LMFDB - L-function 16-147e8-1.1-c6e8-0-3

Dirichlet series

L(s) = 1

− 5·2^-s − 108·3^-s + 54·4^-s + 42·5^-s + 540·6^-s − 593·8^-s + 6.31e3·9^-s − 210·10^-s − 1.07e3·11^-s − 5.83e3·12^-s − 4.53e3·15^-s + 6.97e3·16^-s − 7.21e3·17^-s − 3.15e4·18^-s + 2.46e4·19^-s + 2.26e3·20^-s + 5.35e3·22^-s − 1.52e4·23^-s + 6.40e4·24^-s − 1.92e4·25^-s − 2.62e5·27^-s + 3.25e4·29^-s + 2.26e4·30^-s − 4.02e4·31^-s − 3.24e4·32^-s + 1.15e5·33^-s + 3.60e4·34^-s + ⋯

L(s) = 1

− 5/8·2^-s − 4·3^-s + 0.843·4^-s + 0.335·5^-s + 5/2·6^-s − 1.15·8^-s + 26/3·9^-s − 0.209·10^-s − 0.803·11^-s − 3.37·12^-s − 1.34·15^-s + 1.70·16^-s − 1.46·17^-s − 5.41·18^-s + 3.58·19^-s + 0.283·20^-s + 0.502·22^-s − 1.25·23^-s + 4.63·24^-s − 1.23·25^-s − 13.3·27^-s + 1.33·29^-s + 0.839·30^-s − 1.34·31^-s − 0.991·32^-s + 3.21·33^-s + 0.917·34^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$16$
Conductor:	$3^{8} \cdot 7^{16}$
Sign:	$1$
Analytic conductor:	$1.71071\times 10^{12}$
Root analytic conductor:	$5.81532$
Motivic weight:	$6$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(16,\ 3^{8} \cdot 7^{16} ,\ ( \ : [3]^{8} ),\ 1 )$

Particular Values

$L(\frac{7}{2})$	$\approx$	$0.5749517820$
$L(\frac12)$	$\approx$	$0.5749517820$
$L(4)$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ ( 1 + p^{3} T + p^{5} T^{2} )^{4} $
	7	$ 1 $
good	2	$ 1 + 5 T - 29 T^{2} + 89 p T^{3} - 777 p T^{4} - 9243 p^{2} T^{5} + 50201 p^{2} T^{6} + 71679 p^{5} T^{7} + 17 p^{17} T^{8} + 71679 p^{11} T^{9} + 50201 p^{14} T^{10} - 9243 p^{20} T^{11} - 777 p^{25} T^{12} + 89 p^{31} T^{13} - 29 p^{36} T^{14} + 5 p^{42} T^{15} + p^{48} T^{16} $
	5	$ 1 - 42 T + 4199 p T^{2} - 857094 T^{3} + 59255749 T^{4} - 71357004 p^{3} T^{5} - 55623489398 p^{2} T^{6} - 110556133392 p^{5} T^{7} - 1151873646254 p^{4} T^{8} - 110556133392 p^{11} T^{9} - 55623489398 p^{14} T^{10} - 71357004 p^{21} T^{11} + 59255749 p^{24} T^{12} - 857094 p^{30} T^{13} + 4199 p^{37} T^{14} - 42 p^{42} T^{15} + p^{48} T^{16} $
	11	$ 1 + 1070 T - 516103 p T^{2} - 3448075790 T^{3} + 22836612400425 T^{4} + 7759923983215740 T^{5} - 60967102988727183826 T^{6} - $$45\!\cdots\!20$$ T^{7} + $$12\!\cdots\!86$$ T^{8} - $$45\!\cdots\!20$$ p^{6} T^{9} - 60967102988727183826 p^{12} T^{10} + 7759923983215740 p^{18} T^{11} + 22836612400425 p^{24} T^{12} - 3448075790 p^{30} T^{13} - 516103 p^{37} T^{14} + 1070 p^{42} T^{15} + p^{48} T^{16} $
	13	$ 1 - 16298786 T^{2} + 111852524426401 T^{4} - 37539779661680550026 p T^{6} + $$20\!\cdots\!88$$ T^{8} - 37539779661680550026 p^{13} T^{10} + 111852524426401 p^{24} T^{12} - 16298786 p^{36} T^{14} + p^{48} T^{16} $
	17	$ 1 + 7212 T + 37951384 T^{2} + 148666264032 T^{3} - 401372098722782 T^{4} - 760743811350012996 T^{5} + $$67\!\cdots\!80$$ T^{6} + $$10\!\cdots\!52$$ T^{7} + $$11\!\cdots\!47$$ T^{8} + $$10\!\cdots\!52$$ p^{6} T^{9} + $$67\!\cdots\!80$$ p^{12} T^{10} - 760743811350012996 p^{18} T^{11} - 401372098722782 p^{24} T^{12} + 148666264032 p^{30} T^{13} + 37951384 p^{36} T^{14} + 7212 p^{42} T^{15} + p^{48} T^{16} $
	19	$ 1 - 24606 T + 362733151 T^{2} - 3959468067834 T^{3} + 36560849427203149 T^{4} - $$28\!\cdots\!40$$ T^{5} + $$19\!\cdots\!30$$ T^{6} - $$12\!\cdots\!40$$ T^{7} + $$82\!\cdots\!38$$ T^{8} - $$12\!\cdots\!40$$ p^{6} T^{9} + $$19\!\cdots\!30$$ p^{12} T^{10} - $$28\!\cdots\!40$$ p^{18} T^{11} + 36560849427203149 p^{24} T^{12} - 3959468067834 p^{30} T^{13} + 362733151 p^{36} T^{14} - 24606 p^{42} T^{15} + p^{48} T^{16} $
	23	$ 1 + 15224 T - 71404700 T^{2} - 3251082330032 T^{3} - 39028069690313382 T^{4} - $$38\!\cdots\!08$$ T^{5} - $$16\!\cdots\!68$$ T^{6} + $$76\!\cdots\!84$$ T^{7} + $$17\!\cdots\!71$$ T^{8} + $$76\!\cdots\!84$$ p^{6} T^{9} - $$16\!\cdots\!68$$ p^{12} T^{10} - $$38\!\cdots\!08$$ p^{18} T^{11} - 39028069690313382 p^{24} T^{12} - 3251082330032 p^{30} T^{13} - 71404700 p^{36} T^{14} + 15224 p^{42} T^{15} + p^{48} T^{16} $
	29	$ ( 1 - 16262 T + 1936712745 T^{2} - 19588912728706 T^{3} + 1557131395176891808 T^{4} - 19588912728706 p^{6} T^{5} + 1936712745 p^{12} T^{6} - 16262 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	31	$ 1 + 40200 T + 2343974770 T^{2} + 72572849754000 T^{3} + 2084077897304297293 T^{4} + $$75\!\cdots\!20$$ T^{5} + $$25\!\cdots\!50$$ T^{6} + $$10\!\cdots\!60$$ T^{7} + $$32\!\cdots\!28$$ T^{8} + $$10\!\cdots\!60$$ p^{6} T^{9} + $$25\!\cdots\!50$$ p^{12} T^{10} + $$75\!\cdots\!20$$ p^{18} T^{11} + 2084077897304297293 p^{24} T^{12} + 72572849754000 p^{30} T^{13} + 2343974770 p^{36} T^{14} + 40200 p^{42} T^{15} + p^{48} T^{16} $
	37	$ 1 + 45670 T - 5087844981 T^{2} - 327542380882710 T^{3} + 12351146188805096265 T^{4} + $$98\!\cdots\!60$$ T^{5} - $$98\!\cdots\!54$$ T^{6} - $$12\!\cdots\!80$$ T^{7} - $$38\!\cdots\!86$$ T^{8} - $$12\!\cdots\!80$$ p^{6} T^{9} - $$98\!\cdots\!54$$ p^{12} T^{10} + $$98\!\cdots\!60$$ p^{18} T^{11} + 12351146188805096265 p^{24} T^{12} - 327542380882710 p^{30} T^{13} - 5087844981 p^{36} T^{14} + 45670 p^{42} T^{15} + p^{48} T^{16} $
	41	$ 1 - 26005847576 T^{2} + $$34\!\cdots\!60$$ T^{4} - $$28\!\cdots\!64$$ T^{6} + $$15\!\cdots\!14$$ T^{8} - $$28\!\cdots\!64$$ p^{12} T^{10} + $$34\!\cdots\!60$$ p^{24} T^{12} - 26005847576 p^{36} T^{14} + p^{48} T^{16} $
	43	$ ( 1 + 222830 T + 23708395369 T^{2} + 1380667882433470 T^{3} + 73296626267097035876 T^{4} + 1380667882433470 p^{6} T^{5} + 23708395369 p^{12} T^{6} + 222830 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	47	$ 1 + 82884 T + 33030950488 T^{2} + 2547939641253024 T^{3} + $$61\!\cdots\!34$$ T^{4} + $$67\!\cdots\!36$$ T^{5} + $$88\!\cdots\!56$$ T^{6} + $$10\!\cdots\!28$$ T^{7} + $$95\!\cdots\!35$$ T^{8} + $$10\!\cdots\!28$$ p^{6} T^{9} + $$88\!\cdots\!56$$ p^{12} T^{10} + $$67\!\cdots\!36$$ p^{18} T^{11} + $$61\!\cdots\!34$$ p^{24} T^{12} + 2547939641253024 p^{30} T^{13} + 33030950488 p^{36} T^{14} + 82884 p^{42} T^{15} + p^{48} T^{16} $
	53	$ 1 + 13034 T - 37911011717 T^{2} + 368806876088782 T^{3} + $$25\!\cdots\!77$$ T^{4} - $$27\!\cdots\!36$$ T^{5} - $$79\!\cdots\!10$$ T^{6} + $$35\!\cdots\!88$$ T^{7} + $$46\!\cdots\!02$$ T^{8} + $$35\!\cdots\!88$$ p^{6} T^{9} - $$79\!\cdots\!10$$ p^{12} T^{10} - $$27\!\cdots\!36$$ p^{18} T^{11} + $$25\!\cdots\!77$$ p^{24} T^{12} + 368806876088782 p^{30} T^{13} - 37911011717 p^{36} T^{14} + 13034 p^{42} T^{15} + p^{48} T^{16} $
	59	$ 1 + 1810362 T + 1656169621891 T^{2} + 1020500030287048566 T^{3} + $$47\!\cdots\!61$$ T^{4} + $$17\!\cdots\!16$$ T^{5} + $$54\!\cdots\!70$$ T^{6} + $$14\!\cdots\!12$$ T^{7} + $$31\!\cdots\!62$$ T^{8} + $$14\!\cdots\!12$$ p^{6} T^{9} + $$54\!\cdots\!70$$ p^{12} T^{10} + $$17\!\cdots\!16$$ p^{18} T^{11} + $$47\!\cdots\!61$$ p^{24} T^{12} + 1020500030287048566 p^{30} T^{13} + 1656169621891 p^{36} T^{14} + 1810362 p^{42} T^{15} + p^{48} T^{16} $
	61	$ 1 - 392856 T + 253766894356 T^{2} - 79483260556868064 T^{3} + $$32\!\cdots\!58$$ T^{4} - $$85\!\cdots\!52$$ T^{5} + $$27\!\cdots\!00$$ T^{6} - $$61\!\cdots\!24$$ T^{7} + $$16\!\cdots\!47$$ T^{8} - $$61\!\cdots\!24$$ p^{6} T^{9} + $$27\!\cdots\!00$$ p^{12} T^{10} - $$85\!\cdots\!52$$ p^{18} T^{11} + $$32\!\cdots\!58$$ p^{24} T^{12} - 79483260556868064 p^{30} T^{13} + 253766894356 p^{36} T^{14} - 392856 p^{42} T^{15} + p^{48} T^{16} $
	67	$ 1 - 384094 T - 169408100025 T^{2} + 57304941612054286 T^{3} + $$24\!\cdots\!05$$ T^{4} - $$42\!\cdots\!12$$ T^{5} - $$32\!\cdots\!54$$ T^{6} + $$98\!\cdots\!00$$ T^{7} + $$36\!\cdots\!22$$ T^{8} + $$98\!\cdots\!00$$ p^{6} T^{9} - $$32\!\cdots\!54$$ p^{12} T^{10} - $$42\!\cdots\!12$$ p^{18} T^{11} + $$24\!\cdots\!05$$ p^{24} T^{12} + 57304941612054286 p^{30} T^{13} - 169408100025 p^{36} T^{14} - 384094 p^{42} T^{15} + p^{48} T^{16} $
	71	$ ( 1 - 112844 T + 324448495296 T^{2} + 1073384954931548 T^{3} + $$47\!\cdots\!38$$ T^{4} + 1073384954931548 p^{6} T^{5} + 324448495296 p^{12} T^{6} - 112844 p^{18} T^{7} + p^{24} T^{8} )^{2} $
	73	$ 1 + 903078 T + 897300437755 T^{2} + 564830568330899706 T^{3} + $$36\!\cdots\!49$$ T^{4} + $$18\!\cdots\!76$$ T^{5} + $$93\!\cdots\!70$$ T^{6} + $$39\!\cdots\!56$$ T^{7} + $$16\!\cdots\!62$$ T^{8} + $$39\!\cdots\!56$$ p^{6} T^{9} + $$93\!\cdots\!70$$ p^{12} T^{10} + $$18\!\cdots\!76$$ p^{18} T^{11} + $$36\!\cdots\!49$$ p^{24} T^{12} + 564830568330899706 p^{30} T^{13} + 897300437755 p^{36} T^{14} + 903078 p^{42} T^{15} + p^{48} T^{16} $
	79	$ 1 + 559592 T - 192861426366 T^{2} - 38884411164730496 T^{3} + $$45\!\cdots\!81$$ T^{4} - $$13\!\cdots\!84$$ T^{5} + $$30\!\cdots\!58$$ T^{6} + $$45\!\cdots\!96$$ T^{7} - $$14\!\cdots\!88$$ T^{8} + $$45\!\cdots\!96$$ p^{6} T^{9} + $$30\!\cdots\!58$$ p^{12} T^{10} - $$13\!\cdots\!84$$ p^{18} T^{11} + $$45\!\cdots\!81$$ p^{24} T^{12} - 38884411164730496 p^{30} T^{13} - 192861426366 p^{36} T^{14} + 559592 p^{42} T^{15} + p^{48} T^{16} $
	83	$ 1 - 577456669706 T^{2} + $$27\!\cdots\!89$$ T^{4} - $$12\!\cdots\!18$$ T^{6} + $$42\!\cdots\!16$$ T^{8} - $$12\!\cdots\!18$$ p^{12} T^{10} + $$27\!\cdots\!89$$ p^{24} T^{12} - 577456669706 p^{36} T^{14} + p^{48} T^{16} $
	89	$ 1 - 1770036 T + 3412957024480 T^{2} - 4192533013088545728 T^{3} + $$53\!\cdots\!74$$ T^{4} - $$51\!\cdots\!96$$ T^{5} + $$49\!\cdots\!64$$ T^{6} - $$38\!\cdots\!04$$ T^{7} + $$30\!\cdots\!35$$ T^{8} - $$38\!\cdots\!04$$ p^{6} T^{9} + $$49\!\cdots\!64$$ p^{12} T^{10} - $$51\!\cdots\!96$$ p^{18} T^{11} + $$53\!\cdots\!74$$ p^{24} T^{12} - 4192533013088545728 p^{30} T^{13} + 3412957024480 p^{36} T^{14} - 1770036 p^{42} T^{15} + p^{48} T^{16} $
	97	$ 1 - 4315064900690 T^{2} + $$91\!\cdots\!41$$ T^{4} - $$12\!\cdots\!58$$ T^{6} + $$12\!\cdots\!60$$ T^{8} - $$12\!\cdots\!58$$ p^{12} T^{10} + $$91\!\cdots\!41$$ p^{24} T^{12} - 4315064900690 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

−4.89694826542435441375125652100, −4.73317129452206101877038560152, −4.55959959749671389882157872141, −4.31128505712049518903841859410, −4.26480057395925574136299287944, −4.24782319660920900322819209227, −3.65810348472996861308097978455, −3.33935511872075694349662514298, −3.30113759255934728066338987884, −3.21497055686409498416024634482, −3.03994015497324337692851002495, −2.95810999235717902687315655922, −2.89248258070465242086191437765, −2.02367135822062937674664547972, −1.84334088317644376309649117863, −1.84187715147097261076059135143, −1.79186203132370454970928035520, −1.61115997845326732574353243943, −1.53305366681238357444478499131, −0.923014931527965953673955345324, −0.897265493756241526355295301837, −0.64642817597606383459126205931, −0.32130604628070188230093502892, −0.30972774640889175760339065476, −0.23560527128413133065899137184, 0.23560527128413133065899137184, 0.30972774640889175760339065476, 0.32130604628070188230093502892, 0.64642817597606383459126205931, 0.897265493756241526355295301837, 0.923014931527965953673955345324, 1.53305366681238357444478499131, 1.61115997845326732574353243943, 1.79186203132370454970928035520, 1.84187715147097261076059135143, 1.84334088317644376309649117863, 2.02367135822062937674664547972, 2.89248258070465242086191437765, 2.95810999235717902687315655922, 3.03994015497324337692851002495, 3.21497055686409498416024634482, 3.30113759255934728066338987884, 3.33935511872075694349662514298, 3.65810348472996861308097978455, 4.24782319660920900322819209227, 4.26480057395925574136299287944, 4.31128505712049518903841859410, 4.55959959749671389882157872141, 4.73317129452206101877038560152, 4.89694826542435441375125652100

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 $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(0.5749517820\)
\(L(\frac12)\)	\(\approx\)	\(0.5749517820\)
\(L(4)\)		not available
\(L(1)\)		not available

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