LMFDB - L-function 16-1100e8-1.1-c5e8-0-1

Dirichlet series

L(s) = 1

+ 27·3^-s + 95·7^-s − 419·9^-s + 968·11^-s + 294·13^-s + 515·17^-s − 354·19^-s + 2.56e3·21^-s + 3.07e3·23^-s − 1.72e4·27^-s + 2.74e3·29^-s + 3.76e3·31^-s + 2.61e4·33^-s + 9.25e3·37^-s + 7.93e3·39^-s + 8.75e3·41^-s + 1.52e4·43^-s + 8.51e3·47^-s − 4.90e4·49^-s + 1.39e4·51^-s + 3.04e4·53^-s − 9.55e3·57^-s + 2.04e4·59^-s + 1.05e4·61^-s − 3.98e4·63^-s − 616·67^-s + 8.29e4·69^-s + ⋯

L(s) = 1

+ 1.73·3^-s + 0.732·7^-s − 1.72·9^-s + 2.41·11^-s + 0.482·13^-s + 0.432·17^-s − 0.224·19^-s + 1.26·21^-s + 1.21·23^-s − 4.54·27^-s + 0.605·29^-s + 0.704·31^-s + 4.17·33^-s + 1.11·37^-s + 0.835·39^-s + 0.813·41^-s + 1.26·43^-s + 0.562·47^-s − 2.91·49^-s + 0.748·51^-s + 1.49·53^-s − 0.389·57^-s + 0.762·59^-s + 0.361·61^-s − 1.26·63^-s − 0.0167·67^-s + 2.09·69^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$16$
Conductor:	$2^{16} \cdot 5^{16} \cdot 11^{8}$
Sign:	$1$
Analytic conductor:	$9.38481\times 10^{17}$
Root analytic conductor:	$13.2824$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 2^{16} \cdot 5^{16} \cdot 11^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$47.02327057$
$L(\frac12)$	$\approx$	$47.02327057$
$L(\frac{7}{2})$		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 $
	5	$ 1 $
	11	$ ( 1 - p^{2} T )^{8} $
good	3	$ 1 - p^{3} T + 1148 T^{2} - 25082 T^{3} + 218948 p T^{4} - 12328543 T^{5} + 85491041 p T^{6} - 450348068 p^{2} T^{7} + 2675536180 p^{3} T^{8} - 450348068 p^{7} T^{9} + 85491041 p^{11} T^{10} - 12328543 p^{15} T^{11} + 218948 p^{21} T^{12} - 25082 p^{25} T^{13} + 1148 p^{30} T^{14} - p^{38} T^{15} + p^{40} T^{16} $
	7	$ 1 - 95 T + 58070 T^{2} - 4990750 T^{3} + 1963202942 T^{4} - 131770772635 T^{5} + 45344925083691 T^{6} - 2588052561885180 T^{7} + 119012249052757168 p T^{8} - 2588052561885180 p^{5} T^{9} + 45344925083691 p^{10} T^{10} - 131770772635 p^{15} T^{11} + 1963202942 p^{20} T^{12} - 4990750 p^{25} T^{13} + 58070 p^{30} T^{14} - 95 p^{35} T^{15} + p^{40} T^{16} $
	13	$ 1 - 294 T + 1646986 T^{2} - 299202128 T^{3} + 1300913544220 T^{4} - 131062317794990 T^{5} + 679120853748309572 T^{6} - 32873792682418053086 T^{7} + $$27\!\cdots\!09$$ T^{8} - 32873792682418053086 p^{5} T^{9} + 679120853748309572 p^{10} T^{10} - 131062317794990 p^{15} T^{11} + 1300913544220 p^{20} T^{12} - 299202128 p^{25} T^{13} + 1646986 p^{30} T^{14} - 294 p^{35} T^{15} + p^{40} T^{16} $
	17	$ 1 - 515 T + 8070674 T^{2} - 3378374400 T^{3} + 30803153604746 T^{4} - 10383474743296645 T^{5} + 73615915722511852347 T^{6} - $$20\!\cdots\!60$$ T^{7} + $$12\!\cdots\!20$$ T^{8} - $$20\!\cdots\!60$$ p^{5} T^{9} + 73615915722511852347 p^{10} T^{10} - 10383474743296645 p^{15} T^{11} + 30803153604746 p^{20} T^{12} - 3378374400 p^{25} T^{13} + 8070674 p^{30} T^{14} - 515 p^{35} T^{15} + p^{40} T^{16} $
	19	$ 1 + 354 T + 8500786 T^{2} + 16451228 T^{3} + 31452634933204 T^{4} - 27251998693430050 T^{5} + 62676039577996743348 T^{6} - $$81\!\cdots\!46$$ p T^{7} + $$10\!\cdots\!33$$ T^{8} - $$81\!\cdots\!46$$ p^{6} T^{9} + 62676039577996743348 p^{10} T^{10} - 27251998693430050 p^{15} T^{11} + 31452634933204 p^{20} T^{12} + 16451228 p^{25} T^{13} + 8500786 p^{30} T^{14} + 354 p^{35} T^{15} + p^{40} T^{16} $
	23	$ 1 - 3073 T + 32764513 T^{2} - 70606306181 T^{3} + 485395074193054 T^{4} - 794572568281071813 T^{5} + $$46\!\cdots\!34$$ T^{6} - $$63\!\cdots\!92$$ T^{7} + $$34\!\cdots\!35$$ T^{8} - $$63\!\cdots\!92$$ p^{5} T^{9} + $$46\!\cdots\!34$$ p^{10} T^{10} - 794572568281071813 p^{15} T^{11} + 485395074193054 p^{20} T^{12} - 70606306181 p^{25} T^{13} + 32764513 p^{30} T^{14} - 3073 p^{35} T^{15} + p^{40} T^{16} $
	29	$ 1 - 2743 T + 106386611 T^{2} - 159354195085 T^{3} + 5305993204724862 T^{4} - 4472618667049547707 T^{5} + $$17\!\cdots\!36$$ T^{6} - $$10\!\cdots\!10$$ T^{7} + $$42\!\cdots\!61$$ T^{8} - $$10\!\cdots\!10$$ p^{5} T^{9} + $$17\!\cdots\!36$$ p^{10} T^{10} - 4472618667049547707 p^{15} T^{11} + 5305993204724862 p^{20} T^{12} - 159354195085 p^{25} T^{13} + 106386611 p^{30} T^{14} - 2743 p^{35} T^{15} + p^{40} T^{16} $
	31	$ 1 - 3768 T + 124773304 T^{2} - 509812485728 T^{3} + 8212940005839430 T^{4} - 31651081416180785816 T^{5} + $$36\!\cdots\!96$$ T^{6} - $$12\!\cdots\!64$$ T^{7} + $$12\!\cdots\!63$$ T^{8} - $$12\!\cdots\!64$$ p^{5} T^{9} + $$36\!\cdots\!96$$ p^{10} T^{10} - 31651081416180785816 p^{15} T^{11} + 8212940005839430 p^{20} T^{12} - 509812485728 p^{25} T^{13} + 124773304 p^{30} T^{14} - 3768 p^{35} T^{15} + p^{40} T^{16} $
	37	$ 1 - 9252 T + 389364445 T^{2} - 2924994188780 T^{3} + 66964745580569125 T^{4} - $$42\!\cdots\!92$$ T^{5} + $$70\!\cdots\!90$$ T^{6} - $$38\!\cdots\!96$$ T^{7} + $$55\!\cdots\!26$$ T^{8} - $$38\!\cdots\!96$$ p^{5} T^{9} + $$70\!\cdots\!90$$ p^{10} T^{10} - $$42\!\cdots\!92$$ p^{15} T^{11} + 66964745580569125 p^{20} T^{12} - 2924994188780 p^{25} T^{13} + 389364445 p^{30} T^{14} - 9252 p^{35} T^{15} + p^{40} T^{16} $
	41	$ 1 - 8756 T + 571131337 T^{2} - 5219543555448 T^{3} + 173989147503502253 T^{4} - $$14\!\cdots\!92$$ T^{5} + $$34\!\cdots\!94$$ T^{6} - $$24\!\cdots\!08$$ T^{7} + $$46\!\cdots\!54$$ T^{8} - $$24\!\cdots\!08$$ p^{5} T^{9} + $$34\!\cdots\!94$$ p^{10} T^{10} - $$14\!\cdots\!92$$ p^{15} T^{11} + 173989147503502253 p^{20} T^{12} - 5219543555448 p^{25} T^{13} + 571131337 p^{30} T^{14} - 8756 p^{35} T^{15} + p^{40} T^{16} $
	43	$ 1 - 15290 T + 679752677 T^{2} - 6980272019150 T^{3} + 179279007856110034 T^{4} - $$10\!\cdots\!90$$ T^{5} + $$24\!\cdots\!83$$ T^{6} - $$37\!\cdots\!50$$ T^{7} + $$29\!\cdots\!50$$ T^{8} - $$37\!\cdots\!50$$ p^{5} T^{9} + $$24\!\cdots\!83$$ p^{10} T^{10} - $$10\!\cdots\!90$$ p^{15} T^{11} + 179279007856110034 p^{20} T^{12} - 6980272019150 p^{25} T^{13} + 679752677 p^{30} T^{14} - 15290 p^{35} T^{15} + p^{40} T^{16} $
	47	$ 1 - 8516 T + 353905213 T^{2} - 2702164222340 T^{3} + 154084054928339793 T^{4} - $$67\!\cdots\!40$$ T^{5} + $$25\!\cdots\!50$$ T^{6} - $$79\!\cdots\!36$$ T^{7} + $$79\!\cdots\!02$$ T^{8} - $$79\!\cdots\!36$$ p^{5} T^{9} + $$25\!\cdots\!50$$ p^{10} T^{10} - $$67\!\cdots\!40$$ p^{15} T^{11} + 154084054928339793 p^{20} T^{12} - 2702164222340 p^{25} T^{13} + 353905213 p^{30} T^{14} - 8516 p^{35} T^{15} + p^{40} T^{16} $
	53	$ 1 - 575 p T + 2558062970 T^{2} - 69023752837000 T^{3} + 3169575847517452230 T^{4} - $$73\!\cdots\!25$$ T^{5} + $$24\!\cdots\!27$$ T^{6} - $$47\!\cdots\!00$$ T^{7} + $$12\!\cdots\!60$$ T^{8} - $$47\!\cdots\!00$$ p^{5} T^{9} + $$24\!\cdots\!27$$ p^{10} T^{10} - $$73\!\cdots\!25$$ p^{15} T^{11} + 3169575847517452230 p^{20} T^{12} - 69023752837000 p^{25} T^{13} + 2558062970 p^{30} T^{14} - 575 p^{36} T^{15} + p^{40} T^{16} $
	59	$ 1 - 20400 T + 2673515925 T^{2} - 60048917751812 T^{3} + 4040751400422601561 T^{4} - $$93\!\cdots\!72$$ T^{5} + $$42\!\cdots\!78$$ T^{6} - $$94\!\cdots\!44$$ T^{7} + $$34\!\cdots\!06$$ T^{8} - $$94\!\cdots\!44$$ p^{5} T^{9} + $$42\!\cdots\!78$$ p^{10} T^{10} - $$93\!\cdots\!72$$ p^{15} T^{11} + 4040751400422601561 p^{20} T^{12} - 60048917751812 p^{25} T^{13} + 2673515925 p^{30} T^{14} - 20400 p^{35} T^{15} + p^{40} T^{16} $
	61	$ 1 - 10511 T + 4430862386 T^{2} - 41480469699068 T^{3} + 9044373250333245454 T^{4} - $$70\!\cdots\!81$$ T^{5} + $$11\!\cdots\!31$$ T^{6} - $$75\!\cdots\!72$$ T^{7} + $$11\!\cdots\!28$$ T^{8} - $$75\!\cdots\!72$$ p^{5} T^{9} + $$11\!\cdots\!31$$ p^{10} T^{10} - $$70\!\cdots\!81$$ p^{15} T^{11} + 9044373250333245454 p^{20} T^{12} - 41480469699068 p^{25} T^{13} + 4430862386 p^{30} T^{14} - 10511 p^{35} T^{15} + p^{40} T^{16} $
	67	$ 1 + 616 T + 6992153588 T^{2} + 60819047841672 T^{3} + 23295577694480225076 T^{4} + $$32\!\cdots\!72$$ T^{5} + $$50\!\cdots\!36$$ T^{6} + $$78\!\cdots\!44$$ T^{7} + $$78\!\cdots\!46$$ T^{8} + $$78\!\cdots\!44$$ p^{5} T^{9} + $$50\!\cdots\!36$$ p^{10} T^{10} + $$32\!\cdots\!72$$ p^{15} T^{11} + 23295577694480225076 p^{20} T^{12} + 60819047841672 p^{25} T^{13} + 6992153588 p^{30} T^{14} + 616 p^{35} T^{15} + p^{40} T^{16} $
	71	$ 1 - 31862 T + 4763146327 T^{2} - 207973744950032 T^{3} + 17765665956346644114 T^{4} - $$73\!\cdots\!08$$ T^{5} + $$49\!\cdots\!37$$ T^{6} - $$17\!\cdots\!10$$ T^{7} + $$10\!\cdots\!10$$ T^{8} - $$17\!\cdots\!10$$ p^{5} T^{9} + $$49\!\cdots\!37$$ p^{10} T^{10} - $$73\!\cdots\!08$$ p^{15} T^{11} + 17765665956346644114 p^{20} T^{12} - 207973744950032 p^{25} T^{13} + 4763146327 p^{30} T^{14} - 31862 p^{35} T^{15} + p^{40} T^{16} $
	73	$ 1 + 37377 T + 9691718398 T^{2} + 250095109755704 T^{3} + 38779425301856091094 T^{4} + $$70\!\cdots\!67$$ T^{5} + $$13\!\cdots\!23$$ p T^{6} + $$13\!\cdots\!68$$ T^{7} + $$19\!\cdots\!40$$ T^{8} + $$13\!\cdots\!68$$ p^{5} T^{9} + $$13\!\cdots\!23$$ p^{11} T^{10} + $$70\!\cdots\!67$$ p^{15} T^{11} + 38779425301856091094 p^{20} T^{12} + 250095109755704 p^{25} T^{13} + 9691718398 p^{30} T^{14} + 37377 p^{35} T^{15} + p^{40} T^{16} $
	79	$ 1 + 71679 T + 11592600248 T^{2} + 343428184255746 T^{3} + 50334349577414442244 T^{4} + $$19\!\cdots\!11$$ T^{5} + $$13\!\cdots\!43$$ T^{6} - $$42\!\cdots\!60$$ T^{7} + $$31\!\cdots\!20$$ T^{8} - $$42\!\cdots\!60$$ p^{5} T^{9} + $$13\!\cdots\!43$$ p^{10} T^{10} + $$19\!\cdots\!11$$ p^{15} T^{11} + 50334349577414442244 p^{20} T^{12} + 343428184255746 p^{25} T^{13} + 11592600248 p^{30} T^{14} + 71679 p^{35} T^{15} + p^{40} T^{16} $
	83	$ 1 + 61543 T + 21354644495 T^{2} + 1036154948064317 T^{3} + $$21\!\cdots\!00$$ T^{4} + $$85\!\cdots\!75$$ T^{5} + $$13\!\cdots\!34$$ T^{6} + $$45\!\cdots\!04$$ T^{7} + $$64\!\cdots\!99$$ T^{8} + $$45\!\cdots\!04$$ p^{5} T^{9} + $$13\!\cdots\!34$$ p^{10} T^{10} + $$85\!\cdots\!75$$ p^{15} T^{11} + $$21\!\cdots\!00$$ p^{20} T^{12} + 1036154948064317 p^{25} T^{13} + 21354644495 p^{30} T^{14} + 61543 p^{35} T^{15} + p^{40} T^{16} $
	89	$ 1 + 14961 T + 24626385913 T^{2} + 1040589268863261 T^{3} + $$29\!\cdots\!20$$ T^{4} + $$16\!\cdots\!89$$ T^{5} + $$26\!\cdots\!48$$ T^{6} + $$13\!\cdots\!22$$ T^{7} + $$17\!\cdots\!17$$ T^{8} + $$13\!\cdots\!22$$ p^{5} T^{9} + $$26\!\cdots\!48$$ p^{10} T^{10} + $$16\!\cdots\!89$$ p^{15} T^{11} + $$29\!\cdots\!20$$ p^{20} T^{12} + 1040589268863261 p^{25} T^{13} + 24626385913 p^{30} T^{14} + 14961 p^{35} T^{15} + p^{40} T^{16} $
	97	$ 1 - 186419 T + 12788834187 T^{2} + 277956286846529 T^{3} + 3573328297180559084 T^{4} - $$10\!\cdots\!11$$ T^{5} + $$14\!\cdots\!42$$ T^{6} - $$73\!\cdots\!66$$ T^{7} + $$56\!\cdots\!95$$ T^{8} - $$73\!\cdots\!66$$ p^{5} T^{9} + $$14\!\cdots\!42$$ p^{10} T^{10} - $$10\!\cdots\!11$$ p^{15} T^{11} + 3573328297180559084 p^{20} T^{12} + 277956286846529 p^{25} T^{13} + 12788834187 p^{30} T^{14} - 186419 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

−3.56251338598277829399163215825, −3.00458527039790607141616736370, −2.99535311315805961561638383828, −2.93562146018154095751748581758, −2.91425596064450502815069306259, −2.91275310933124350273586983549, −2.88725794584531715391114905387, −2.73854356549439115407926822824, −2.52563476474985293456710137724, −2.17017840287310172892438214336, −2.09653614068863883264068975156, −1.93202248989083001425158108251, −1.76701432019949715882330903078, −1.76059418641171176093122128075, −1.75881736468351861764410349425, −1.56549028518069819016539282236, −1.31877067029414172646003803409, −1.14256593901207743922366141036, −0.71965051475183915677881963072, −0.70549675726950669017363838456, −0.67554413549722652663076284114, −0.64241412596652426288203802490, −0.62673361611550052738025976653, −0.53580571162123177534505436548, −0.10200329936327162049398419941, 0.10200329936327162049398419941, 0.53580571162123177534505436548, 0.62673361611550052738025976653, 0.64241412596652426288203802490, 0.67554413549722652663076284114, 0.70549675726950669017363838456, 0.71965051475183915677881963072, 1.14256593901207743922366141036, 1.31877067029414172646003803409, 1.56549028518069819016539282236, 1.75881736468351861764410349425, 1.76059418641171176093122128075, 1.76701432019949715882330903078, 1.93202248989083001425158108251, 2.09653614068863883264068975156, 2.17017840287310172892438214336, 2.52563476474985293456710137724, 2.73854356549439115407926822824, 2.88725794584531715391114905387, 2.91275310933124350273586983549, 2.91425596064450502815069306259, 2.93562146018154095751748581758, 2.99535311315805961561638383828, 3.00458527039790607141616736370, 3.56251338598277829399163215825

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(47.02327057\)
\(L(\frac12)\)	\(\approx\)	\(47.02327057\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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