LMFDB - L-function 16-63e8-1.1-c7e8-0-0

Dirichlet series

L(s) = 1

− 6·2^-s + 100·4^-s + 252·5^-s + 672·7^-s − 384·8^-s − 1.51e3·10^-s − 3.97e3·11^-s − 2.35e3·13^-s − 4.03e3·14^-s + 848·16^-s + 5.63e4·17^-s − 4.17e4·19^-s + 2.52e4·20^-s + 2.38e4·22^-s + 1.31e5·23^-s + 2.22e5·25^-s + 1.41e4·26^-s + 6.72e4·28^-s − 6.81e4·29^-s − 4.01e5·31^-s − 5.52e4·32^-s − 3.38e5·34^-s + 1.69e5·35^-s + 5.39e3·37^-s + 2.50e5·38^-s − 9.67e4·40^-s − 8.20e5·41^-s + ⋯

L(s) = 1

− 0.530·2^-s + 0.781·4^-s + 0.901·5^-s + 0.740·7^-s − 0.265·8^-s − 0.478·10^-s − 0.899·11^-s − 0.296·13^-s − 0.392·14^-s + 0.0517·16^-s + 2.78·17^-s − 1.39·19^-s + 0.704·20^-s + 0.477·22^-s + 2.25·23^-s + 2.84·25^-s + 0.157·26^-s + 0.578·28^-s − 0.518·29^-s − 2.41·31^-s − 0.297·32^-s − 1.47·34^-s + 0.667·35^-s + 0.0175·37^-s + 0.740·38^-s − 0.239·40^-s − 1.86·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$3^{16} \cdot 7^{8}$
Sign:	$1$
Analytic conductor:	$2.25032\times 10^{10}$
Root analytic conductor:	$4.43624$
Motivic weight:	$7$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 3^{16} \cdot 7^{8} ,\ ( \ : [7/2]^{8} ),\ 1 )$

Particular Values

$L(4)$	$\approx$	$7.766230530$
$L(\frac12)$	$\approx$	$7.766230530$
$L(\frac{9}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	7	$ 1 - 96 p T + 33116 p^{2} T^{2} - 158880 p^{4} T^{3} + 1507914 p^{7} T^{4} - 158880 p^{11} T^{5} + 33116 p^{16} T^{6} - 96 p^{22} T^{7} + p^{28} T^{8} $
good	2	$ 1 + 3 p T - p^{6} T^{2} - 75 p^{3} T^{3} + 133 p^{5} T^{4} + 3471 p^{5} T^{5} + 46275 p^{6} T^{6} + 32577 p^{8} T^{7} - 315199 p^{10} T^{8} + 32577 p^{15} T^{9} + 46275 p^{20} T^{10} + 3471 p^{26} T^{11} + 133 p^{33} T^{12} - 75 p^{38} T^{13} - p^{48} T^{14} + 3 p^{50} T^{15} + p^{56} T^{16} $
	5	$ 1 - 252 T - 158506 T^{2} + 4582536 p T^{3} + 674342201 p^{2} T^{4} - 795864888 p^{4} T^{5} - 2408486026458 p^{4} T^{6} + 6152884831212 p^{5} T^{7} + 6846672196907924 p^{6} T^{8} + 6152884831212 p^{12} T^{9} - 2408486026458 p^{18} T^{10} - 795864888 p^{25} T^{11} + 674342201 p^{30} T^{12} + 4582536 p^{36} T^{13} - 158506 p^{42} T^{14} - 252 p^{49} T^{15} + p^{56} T^{16} $
	11	$ 1 + 3972 T - 44545426 T^{2} - 16974598392 p T^{3} + 1193279820296585 T^{4} + 4311182308727377824 T^{5} - $$21\!\cdots\!46$$ T^{6} - $$39\!\cdots\!00$$ T^{7} + $$37\!\cdots\!60$$ T^{8} - $$39\!\cdots\!00$$ p^{7} T^{9} - $$21\!\cdots\!46$$ p^{14} T^{10} + 4311182308727377824 p^{21} T^{11} + 1193279820296585 p^{28} T^{12} - 16974598392 p^{36} T^{13} - 44545426 p^{42} T^{14} + 3972 p^{49} T^{15} + p^{56} T^{16} $
	13	$ ( 1 + 1176 T + 151551116 T^{2} + 19538457960 p T^{3} + 75518676758118 p^{2} T^{4} + 19538457960 p^{8} T^{5} + 151551116 p^{14} T^{6} + 1176 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	17	$ 1 - 56364 T + 709129558 T^{2} + 4712531411112 T^{3} + 360653231962445473 T^{4} - $$17\!\cdots\!28$$ T^{5} + $$20\!\cdots\!10$$ T^{6} - $$22\!\cdots\!96$$ T^{7} + $$74\!\cdots\!84$$ T^{8} - $$22\!\cdots\!96$$ p^{7} T^{9} + $$20\!\cdots\!10$$ p^{14} T^{10} - $$17\!\cdots\!28$$ p^{21} T^{11} + 360653231962445473 p^{28} T^{12} + 4712531411112 p^{35} T^{13} + 709129558 p^{42} T^{14} - 56364 p^{49} T^{15} + p^{56} T^{16} $
	19	$ 1 + 41748 T - 2001962786 T^{2} - 58487933247240 T^{3} + 4448060922484490329 T^{4} + $$73\!\cdots\!44$$ T^{5} - $$55\!\cdots\!70$$ T^{6} - $$17\!\cdots\!88$$ T^{7} + $$62\!\cdots\!72$$ T^{8} - $$17\!\cdots\!88$$ p^{7} T^{9} - $$55\!\cdots\!70$$ p^{14} T^{10} + $$73\!\cdots\!44$$ p^{21} T^{11} + 4448060922484490329 p^{28} T^{12} - 58487933247240 p^{35} T^{13} - 2001962786 p^{42} T^{14} + 41748 p^{49} T^{15} + p^{56} T^{16} $
	23	$ 1 - 131748 T + 1744906414 T^{2} + 95298488634120 T^{3} + 26557457139118316761 T^{4} - $$59\!\cdots\!56$$ T^{5} - $$14\!\cdots\!10$$ T^{6} + $$42\!\cdots\!36$$ T^{7} + $$15\!\cdots\!44$$ T^{8} + $$42\!\cdots\!36$$ p^{7} T^{9} - $$14\!\cdots\!10$$ p^{14} T^{10} - $$59\!\cdots\!56$$ p^{21} T^{11} + 26557457139118316761 p^{28} T^{12} + 95298488634120 p^{35} T^{13} + 1744906414 p^{42} T^{14} - 131748 p^{49} T^{15} + p^{56} T^{16} $
	29	$ ( 1 + 34056 T + 37434723340 T^{2} - 965899818999528 T^{3} + $$63\!\cdots\!58$$ T^{4} - 965899818999528 p^{7} T^{5} + 37434723340 p^{14} T^{6} + 34056 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	31	$ 1 + 401212 T + 20995911038 T^{2} - 7710914411317432 T^{3} + $$63\!\cdots\!53$$ T^{4} + $$41\!\cdots\!68$$ T^{5} + $$19\!\cdots\!38$$ T^{6} + $$11\!\cdots\!88$$ T^{7} + $$99\!\cdots\!68$$ T^{8} + $$11\!\cdots\!88$$ p^{7} T^{9} + $$19\!\cdots\!38$$ p^{14} T^{10} + $$41\!\cdots\!68$$ p^{21} T^{11} + $$63\!\cdots\!53$$ p^{28} T^{12} - 7710914411317432 p^{35} T^{13} + 20995911038 p^{42} T^{14} + 401212 p^{49} T^{15} + p^{56} T^{16} $
	37	$ 1 - 5396 T - 363098326522 T^{2} + 1692295529079608 T^{3} + $$80\!\cdots\!33$$ T^{4} - $$28\!\cdots\!72$$ T^{5} - $$11\!\cdots\!30$$ T^{6} + $$10\!\cdots\!56$$ T^{7} + $$13\!\cdots\!04$$ T^{8} + $$10\!\cdots\!56$$ p^{7} T^{9} - $$11\!\cdots\!30$$ p^{14} T^{10} - $$28\!\cdots\!72$$ p^{21} T^{11} + $$80\!\cdots\!33$$ p^{28} T^{12} + 1692295529079608 p^{35} T^{13} - 363098326522 p^{42} T^{14} - 5396 p^{49} T^{15} + p^{56} T^{16} $
	41	$ ( 1 + 410424 T + 514131659548 T^{2} + 263929507132312392 T^{3} + $$12\!\cdots\!54$$ T^{4} + 263929507132312392 p^{7} T^{5} + 514131659548 p^{14} T^{6} + 410424 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	43	$ ( 1 - 46544 T + 620213285356 T^{2} - 12386906008354640 T^{3} + $$22\!\cdots\!42$$ T^{4} - 12386906008354640 p^{7} T^{5} + 620213285356 p^{14} T^{6} - 46544 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	47	$ 1 - 1470084 T - 61407153218 T^{2} + 676586115609875016 T^{3} + $$33\!\cdots\!21$$ T^{4} - $$36\!\cdots\!44$$ T^{5} - $$22\!\cdots\!18$$ T^{6} + $$49\!\cdots\!56$$ T^{7} + $$16\!\cdots\!52$$ T^{8} + $$49\!\cdots\!56$$ p^{7} T^{9} - $$22\!\cdots\!18$$ p^{14} T^{10} - $$36\!\cdots\!44$$ p^{21} T^{11} + $$33\!\cdots\!21$$ p^{28} T^{12} + 676586115609875016 p^{35} T^{13} - 61407153218 p^{42} T^{14} - 1470084 p^{49} T^{15} + p^{56} T^{16} $
	53	$ 1 + 642372 T - 2597575978538 T^{2} - 2359978517441309784 T^{3} + $$29\!\cdots\!33$$ T^{4} + $$28\!\cdots\!24$$ T^{5} - $$30\!\cdots\!50$$ T^{6} - $$12\!\cdots\!88$$ T^{7} + $$40\!\cdots\!84$$ T^{8} - $$12\!\cdots\!88$$ p^{7} T^{9} - $$30\!\cdots\!50$$ p^{14} T^{10} + $$28\!\cdots\!24$$ p^{21} T^{11} + $$29\!\cdots\!33$$ p^{28} T^{12} - 2359978517441309784 p^{35} T^{13} - 2597575978538 p^{42} T^{14} + 642372 p^{49} T^{15} + p^{56} T^{16} $
	59	$ 1 - 752220 T - 3986199031874 T^{2} - 1265706973401854760 T^{3} + $$80\!\cdots\!61$$ T^{4} + $$10\!\cdots\!20$$ T^{5} + $$13\!\cdots\!18$$ T^{6} - $$21\!\cdots\!40$$ T^{7} - $$10\!\cdots\!32$$ T^{8} - $$21\!\cdots\!40$$ p^{7} T^{9} + $$13\!\cdots\!18$$ p^{14} T^{10} + $$10\!\cdots\!20$$ p^{21} T^{11} + $$80\!\cdots\!61$$ p^{28} T^{12} - 1265706973401854760 p^{35} T^{13} - 3986199031874 p^{42} T^{14} - 752220 p^{49} T^{15} + p^{56} T^{16} $
	61	$ 1 + 1325772 T - 2658227552762 T^{2} + 8845336598195007288 T^{3} + $$19\!\cdots\!13$$ T^{4} - $$17\!\cdots\!12$$ T^{5} + $$85\!\cdots\!78$$ T^{6} + $$13\!\cdots\!68$$ T^{7} - $$18\!\cdots\!92$$ T^{8} + $$13\!\cdots\!68$$ p^{7} T^{9} + $$85\!\cdots\!78$$ p^{14} T^{10} - $$17\!\cdots\!12$$ p^{21} T^{11} + $$19\!\cdots\!13$$ p^{28} T^{12} + 8845336598195007288 p^{35} T^{13} - 2658227552762 p^{42} T^{14} + 1325772 p^{49} T^{15} + p^{56} T^{16} $
	67	$ 1 - 290916 T - 10943334257858 T^{2} - 8785742150796028056 T^{3} + $$62\!\cdots\!41$$ T^{4} + $$82\!\cdots\!44$$ T^{5} + $$15\!\cdots\!02$$ T^{6} - $$38\!\cdots\!96$$ T^{7} - $$21\!\cdots\!88$$ T^{8} - $$38\!\cdots\!96$$ p^{7} T^{9} + $$15\!\cdots\!02$$ p^{14} T^{10} + $$82\!\cdots\!44$$ p^{21} T^{11} + $$62\!\cdots\!41$$ p^{28} T^{12} - 8785742150796028056 p^{35} T^{13} - 10943334257858 p^{42} T^{14} - 290916 p^{49} T^{15} + p^{56} T^{16} $
	71	$ ( 1 + 3377760 T + 21455628109916 T^{2} + 39866994142106473440 T^{3} + $$20\!\cdots\!26$$ T^{4} + 39866994142106473440 p^{7} T^{5} + 21455628109916 p^{14} T^{6} + 3377760 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	73	$ 1 + 6706588 T - 4081288956538 T^{2} - 75839108854686445576 T^{3} + $$20\!\cdots\!73$$ T^{4} + $$78\!\cdots\!56$$ T^{5} - $$36\!\cdots\!10$$ T^{6} + $$16\!\cdots\!88$$ T^{7} + $$80\!\cdots\!84$$ T^{8} + $$16\!\cdots\!88$$ p^{7} T^{9} - $$36\!\cdots\!10$$ p^{14} T^{10} + $$78\!\cdots\!56$$ p^{21} T^{11} + $$20\!\cdots\!73$$ p^{28} T^{12} - 75839108854686445576 p^{35} T^{13} - 4081288956538 p^{42} T^{14} + 6706588 p^{49} T^{15} + p^{56} T^{16} $
	79	$ 1 + 3946244 T - 34598478524562 T^{2} + 37240845779266596472 T^{3} + $$11\!\cdots\!45$$ T^{4} - $$24\!\cdots\!12$$ T^{5} - $$13\!\cdots\!86$$ T^{6} + $$27\!\cdots\!48$$ T^{7} + $$56\!\cdots\!44$$ T^{8} + $$27\!\cdots\!48$$ p^{7} T^{9} - $$13\!\cdots\!86$$ p^{14} T^{10} - $$24\!\cdots\!12$$ p^{21} T^{11} + $$11\!\cdots\!45$$ p^{28} T^{12} + 37240845779266596472 p^{35} T^{13} - 34598478524562 p^{42} T^{14} + 3946244 p^{49} T^{15} + p^{56} T^{16} $
	83	$ ( 1 + 9542064 T + 123192866313548 T^{2} + $$78\!\cdots\!08$$ T^{3} + $$52\!\cdots\!18$$ T^{4} + $$78\!\cdots\!08$$ p^{7} T^{5} + 123192866313548 p^{14} T^{6} + 9542064 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	89	$ 1 - 16165212 T + 27735348124934 T^{2} + $$31\!\cdots\!60$$ T^{3} + $$50\!\cdots\!09$$ T^{4} - $$34\!\cdots\!96$$ T^{5} - $$22\!\cdots\!10$$ T^{6} - $$27\!\cdots\!28$$ T^{7} + $$22\!\cdots\!72$$ T^{8} - $$27\!\cdots\!28$$ p^{7} T^{9} - $$22\!\cdots\!10$$ p^{14} T^{10} - $$34\!\cdots\!96$$ p^{21} T^{11} + $$50\!\cdots\!09$$ p^{28} T^{12} + $$31\!\cdots\!60$$ p^{35} T^{13} + 27735348124934 p^{42} T^{14} - 16165212 p^{49} T^{15} + p^{56} T^{16} $
	97	$ ( 1 - 1533112 T + 125450995201724 T^{2} - $$51\!\cdots\!40$$ T^{3} + $$11\!\cdots\!22$$ T^{4} - $$51\!\cdots\!40$$ p^{7} T^{5} + 125450995201724 p^{14} T^{6} - 1533112 p^{21} T^{7} + p^{28} 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

−5.72621728308962029092777945320, −5.15724005852326471107859818374, −5.03665198918309692411346820485, −5.01474640602960020721431775999, −4.93501424688376654475193128781, −4.61528469963346273202425314801, −4.42789434996066925747451667044, −4.27462289322728508279575309890, −3.80161941487494090387868822406, −3.61473888892662798676808791704, −3.38615012279172161183386332492, −3.13049689857365583796264799284, −3.03458279700566408320190533108, −2.80296074170424010546810510731, −2.78787933978682615178016483590, −2.31504757861725879384692752636, −2.09469420672859961714545260655, −1.78338488859347101146990309885, −1.59247898814048467020197075853, −1.48182801540446205378018130177, −1.21392527784942589661141775267, −1.15080405365266326477142874873, −0.55440139509748247294022404484, −0.47131400495982704093924976253, −0.23285641499744925530502513372, 0.23285641499744925530502513372, 0.47131400495982704093924976253, 0.55440139509748247294022404484, 1.15080405365266326477142874873, 1.21392527784942589661141775267, 1.48182801540446205378018130177, 1.59247898814048467020197075853, 1.78338488859347101146990309885, 2.09469420672859961714545260655, 2.31504757861725879384692752636, 2.78787933978682615178016483590, 2.80296074170424010546810510731, 3.03458279700566408320190533108, 3.13049689857365583796264799284, 3.38615012279172161183386332492, 3.61473888892662798676808791704, 3.80161941487494090387868822406, 4.27462289322728508279575309890, 4.42789434996066925747451667044, 4.61528469963346273202425314801, 4.93501424688376654475193128781, 5.01474640602960020721431775999, 5.03665198918309692411346820485, 5.15724005852326471107859818374, 5.72621728308962029092777945320

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(7.766230530\)
\(L(\frac12)\)	\(\approx\)	\(7.766230530\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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