667858428, −1.66505149167892677930954589670, −1.41467819292916183012731991086, −1.19304548048622912135470727506, −1.12292226831489064370892039816, −0.826409653357325263678098614459, −0.77291856939982006494036001325, −0.57993523984438617448134779424, −0.37599541572260257325003154814, −0.21610767831528016357935567671, 0.21610767831528016357935567671, 0.37599541572260257325003154814, 0.57993523984438617448134779424, 0.77291856939982006494036001325, 0.826409653357325263678098614459, 1.12292226831489064370892039816, 1.19304548048622912135470727506, 1.41467819292916183012731991086, 1.66505149167892677930954589670, 1.71643712206865327308667858428, 2.15424717468343957309587729491, 2.18996415811791980342764399688, 2.24624557512753211269994683361, 2.37363659432115446750852436546, 2.39458375571346075597582544937, 2.66210759459276165913189991512, 2.91077994180661801034459583059, 2.94725774710829250650097978635, 3.17558487411633074234809689886, 3.29007433898478648829891145914, 3.33242808122474221995327403571, 3.40689461310276343446424631652, 3.44305395823054184320622474228, 3.94465296532921731932075514677, 4.21389574580403756123017196531

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 $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(122.0256330\)
\(L(\frac12)\)	\(\approx\)	\(122.0256330\)
\(L(4)\)		not available
\(L(1)\)		not available

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