LMFDB - L-function 16-14e8-1.1-c6e8-0-0

Dirichlet series

L(s) = 1

− 64·4^-s − 336·5^-s + 652·7^-s − 1.08e3·9^-s − 1.35e3·11^-s + 1.02e3·16^-s − 1.73e4·17^-s − 3.20e4·19^-s + 2.15e4·20^-s − 4.12e3·23^-s + 2.75e4·25^-s − 4.17e4·28^-s − 3.03e4·29^-s − 3.10e3·31^-s − 2.19e5·35^-s + 6.91e4·36^-s − 6.12e3·37^-s − 2.97e5·43^-s + 8.67e4·44^-s + 3.62e5·45^-s + 3.13e5·47^-s + 2.28e5·49^-s + 2.78e5·53^-s + 4.55e5·55^-s − 8.35e5·59^-s − 9.95e5·61^-s − 7.04e5·63^-s + ⋯

L(s) = 1

− 4^-s − 2.68·5^-s + 1.90·7^-s − 1.48·9^-s − 1.01·11^-s + 1/4·16^-s − 3.52·17^-s − 4.66·19^-s + 2.68·20^-s − 0.339·23^-s + 1.76·25^-s − 1.90·28^-s − 1.24·29^-s − 0.104·31^-s − 5.10·35^-s + 1.48·36^-s − 0.120·37^-s − 3.74·43^-s + 1.01·44^-s + 3.98·45^-s + 3.02·47^-s + 1.94·49^-s + 1.87·53^-s + 2.73·55^-s − 4.06·59^-s − 4.38·61^-s − 2.81·63^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$16$
Conductor:	$2^{8} \cdot 7^{8}$
Sign:	$1$
Analytic conductor:	$11578.8$
Root analytic conductor:	$1.79464$
Motivic weight:	$6$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{8} \cdot 7^{8} ,\ ( \ : [3]^{8} ),\ 1 )$

Particular Values

$L(\frac{7}{2})$	$\approx$	$0.0006378165735$
$L(\frac12)$	$\approx$	$0.0006378165735$
$L(4)$		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 + p^{5} T^{2} + p^{10} T^{4} )^{2} $
	7	$ 1 - 652 T + 28048 p T^{2} + 63236 p^{3} T^{3} - 1332974 p^{5} T^{4} + 63236 p^{9} T^{5} + 28048 p^{13} T^{6} - 652 p^{18} T^{7} + p^{24} T^{8} $
good	3	$ 1 + 40 p^{3} T^{2} + 16771 p^{2} T^{4} + 25844 p^{5} T^{5} + 2055304 p^{3} T^{6} + 224304052 p^{4} T^{7} + 2551219432 p^{4} T^{8} + 224304052 p^{10} T^{9} + 2055304 p^{15} T^{10} + 25844 p^{23} T^{11} + 16771 p^{26} T^{12} + 40 p^{39} T^{14} + p^{48} T^{16} $
	5	$ 1 + 336 T + 85366 T^{2} + 16038624 T^{3} + 2456362441 T^{4} + 65605622544 p T^{5} + 1589301396406 p^{2} T^{6} + 37401029748096 p^{3} T^{7} + 919144374631636 p^{4} T^{8} + 37401029748096 p^{9} T^{9} + 1589301396406 p^{14} T^{10} + 65605622544 p^{19} T^{11} + 2456362441 p^{24} T^{12} + 16038624 p^{30} T^{13} + 85366 p^{36} T^{14} + 336 p^{42} T^{15} + p^{48} T^{16} $
	11	$ 1 + 1356 T - 122600 p T^{2} - 5127563208 T^{3} - 3659896680029 T^{4} + 3327928404358428 T^{5} + 7354942466938909256 T^{6} + $$27\!\cdots\!04$$ T^{7} - $$58\!\cdots\!00$$ T^{8} + $$27\!\cdots\!04$$ p^{6} T^{9} + 7354942466938909256 p^{12} T^{10} + 3327928404358428 p^{18} T^{11} - 3659896680029 p^{24} T^{12} - 5127563208 p^{30} T^{13} - 122600 p^{37} T^{14} + 1356 p^{42} T^{15} + p^{48} T^{16} $
	13	$ 1 - 26799680 T^{2} + 345775979967292 T^{4} - $$28\!\cdots\!12$$ T^{6} + $$16\!\cdots\!94$$ T^{8} - $$28\!\cdots\!12$$ p^{12} T^{10} + 345775979967292 p^{24} T^{12} - 26799680 p^{36} T^{14} + p^{48} T^{16} $
	17	$ 1 + 17304 T + 206140798 T^{2} + 1839957265104 T^{3} + 14302857981101617 T^{4} + 99192647737413753264 T^{5} + $$61\!\cdots\!82$$ T^{6} + $$34\!\cdots\!60$$ T^{7} + $$17\!\cdots\!64$$ T^{8} + $$34\!\cdots\!60$$ p^{6} T^{9} + $$61\!\cdots\!82$$ p^{12} T^{10} + 99192647737413753264 p^{18} T^{11} + 14302857981101617 p^{24} T^{12} + 1839957265104 p^{30} T^{13} + 206140798 p^{36} T^{14} + 17304 p^{42} T^{15} + p^{48} T^{16} $
	19	$ 1 + 32004 T + 616491088 T^{2} + 8803417601664 T^{3} + 103939442799115627 T^{4} + $$10\!\cdots\!20$$ T^{5} + $$95\!\cdots\!20$$ T^{6} + $$77\!\cdots\!72$$ T^{7} + $$55\!\cdots\!72$$ T^{8} + $$77\!\cdots\!72$$ p^{6} T^{9} + $$95\!\cdots\!20$$ p^{12} T^{10} + $$10\!\cdots\!20$$ p^{18} T^{11} + 103939442799115627 p^{24} T^{12} + 8803417601664 p^{30} T^{13} + 616491088 p^{36} T^{14} + 32004 p^{42} T^{15} + p^{48} T^{16} $
	23	$ 1 + 4128 T - 386373856 T^{2} + 223897068600 T^{3} + 83079437636729539 T^{4} - $$23\!\cdots\!96$$ T^{5} - $$13\!\cdots\!80$$ T^{6} + $$19\!\cdots\!12$$ T^{7} + $$18\!\cdots\!12$$ T^{8} + $$19\!\cdots\!12$$ p^{6} T^{9} - $$13\!\cdots\!80$$ p^{12} T^{10} - $$23\!\cdots\!96$$ p^{18} T^{11} + 83079437636729539 p^{24} T^{12} + 223897068600 p^{30} T^{13} - 386373856 p^{36} T^{14} + 4128 p^{42} T^{15} + p^{48} T^{16} $
	29	$ ( 1 + 15156 T + 1003136368 T^{2} - 3639995041860 T^{3} + 336439222276101918 T^{4} - 3639995041860 p^{6} T^{5} + 1003136368 p^{12} T^{6} + 15156 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	31	$ 1 + 3108 T + 791635696 T^{2} + 79045042944 p T^{3} + 26109511358578261 p T^{4} - $$14\!\cdots\!36$$ p T^{5} - $$69\!\cdots\!80$$ T^{6} - $$10\!\cdots\!12$$ T^{7} - $$47\!\cdots\!88$$ T^{8} - $$10\!\cdots\!12$$ p^{6} T^{9} - $$69\!\cdots\!80$$ p^{12} T^{10} - $$14\!\cdots\!36$$ p^{19} T^{11} + 26109511358578261 p^{25} T^{12} + 79045042944 p^{31} T^{13} + 791635696 p^{36} T^{14} + 3108 p^{42} T^{15} + p^{48} T^{16} $
	37	$ 1 + 6124 T - 2410851594 T^{2} + 105511123683224 T^{3} + 5352778640202062561 T^{4} - $$23\!\cdots\!04$$ T^{5} + $$32\!\cdots\!86$$ T^{6} + $$72\!\cdots\!72$$ T^{7} - $$85\!\cdots\!20$$ T^{8} + $$72\!\cdots\!72$$ p^{6} T^{9} + $$32\!\cdots\!86$$ p^{12} T^{10} - $$23\!\cdots\!04$$ p^{18} T^{11} + 5352778640202062561 p^{24} T^{12} + 105511123683224 p^{30} T^{13} - 2410851594 p^{36} T^{14} + 6124 p^{42} T^{15} + p^{48} T^{16} $
	41	$ 1 - 12858425408 T^{2} + $$12\!\cdots\!52$$ T^{4} - $$77\!\cdots\!04$$ T^{6} + $$42\!\cdots\!02$$ T^{8} - $$77\!\cdots\!04$$ p^{12} T^{10} + $$12\!\cdots\!52$$ p^{24} T^{12} - 12858425408 p^{36} T^{14} + p^{48} T^{16} $
	43	$ ( 1 + 148688 T + 16174962100 T^{2} + 944168320869968 T^{3} + 67784308274406325750 T^{4} + 944168320869968 p^{6} T^{5} + 16174962100 p^{12} T^{6} + 148688 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	47	$ 1 - 313908 T + 61402539880 T^{2} - 8964101996547936 T^{3} + 20135717581678287629 p T^{4} - $$68\!\cdots\!56$$ T^{5} + $$17\!\cdots\!12$$ T^{6} + $$38\!\cdots\!92$$ T^{7} - $$62\!\cdots\!88$$ T^{8} + $$38\!\cdots\!92$$ p^{6} T^{9} + $$17\!\cdots\!12$$ p^{12} T^{10} - $$68\!\cdots\!56$$ p^{18} T^{11} + 20135717581678287629 p^{25} T^{12} - 8964101996547936 p^{30} T^{13} + 61402539880 p^{36} T^{14} - 313908 p^{42} T^{15} + p^{48} T^{16} $
	53	$ 1 - 278484 T - 12243422506 T^{2} + 6978883270647576 T^{3} + $$54\!\cdots\!05$$ T^{4} - $$81\!\cdots\!24$$ T^{5} - $$30\!\cdots\!46$$ T^{6} - $$74\!\cdots\!84$$ T^{7} + $$12\!\cdots\!48$$ T^{8} - $$74\!\cdots\!84$$ p^{6} T^{9} - $$30\!\cdots\!46$$ p^{12} T^{10} - $$81\!\cdots\!24$$ p^{18} T^{11} + $$54\!\cdots\!05$$ p^{24} T^{12} + 6978883270647576 p^{30} T^{13} - 12243422506 p^{36} T^{14} - 278484 p^{42} T^{15} + p^{48} T^{16} $
	59	$ 1 + 835464 T + 435025459648 T^{2} + 169063460080564224 T^{3} + $$54\!\cdots\!31$$ T^{4} + $$26\!\cdots\!00$$ p T^{5} + $$39\!\cdots\!40$$ T^{6} + $$15\!\cdots\!40$$ p T^{7} + $$20\!\cdots\!20$$ T^{8} + $$15\!\cdots\!40$$ p^{7} T^{9} + $$39\!\cdots\!40$$ p^{12} T^{10} + $$26\!\cdots\!00$$ p^{19} T^{11} + $$54\!\cdots\!31$$ p^{24} T^{12} + 169063460080564224 p^{30} T^{13} + 435025459648 p^{36} T^{14} + 835464 p^{42} T^{15} + p^{48} T^{16} $
	61	$ 1 + 995316 T + 651608313382 T^{2} + 319884941108213880 T^{3} + $$13\!\cdots\!29$$ T^{4} + $$45\!\cdots\!36$$ T^{5} + $$13\!\cdots\!74$$ T^{6} + $$37\!\cdots\!40$$ T^{7} + $$90\!\cdots\!88$$ T^{8} + $$37\!\cdots\!40$$ p^{6} T^{9} + $$13\!\cdots\!74$$ p^{12} T^{10} + $$45\!\cdots\!36$$ p^{18} T^{11} + $$13\!\cdots\!29$$ p^{24} T^{12} + 319884941108213880 p^{30} T^{13} + 651608313382 p^{36} T^{14} + 995316 p^{42} T^{15} + p^{48} T^{16} $
	67	$ 1 - 648808 T + 6714089688 T^{2} + 44922752496936664 T^{3} + $$11\!\cdots\!35$$ T^{4} - $$40\!\cdots\!72$$ T^{5} - $$17\!\cdots\!32$$ T^{6} - $$18\!\cdots\!00$$ T^{7} + $$27\!\cdots\!80$$ T^{8} - $$18\!\cdots\!00$$ p^{6} T^{9} - $$17\!\cdots\!32$$ p^{12} T^{10} - $$40\!\cdots\!72$$ p^{18} T^{11} + $$11\!\cdots\!35$$ p^{24} T^{12} + 44922752496936664 p^{30} T^{13} + 6714089688 p^{36} T^{14} - 648808 p^{42} T^{15} + p^{48} T^{16} $
	71	$ ( 1 - 95064 T + 220277516404 T^{2} + 16633830011187384 T^{3} + $$29\!\cdots\!42$$ T^{4} + 16633830011187384 p^{6} T^{5} + 220277516404 p^{12} T^{6} - 95064 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	73	$ 1 + 1617084 T + 1664708471134 T^{2} + 1282436417049503688 T^{3} + $$82\!\cdots\!85$$ T^{4} + $$46\!\cdots\!92$$ T^{5} + $$23\!\cdots\!42$$ T^{6} + $$10\!\cdots\!80$$ T^{7} + $$42\!\cdots\!88$$ T^{8} + $$10\!\cdots\!80$$ p^{6} T^{9} + $$23\!\cdots\!42$$ p^{12} T^{10} + $$46\!\cdots\!92$$ p^{18} T^{11} + $$82\!\cdots\!85$$ p^{24} T^{12} + 1282436417049503688 p^{30} T^{13} + 1664708471134 p^{36} T^{14} + 1617084 p^{42} T^{15} + p^{48} T^{16} $
	79	$ 1 - 70096 T - 570937949040 T^{2} - 91349453058129800 T^{3} + $$16\!\cdots\!43$$ T^{4} + $$44\!\cdots\!64$$ T^{5} - $$29\!\cdots\!92$$ T^{6} - $$61\!\cdots\!56$$ T^{7} + $$61\!\cdots\!96$$ T^{8} - $$61\!\cdots\!56$$ p^{6} T^{9} - $$29\!\cdots\!92$$ p^{12} T^{10} + $$44\!\cdots\!64$$ p^{18} T^{11} + $$16\!\cdots\!43$$ p^{24} T^{12} - 91349453058129800 p^{30} T^{13} - 570937949040 p^{36} T^{14} - 70096 p^{42} T^{15} + p^{48} T^{16} $
	83	$ 1 - 1511767571240 T^{2} + $$11\!\cdots\!56$$ T^{4} - $$60\!\cdots\!20$$ T^{6} + $$23\!\cdots\!90$$ T^{8} - $$60\!\cdots\!20$$ p^{12} T^{10} + $$11\!\cdots\!56$$ p^{24} T^{12} - 1511767571240 p^{36} T^{14} + p^{48} T^{16} $
	89	$ 1 - 739116 T + 1962144789862 T^{2} - 1315661442189804360 T^{3} + $$21\!\cdots\!93$$ T^{4} - $$11\!\cdots\!40$$ T^{5} + $$15\!\cdots\!30$$ T^{6} - $$72\!\cdots\!64$$ T^{7} + $$84\!\cdots\!64$$ T^{8} - $$72\!\cdots\!64$$ p^{6} T^{9} + $$15\!\cdots\!30$$ p^{12} T^{10} - $$11\!\cdots\!40$$ p^{18} T^{11} + $$21\!\cdots\!93$$ p^{24} T^{12} - 1315661442189804360 p^{30} T^{13} + 1962144789862 p^{36} T^{14} - 739116 p^{42} T^{15} + p^{48} T^{16} $
	97	$ 1 - 2264519987840 T^{2} + $$25\!\cdots\!68$$ T^{4} - $$26\!\cdots\!00$$ T^{6} + $$24\!\cdots\!18$$ T^{8} - $$26\!\cdots\!00$$ p^{12} T^{10} + $$25\!\cdots\!68$$ p^{24} T^{12} - 2264519987840 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

−8.479157816787994468742971312205, −8.440769700455981893589758909666, −8.052674674244446380541883275511, −7.997149627071705271107016265200, −7.85178318415484875848240046217, −7.55035587508063884003396003026, −7.23144016802011448639306366245, −6.79581831359207248185149357452, −6.69292121467620542601053896388, −6.40064284635542909788011158859, −5.88485815721681064297071919201, −5.66345889003333732125204759490, −5.58322959317473051261828330436, −4.57558884213832782254869380218, −4.56593239874600100182625789833, −4.54020170766729678799218717100, −4.30491369981107068926182868784, −4.26718284075360051888313407319, −3.37708825232127412988317115454, −3.35840157923930725669284949627, −2.25381316288422032692063158545, −2.13848555479060199688344947858, −1.89994797709191889328601166265, −0.07782869679193240682227583333, −0.06301963162225269464454508655, 0.06301963162225269464454508655, 0.07782869679193240682227583333, 1.89994797709191889328601166265, 2.13848555479060199688344947858, 2.25381316288422032692063158545, 3.35840157923930725669284949627, 3.37708825232127412988317115454, 4.26718284075360051888313407319, 4.30491369981107068926182868784, 4.54020170766729678799218717100, 4.56593239874600100182625789833, 4.57558884213832782254869380218, 5.58322959317473051261828330436, 5.66345889003333732125204759490, 5.88485815721681064297071919201, 6.40064284635542909788011158859, 6.69292121467620542601053896388, 6.79581831359207248185149357452, 7.23144016802011448639306366245, 7.55035587508063884003396003026, 7.85178318415484875848240046217, 7.997149627071705271107016265200, 8.052674674244446380541883275511, 8.440769700455981893589758909666, 8.479157816787994468742971312205

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 $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(0.0006378165735\)
\(L(\frac12)\)	\(\approx\)	\(0.0006378165735\)
\(L(4)\)		not available
\(L(1)\)		not available

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