LMFDB - L-function 16-252e8-1.1-c7e8-0-0

Dirichlet series

L(s) = 1

+ 196·5^-s − 434·7^-s − 406·11^-s + 3.94e3·13^-s + 7.43e3·17^-s + 1.58e4·19^-s + 6.78e3·23^-s + 2.10e5·25^-s + 1.89e5·29^-s − 5.58e4·31^-s − 8.50e4·35^-s + 9.37e4·37^-s − 1.39e4·41^-s − 4.87e5·43^-s + 6.50e5·47^-s − 8.23e5·49^-s − 3.68e5·53^-s − 7.95e4·55^-s − 9.60e4·59^-s − 3.61e6·61^-s + 7.73e5·65^-s − 3.16e5·67^-s − 1.24e7·71^-s − 1.28e6·73^-s + 1.76e5·77^-s + 8.18e6·79^-s − 7.38e6·83^-s + ⋯

L(s) = 1

+ 0.701·5^-s − 0.478·7^-s − 0.0919·11^-s + 0.498·13^-s + 0.367·17^-s + 0.530·19^-s + 0.116·23^-s + 2.69·25^-s + 1.43·29^-s − 0.336·31^-s − 0.335·35^-s + 0.304·37^-s − 0.0315·41^-s − 0.935·43^-s + 0.913·47^-s − 0.999·49^-s − 0.340·53^-s − 0.0644·55^-s − 0.0608·59^-s − 2.04·61^-s + 0.349·65^-s − 0.128·67^-s − 4.12·71^-s − 0.387·73^-s + 0.0439·77^-s + 1.86·79^-s − 1.41·83^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$16$
Conductor:	$2^{16} \cdot 3^{16} \cdot 7^{8}$
Sign:	$1$
Analytic conductor:	$1.47477\times 10^{15}$
Root analytic conductor:	$8.87248$
Motivic weight:	$7$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{16} \cdot 3^{16} \cdot 7^{8} ,\ ( \ : [7/2]^{8} ),\ 1 )$

Particular Values

$L(4)$	$\approx$	$0.6675390267$
$L(\frac12)$	$\approx$	$0.6675390267$
$L(\frac{9}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	7	$ 1 + 62 p T + 20644 p^{2} T^{2} + 341944 p^{4} T^{3} + 927659 p^{7} T^{4} + 341944 p^{11} T^{5} + 20644 p^{16} T^{6} + 62 p^{22} T^{7} + p^{28} T^{8} $
good	5	$ 1 - 196 T - 171991 T^{2} + 60428308 T^{3} + 13267243009 T^{4} - 1283882807136 p T^{5} - 423178380922 p^{3} T^{6} + 2051911371257344 p^{3} T^{7} - 48017077629846826 p^{4} T^{8} + 2051911371257344 p^{10} T^{9} - 423178380922 p^{17} T^{10} - 1283882807136 p^{22} T^{11} + 13267243009 p^{28} T^{12} + 60428308 p^{35} T^{13} - 171991 p^{42} T^{14} - 196 p^{49} T^{15} + p^{56} T^{16} $
	11	$ 1 + 406 T - 48633157 T^{2} - 18050768542 T^{3} + 1037248580723101 T^{4} + 207648939251728824 T^{5} - $$25\!\cdots\!58$$ p T^{6} - $$17\!\cdots\!56$$ T^{7} + $$73\!\cdots\!06$$ T^{8} - $$17\!\cdots\!56$$ p^{7} T^{9} - $$25\!\cdots\!58$$ p^{15} T^{10} + 207648939251728824 p^{21} T^{11} + 1037248580723101 p^{28} T^{12} - 18050768542 p^{35} T^{13} - 48633157 p^{42} T^{14} + 406 p^{49} T^{15} + p^{56} T^{16} $
	13	$ ( 1 - 1974 T + 208939117 T^{2} - 412046927534 T^{3} + 18378156938146800 T^{4} - 412046927534 p^{7} T^{5} + 208939117 p^{14} T^{6} - 1974 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	17	$ 1 - 7436 T - 1304530900 T^{2} + 6055545887768 T^{3} + 993238264352979418 T^{4} - $$24\!\cdots\!76$$ T^{5} - $$58\!\cdots\!60$$ T^{6} + $$21\!\cdots\!92$$ p T^{7} + $$94\!\cdots\!27$$ p^{2} T^{8} + $$21\!\cdots\!92$$ p^{8} T^{9} - $$58\!\cdots\!60$$ p^{14} T^{10} - $$24\!\cdots\!76$$ p^{21} T^{11} + 993238264352979418 p^{28} T^{12} + 6055545887768 p^{35} T^{13} - 1304530900 p^{42} T^{14} - 7436 p^{49} T^{15} + p^{56} T^{16} $
	19	$ 1 - 15874 T - 1825616913 T^{2} - 35050750154462 T^{3} + 2528615806290940817 T^{4} + $$60\!\cdots\!32$$ T^{5} - $$41\!\cdots\!74$$ T^{6} - $$39\!\cdots\!00$$ T^{7} - $$47\!\cdots\!54$$ T^{8} - $$39\!\cdots\!00$$ p^{7} T^{9} - $$41\!\cdots\!74$$ p^{14} T^{10} + $$60\!\cdots\!32$$ p^{21} T^{11} + 2528615806290940817 p^{28} T^{12} - 35050750154462 p^{35} T^{13} - 1825616913 p^{42} T^{14} - 15874 p^{49} T^{15} + p^{56} T^{16} $
	23	$ 1 - 6788 T - 193987412 p T^{2} + 45990947182904 T^{3} - 8251585987926346982 T^{4} + $$81\!\cdots\!72$$ T^{5} - $$22\!\cdots\!88$$ T^{6} - $$53\!\cdots\!16$$ T^{7} + $$43\!\cdots\!71$$ T^{8} - $$53\!\cdots\!16$$ p^{7} T^{9} - $$22\!\cdots\!88$$ p^{14} T^{10} + $$81\!\cdots\!72$$ p^{21} T^{11} - 8251585987926346982 p^{28} T^{12} + 45990947182904 p^{35} T^{13} - 193987412 p^{43} T^{14} - 6788 p^{49} T^{15} + p^{56} T^{16} $
	29	$ ( 1 - 94544 T + 39034925795 T^{2} - 6585126243044520 T^{3} + $$72\!\cdots\!16$$ T^{4} - 6585126243044520 p^{7} T^{5} + 39034925795 p^{14} T^{6} - 94544 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	31	$ 1 + 55890 T - 40459286872 T^{2} - 12970120608204280 T^{3} + $$29\!\cdots\!71$$ T^{4} + $$41\!\cdots\!00$$ T^{5} + $$62\!\cdots\!88$$ T^{6} - $$64\!\cdots\!10$$ T^{7} - $$21\!\cdots\!36$$ T^{8} - $$64\!\cdots\!10$$ p^{7} T^{9} + $$62\!\cdots\!88$$ p^{14} T^{10} + $$41\!\cdots\!00$$ p^{21} T^{11} + $$29\!\cdots\!71$$ p^{28} T^{12} - 12970120608204280 p^{35} T^{13} - 40459286872 p^{42} T^{14} + 55890 p^{49} T^{15} + p^{56} T^{16} $
	37	$ 1 - 93742 T - 13501435137 T^{2} - 8046691878917306 T^{3} - $$11\!\cdots\!43$$ T^{4} + $$27\!\cdots\!80$$ T^{5} + $$15\!\cdots\!66$$ T^{6} - $$73\!\cdots\!88$$ T^{7} + $$89\!\cdots\!78$$ T^{8} - $$73\!\cdots\!88$$ p^{7} T^{9} + $$15\!\cdots\!66$$ p^{14} T^{10} + $$27\!\cdots\!80$$ p^{21} T^{11} - $$11\!\cdots\!43$$ p^{28} T^{12} - 8046691878917306 p^{35} T^{13} - 13501435137 p^{42} T^{14} - 93742 p^{49} T^{15} + p^{56} T^{16} $
	41	$ ( 1 + 6972 T + 179204422376 T^{2} - 22687004651735916 T^{3} + $$68\!\cdots\!86$$ T^{4} - 22687004651735916 p^{7} T^{5} + 179204422376 p^{14} T^{6} + 6972 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	43	$ ( 1 + 243922 T + 668754369745 T^{2} + 172816838450048086 T^{3} + $$21\!\cdots\!36$$ T^{4} + 172816838450048086 p^{7} T^{5} + 668754369745 p^{14} T^{6} + 243922 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	47	$ 1 - 650484 T - 1044494449676 T^{2} + 955011641322928248 T^{3} + $$54\!\cdots\!62$$ T^{4} - $$60\!\cdots\!60$$ T^{5} - $$93\!\cdots\!16$$ T^{6} + $$15\!\cdots\!56$$ T^{7} - $$31\!\cdots\!65$$ T^{8} + $$15\!\cdots\!56$$ p^{7} T^{9} - $$93\!\cdots\!16$$ p^{14} T^{10} - $$60\!\cdots\!60$$ p^{21} T^{11} + $$54\!\cdots\!62$$ p^{28} T^{12} + 955011641322928248 p^{35} T^{13} - 1044494449676 p^{42} T^{14} - 650484 p^{49} T^{15} + p^{56} T^{16} $
	53	$ 1 + 6960 p T - 2457037969043 T^{2} - 2322920472920175120 T^{3} + $$23\!\cdots\!69$$ T^{4} + $$34\!\cdots\!00$$ T^{5} - $$57\!\cdots\!06$$ T^{6} - $$19\!\cdots\!60$$ T^{7} + $$11\!\cdots\!58$$ T^{8} - $$19\!\cdots\!60$$ p^{7} T^{9} - $$57\!\cdots\!06$$ p^{14} T^{10} + $$34\!\cdots\!00$$ p^{21} T^{11} + $$23\!\cdots\!69$$ p^{28} T^{12} - 2322920472920175120 p^{35} T^{13} - 2457037969043 p^{42} T^{14} + 6960 p^{50} T^{15} + p^{56} T^{16} $
	59	$ 1 + 96026 T - 6920178417697 T^{2} - 5620425672203653874 T^{3} + $$26\!\cdots\!93$$ T^{4} + $$25\!\cdots\!12$$ T^{5} - $$54\!\cdots\!82$$ T^{6} - $$34\!\cdots\!52$$ T^{7} + $$10\!\cdots\!54$$ T^{8} - $$34\!\cdots\!52$$ p^{7} T^{9} - $$54\!\cdots\!82$$ p^{14} T^{10} + $$25\!\cdots\!12$$ p^{21} T^{11} + $$26\!\cdots\!93$$ p^{28} T^{12} - 5620425672203653874 p^{35} T^{13} - 6920178417697 p^{42} T^{14} + 96026 p^{49} T^{15} + p^{56} T^{16} $
	61	$ 1 + 3618156 T - 1321680544372 T^{2} - 12594132072707290648 T^{3} + $$19\!\cdots\!98$$ T^{4} + $$53\!\cdots\!56$$ T^{5} - $$59\!\cdots\!12$$ T^{6} - $$43\!\cdots\!00$$ T^{7} + $$42\!\cdots\!87$$ T^{8} - $$43\!\cdots\!00$$ p^{7} T^{9} - $$59\!\cdots\!12$$ p^{14} T^{10} + $$53\!\cdots\!56$$ p^{21} T^{11} + $$19\!\cdots\!98$$ p^{28} T^{12} - 12594132072707290648 p^{35} T^{13} - 1321680544372 p^{42} T^{14} + 3618156 p^{49} T^{15} + p^{56} T^{16} $
	67	$ 1 + 316006 T - 21543494327325 T^{2} - 920188823290287798 T^{3} + $$27\!\cdots\!13$$ T^{4} - $$11\!\cdots\!04$$ T^{5} - $$24\!\cdots\!30$$ T^{6} + $$36\!\cdots\!36$$ T^{7} + $$17\!\cdots\!78$$ T^{8} + $$36\!\cdots\!36$$ p^{7} T^{9} - $$24\!\cdots\!30$$ p^{14} T^{10} - $$11\!\cdots\!04$$ p^{21} T^{11} + $$27\!\cdots\!13$$ p^{28} T^{12} - 920188823290287798 p^{35} T^{13} - 21543494327325 p^{42} T^{14} + 316006 p^{49} T^{15} + p^{56} T^{16} $
	71	$ ( 1 + 6218156 T + 46176745736588 T^{2} + $$17\!\cdots\!76$$ T^{3} + $$67\!\cdots\!62$$ T^{4} + $$17\!\cdots\!76$$ p^{7} T^{5} + 46176745736588 p^{14} T^{6} + 6218156 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	73	$ 1 + 1287286 T - 36333104982741 T^{2} - 36377980881729081894 T^{3} + $$79\!\cdots\!17$$ T^{4} + $$55\!\cdots\!44$$ T^{5} - $$12\!\cdots\!94$$ T^{6} - $$25\!\cdots\!56$$ T^{7} + $$15\!\cdots\!02$$ T^{8} - $$25\!\cdots\!56$$ p^{7} T^{9} - $$12\!\cdots\!94$$ p^{14} T^{10} + $$55\!\cdots\!44$$ p^{21} T^{11} + $$79\!\cdots\!17$$ p^{28} T^{12} - 36377980881729081894 p^{35} T^{13} - 36333104982741 p^{42} T^{14} + 1287286 p^{49} T^{15} + p^{56} T^{16} $
	79	$ 1 - 8187282 T - 4574229425368 T^{2} + $$32\!\cdots\!92$$ T^{3} - $$61\!\cdots\!49$$ T^{4} - $$85\!\cdots\!00$$ T^{5} + $$45\!\cdots\!40$$ T^{6} + $$64\!\cdots\!50$$ T^{7} - $$10\!\cdots\!00$$ T^{8} + $$64\!\cdots\!50$$ p^{7} T^{9} + $$45\!\cdots\!40$$ p^{14} T^{10} - $$85\!\cdots\!00$$ p^{21} T^{11} - $$61\!\cdots\!49$$ p^{28} T^{12} + $$32\!\cdots\!92$$ p^{35} T^{13} - 4574229425368 p^{42} T^{14} - 8187282 p^{49} T^{15} + p^{56} T^{16} $
	83	$ ( 1 + 3693650 T + 66867395206505 T^{2} + $$20\!\cdots\!94$$ T^{3} + $$22\!\cdots\!04$$ T^{4} + $$20\!\cdots\!94$$ p^{7} T^{5} + 66867395206505 p^{14} T^{6} + 3693650 p^{21} T^{7} + p^{28} T^{8} )^{2} $
	89	$ 1 - 14489928 T + 130849624837552 T^{2} - $$13\!\cdots\!44$$ T^{3} + $$10\!\cdots\!14$$ T^{4} - $$80\!\cdots\!44$$ T^{5} + $$62\!\cdots\!64$$ T^{6} - $$44\!\cdots\!04$$ T^{7} + $$29\!\cdots\!07$$ T^{8} - $$44\!\cdots\!04$$ p^{7} T^{9} + $$62\!\cdots\!64$$ p^{14} T^{10} - $$80\!\cdots\!44$$ p^{21} T^{11} + $$10\!\cdots\!14$$ p^{28} T^{12} - $$13\!\cdots\!44$$ p^{35} T^{13} + 130849624837552 p^{42} T^{14} - 14489928 p^{49} T^{15} + p^{56} T^{16} $
	97	$ ( 1 - 11861290 T + 293699437376665 T^{2} - $$23\!\cdots\!50$$ T^{3} + $$33\!\cdots\!96$$ T^{4} - $$23\!\cdots\!50$$ p^{7} T^{5} + 293699437376665 p^{14} T^{6} - 11861290 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

−4.11640199178377804158937536517, −4.02941160619509091429328695712, −3.63946491012215234161819536615, −3.57525761326963691443238244939, −3.57039623751030723699540068757, −3.26374709063686183364340852005, −3.00604148979713549724028203676, −2.89144378615272882517029530909, −2.86970790358564565307432731827, −2.83664116978822550521456735145, −2.64415035682650806199469710945, −2.53372597823955207811371459545, −2.04602916241435331646228323424, −1.96975271882392169255797763567, −1.79181477035950271719873405015, −1.66549646097759614940660066468, −1.61049876081004190740919787507, −1.27059786092752704755898133808, −1.03403100746666326596827437535, −1.00327583645059533311531871694, −0.960498982499725815952875876201, −0.71836774912779540533256298652, −0.51540624461453908347647768798, −0.12446281543321810294947183492, −0.082667881330453216909519311416, 0.082667881330453216909519311416, 0.12446281543321810294947183492, 0.51540624461453908347647768798, 0.71836774912779540533256298652, 0.960498982499725815952875876201, 1.00327583645059533311531871694, 1.03403100746666326596827437535, 1.27059786092752704755898133808, 1.61049876081004190740919787507, 1.66549646097759614940660066468, 1.79181477035950271719873405015, 1.96975271882392169255797763567, 2.04602916241435331646228323424, 2.53372597823955207811371459545, 2.64415035682650806199469710945, 2.83664116978822550521456735145, 2.86970790358564565307432731827, 2.89144378615272882517029530909, 3.00604148979713549724028203676, 3.26374709063686183364340852005, 3.57039623751030723699540068757, 3.57525761326963691443238244939, 3.63946491012215234161819536615, 4.02941160619509091429328695712, 4.11640199178377804158937536517

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(0.6675390267\)
\(L(\frac12)\)	\(\approx\)	\(0.6675390267\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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