LMFDB - L-function 32-21e16-1.1-c4e16-0-0

Dirichlet series

L(s) = 1

+ 8·3^-s − 25·4^-s − 168·7^-s + 63·9^-s − 200·12^-s − 492·13^-s + 662·16^-s − 1.32e3·19^-s − 1.34e3·21^-s − 1.45e3·25^-s + 676·27^-s + 4.20e3·28^-s − 2.50e3·31^-s − 1.57e3·36^-s + 1.34e3·37^-s − 3.93e3·39^-s + 1.46e3·43^-s + 5.29e3·48^-s + 1.68e4·49^-s + 1.23e4·52^-s − 1.06e4·57^-s + 1.50e4·61^-s − 1.05e4·63^-s − 1.08e4·64^-s + 8.65e3·67^-s + 1.23e4·73^-s − 1.16e4·75^-s + ⋯

L(s) = 1

+ 8/9·3^-s − 1.56·4^-s − 3.42·7^-s + 7/9·9^-s − 1.38·12^-s − 2.91·13^-s + 2.58·16^-s − 3.67·19^-s − 3.04·21^-s − 2.32·25^-s + 0.927·27^-s + 5.35·28^-s − 2.60·31^-s − 1.21·36^-s + 0.980·37^-s − 2.58·39^-s + 0.789·43^-s + 2.29·48^-s + 7.03·49^-s + 4.54·52^-s − 3.26·57^-s + 4.03·61^-s − 8/3·63^-s − 2.65·64^-s + 1.92·67^-s + 2.31·73^-s − 2.06·75^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]

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

Invariants

Degree:	$32$
Conductor:	$3^{16} \cdot 7^{16}$
Sign:	$1$
Analytic conductor:	$243118.$
Root analytic conductor:	$1.47335$
Motivic weight:	$4$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(32,\ 3^{16} \cdot 7^{16} ,\ ( \ : [2]^{16} ),\ 1 )$

Particular Values

$L(\frac{5}{2})$	$\approx$	$0.06611522494$
$L(\frac12)$	$\approx$	$0.06611522494$
$L(3)$		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 - 8 T + T^{2} - 20 p^{2} T^{3} + 311 p^{3} T^{4} - 4256 p^{3} T^{5} + 4562 p^{5} T^{6} - 2276 p^{7} T^{7} + 40570 p^{6} T^{8} - 2276 p^{11} T^{9} + 4562 p^{13} T^{10} - 4256 p^{15} T^{11} + 311 p^{19} T^{12} - 20 p^{22} T^{13} + p^{24} T^{14} - 8 p^{28} T^{15} + p^{32} T^{16} $
	7	$ ( 1 + 12 p T + 305 p T^{2} - 4476 p^{2} T^{3} - 8028 p^{4} T^{4} - 4476 p^{6} T^{5} + 305 p^{9} T^{6} + 12 p^{13} T^{7} + p^{16} T^{8} )^{2} $
good	2	$ 1 + 25 T^{2} - 37 T^{4} - 3305 p T^{6} - 1981 p^{4} T^{8} + 201055 p^{3} T^{10} + 213303 p^{7} T^{12} - 10357035 p^{5} T^{14} - 215848295 p^{6} T^{16} - 10357035 p^{13} T^{18} + 213303 p^{23} T^{20} + 201055 p^{27} T^{22} - 1981 p^{36} T^{24} - 3305 p^{41} T^{26} - 37 p^{48} T^{28} + 25 p^{56} T^{30} + p^{64} T^{32} $
	5	$ 1 + 1453 T^{2} + 134384 T^{4} - 455404063 T^{6} + 252221585189 T^{8} + 341434686490976 T^{10} - 19993501235015814 T^{12} + 15120310701929392626 T^{14} + $$83\!\cdots\!16$$ T^{16} + 15120310701929392626 p^{8} T^{18} - 19993501235015814 p^{16} T^{20} + 341434686490976 p^{24} T^{22} + 252221585189 p^{32} T^{24} - 455404063 p^{40} T^{26} + 134384 p^{48} T^{28} + 1453 p^{56} T^{30} + p^{64} T^{32} $
	11	$ 1 + 67873 T^{2} + 2278097996 T^{4} + 4464504840043 p T^{6} + 794148423045616457 T^{8} + $$12\!\cdots\!24$$ T^{10} + $$19\!\cdots\!02$$ p T^{12} + $$38\!\cdots\!58$$ p T^{14} + $$70\!\cdots\!08$$ T^{16} + $$38\!\cdots\!58$$ p^{9} T^{18} + $$19\!\cdots\!02$$ p^{17} T^{20} + $$12\!\cdots\!24$$ p^{24} T^{22} + 794148423045616457 p^{32} T^{24} + 4464504840043 p^{41} T^{26} + 2278097996 p^{48} T^{28} + 67873 p^{56} T^{30} + p^{64} T^{32} $
	13	$ ( 1 + 123 T + 89198 T^{2} + 8779305 T^{3} + 3578222442 T^{4} + 8779305 p^{4} T^{5} + 89198 p^{8} T^{6} + 123 p^{12} T^{7} + p^{16} T^{8} )^{4} $
	17	$ 1 + 347978 T^{2} + 55185328659 T^{4} + 6141560869607686 T^{6} + $$61\!\cdots\!85$$ T^{8} + $$52\!\cdots\!48$$ T^{10} + $$35\!\cdots\!38$$ T^{12} + $$26\!\cdots\!96$$ T^{14} + $$22\!\cdots\!94$$ T^{16} + $$26\!\cdots\!96$$ p^{8} T^{18} + $$35\!\cdots\!38$$ p^{16} T^{20} + $$52\!\cdots\!48$$ p^{24} T^{22} + $$61\!\cdots\!85$$ p^{32} T^{24} + 6141560869607686 p^{40} T^{26} + 55185328659 p^{48} T^{28} + 347978 p^{56} T^{30} + p^{64} T^{32} $
	19	$ ( 1 + 663 T - 139604 T^{2} - 97518909 T^{3} + 59358171097 T^{4} + 16385266408896 T^{5} - 10617872015267030 T^{6} - 315730595407735542 T^{7} + $$19\!\cdots\!12$$ T^{8} - 315730595407735542 p^{4} T^{9} - 10617872015267030 p^{8} T^{10} + 16385266408896 p^{12} T^{11} + 59358171097 p^{16} T^{12} - 97518909 p^{20} T^{13} - 139604 p^{24} T^{14} + 663 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	23	$ 1 + 781946 T^{2} + 261605266347 T^{4} + 36994082198606062 T^{6} - $$58\!\cdots\!03$$ T^{8} - $$52\!\cdots\!96$$ T^{10} - $$14\!\cdots\!74$$ T^{12} - $$20\!\cdots\!40$$ T^{14} - $$19\!\cdots\!02$$ T^{16} - $$20\!\cdots\!40$$ p^{8} T^{18} - $$14\!\cdots\!74$$ p^{16} T^{20} - $$52\!\cdots\!96$$ p^{24} T^{22} - $$58\!\cdots\!03$$ p^{32} T^{24} + 36994082198606062 p^{40} T^{26} + 261605266347 p^{48} T^{28} + 781946 p^{56} T^{30} + p^{64} T^{32} $
	29	$ ( 1 - 5186035 T^{2} + 12065918207126 T^{4} - 16445421521068230957 T^{6} + $$14\!\cdots\!26$$ T^{8} - 16445421521068230957 p^{8} T^{10} + 12065918207126 p^{16} T^{12} - 5186035 p^{24} T^{14} + p^{32} T^{16} )^{2} $
	31	$ ( 1 + 1252 T - 1970800 T^{2} - 2329320376 T^{3} + 3452122792283 T^{4} + 2969598324146804 T^{5} - 114902947284128928 p T^{6} - $$97\!\cdots\!48$$ T^{7} + $$39\!\cdots\!08$$ T^{8} - $$97\!\cdots\!48$$ p^{4} T^{9} - 114902947284128928 p^{9} T^{10} + 2969598324146804 p^{12} T^{11} + 3452122792283 p^{16} T^{12} - 2329320376 p^{20} T^{13} - 1970800 p^{24} T^{14} + 1252 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	37	$ ( 1 - 671 T - 5967634 T^{2} + 2137891607 T^{3} + 22021467662243 T^{4} - 3896617507037944 T^{5} - 57724308257628892140 T^{6} + $$31\!\cdots\!88$$ T^{7} + $$11\!\cdots\!76$$ T^{8} + $$31\!\cdots\!88$$ p^{4} T^{9} - 57724308257628892140 p^{8} T^{10} - 3896617507037944 p^{12} T^{11} + 22021467662243 p^{16} T^{12} + 2137891607 p^{20} T^{13} - 5967634 p^{24} T^{14} - 671 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	41	$ ( 1 - 12671212 T^{2} + 87553197821348 T^{4} - $$40\!\cdots\!00$$ T^{6} + $$13\!\cdots\!22$$ T^{8} - $$40\!\cdots\!00$$ p^{8} T^{10} + 87553197821348 p^{16} T^{12} - 12671212 p^{24} T^{14} + p^{32} T^{16} )^{2} $
	43	$ ( 1 - 365 T + 9824920 T^{2} - 2005752035 T^{3} + 44687816213950 T^{4} - 2005752035 p^{4} T^{5} + 9824920 p^{8} T^{6} - 365 p^{12} T^{7} + p^{16} T^{8} )^{4} $
	47	$ 1 + 23536202 T^{2} + 297904241979867 T^{4} + $$24\!\cdots\!82$$ T^{6} + $$13\!\cdots\!93$$ T^{8} + $$44\!\cdots\!72$$ T^{10} + $$41\!\cdots\!50$$ T^{12} - $$58\!\cdots\!64$$ T^{14} - $$43\!\cdots\!62$$ T^{16} - $$58\!\cdots\!64$$ p^{8} T^{18} + $$41\!\cdots\!50$$ p^{16} T^{20} + $$44\!\cdots\!72$$ p^{24} T^{22} + $$13\!\cdots\!93$$ p^{32} T^{24} + $$24\!\cdots\!82$$ p^{40} T^{26} + 297904241979867 p^{48} T^{28} + 23536202 p^{56} T^{30} + p^{64} T^{32} $
	53	$ 1 + 32781197 T^{2} + 435598902212592 T^{4} + $$47\!\cdots\!81$$ T^{6} + $$67\!\cdots\!81$$ T^{8} + $$73\!\cdots\!12$$ T^{10} + $$59\!\cdots\!22$$ T^{12} + $$55\!\cdots\!10$$ T^{14} + $$52\!\cdots\!32$$ T^{16} + $$55\!\cdots\!10$$ p^{8} T^{18} + $$59\!\cdots\!22$$ p^{16} T^{20} + $$73\!\cdots\!12$$ p^{24} T^{22} + $$67\!\cdots\!81$$ p^{32} T^{24} + $$47\!\cdots\!81$$ p^{40} T^{26} + 435598902212592 p^{48} T^{28} + 32781197 p^{56} T^{30} + p^{64} T^{32} $
	59	$ 1 + 52671473 T^{2} + 1419131393553684 T^{4} + $$23\!\cdots\!33$$ T^{6} + $$26\!\cdots\!65$$ T^{8} + $$17\!\cdots\!64$$ T^{10} + $$21\!\cdots\!18$$ T^{12} - $$11\!\cdots\!90$$ T^{14} - $$18\!\cdots\!20$$ T^{16} - $$11\!\cdots\!90$$ p^{8} T^{18} + $$21\!\cdots\!18$$ p^{16} T^{20} + $$17\!\cdots\!64$$ p^{24} T^{22} + $$26\!\cdots\!65$$ p^{32} T^{24} + $$23\!\cdots\!33$$ p^{40} T^{26} + 1419131393553684 p^{48} T^{28} + 52671473 p^{56} T^{30} + p^{64} T^{32} $
	61	$ ( 1 - 7506 T - 274613 T^{2} + 109556957610 T^{3} - 2274088167959 T^{4} - 794033137854300204 T^{5} - $$26\!\cdots\!02$$ T^{6} - $$35\!\cdots\!00$$ T^{7} + $$10\!\cdots\!10$$ T^{8} - $$35\!\cdots\!00$$ p^{4} T^{9} - $$26\!\cdots\!02$$ p^{8} T^{10} - 794033137854300204 p^{12} T^{11} - 2274088167959 p^{16} T^{12} + 109556957610 p^{20} T^{13} - 274613 p^{24} T^{14} - 7506 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	67	$ ( 1 - 4329 T - 21537980 T^{2} + 342825870483 T^{3} - 482762461607615 T^{4} - 7093070519454907056 T^{5} + $$37\!\cdots\!62$$ T^{6} + $$58\!\cdots\!22$$ T^{7} - $$89\!\cdots\!12$$ T^{8} + $$58\!\cdots\!22$$ p^{4} T^{9} + $$37\!\cdots\!62$$ p^{8} T^{10} - 7093070519454907056 p^{12} T^{11} - 482762461607615 p^{16} T^{12} + 342825870483 p^{20} T^{13} - 21537980 p^{24} T^{14} - 4329 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	71	$ ( 1 - 48207356 T^{2} + 2564055417989044 T^{4} - $$11\!\cdots\!76$$ p T^{6} + $$24\!\cdots\!58$$ T^{8} - $$11\!\cdots\!76$$ p^{9} T^{10} + 2564055417989044 p^{16} T^{12} - 48207356 p^{24} T^{14} + p^{32} T^{16} )^{2} $
	73	$ ( 1 - 6161 T - 54888448 T^{2} + 146364908363 T^{3} + 2922222915537797 T^{4} - 605652958316299264 T^{5} - $$10\!\cdots\!54$$ T^{6} + $$39\!\cdots\!30$$ T^{7} + $$26\!\cdots\!68$$ T^{8} + $$39\!\cdots\!30$$ p^{4} T^{9} - $$10\!\cdots\!54$$ p^{8} T^{10} - 605652958316299264 p^{12} T^{11} + 2922222915537797 p^{16} T^{12} + 146364908363 p^{20} T^{13} - 54888448 p^{24} T^{14} - 6161 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	79	$ ( 1 + 20084 T + 146293020 T^{2} + 474011182552 T^{3} + 2075088233131319 T^{4} + 14218940535286663908 T^{5} + $$91\!\cdots\!36$$ T^{6} + $$17\!\cdots\!64$$ T^{7} + $$17\!\cdots\!48$$ T^{8} + $$17\!\cdots\!64$$ p^{4} T^{9} + $$91\!\cdots\!36$$ p^{8} T^{10} + 14218940535286663908 p^{12} T^{11} + 2075088233131319 p^{16} T^{12} + 474011182552 p^{20} T^{13} + 146293020 p^{24} T^{14} + 20084 p^{28} T^{15} + p^{32} T^{16} )^{2} $
	83	$ ( 1 - 262853795 T^{2} + 32996349735813814 T^{4} - $$26\!\cdots\!85$$ T^{6} + $$14\!\cdots\!94$$ T^{8} - $$26\!\cdots\!85$$ p^{8} T^{10} + 32996349735813814 p^{16} T^{12} - 262853795 p^{24} T^{14} + p^{32} T^{16} )^{2} $
	89	$ 1 + 299506750 T^{2} + 50244486149850515 T^{4} + $$53\!\cdots\!06$$ T^{6} + $$38\!\cdots\!29$$ T^{8} + $$13\!\cdots\!52$$ T^{10} - $$50\!\cdots\!06$$ T^{12} - $$12\!\cdots\!28$$ T^{14} - $$10\!\cdots\!34$$ T^{16} - $$12\!\cdots\!28$$ p^{8} T^{18} - $$50\!\cdots\!06$$ p^{16} T^{20} + $$13\!\cdots\!52$$ p^{24} T^{22} + $$38\!\cdots\!29$$ p^{32} T^{24} + $$53\!\cdots\!06$$ p^{40} T^{26} + 50244486149850515 p^{48} T^{28} + 299506750 p^{56} T^{30} + p^{64} T^{32} $
	97	$ ( 1 + 7067 T + 323397218 T^{2} + 1714983727929 T^{3} + 41953604440291058 T^{4} + 1714983727929 p^{4} T^{5} + 323397218 p^{8} T^{6} + 7067 p^{12} T^{7} + p^{16} T^{8} )^{4} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−5.41154980973924303426802446441, −5.31586157561232279469080273571, −5.12848509328607877220537272153, −4.96794541554377263893993248540, −4.89243400401016631768142934658, −4.44136481673089485663133267414, −4.43814455197664327380148048106, −4.41152577278281526328731283875, −4.06700037882138160869696443665, −4.03224829856238470519313190070, −4.00552268290051356058457023530, −3.73817843433078285553567782705, −3.52397951808616864513965984695, −3.51141626131272618856608456503, −3.32359564134990896529985002943, −2.99545049964791453773188517067, −2.68424995092003697095955259355, −2.52784668260452574388921526710, −2.33484256665381316649402326706, −2.21356303044645060205519549065, −2.14908222858195243301578847850, −1.48177338138975177424949868852, −0.916316394075687453818861136345, −0.24651093793199083311280767667, −0.13488343211258448805478056331, 0.13488343211258448805478056331, 0.24651093793199083311280767667, 0.916316394075687453818861136345, 1.48177338138975177424949868852, 2.14908222858195243301578847850, 2.21356303044645060205519549065, 2.33484256665381316649402326706, 2.52784668260452574388921526710, 2.68424995092003697095955259355, 2.99545049964791453773188517067, 3.32359564134990896529985002943, 3.51141626131272618856608456503, 3.52397951808616864513965984695, 3.73817843433078285553567782705, 4.00552268290051356058457023530, 4.03224829856238470519313190070, 4.06700037882138160869696443665, 4.41152577278281526328731283875, 4.43814455197664327380148048106, 4.44136481673089485663133267414, 4.89243400401016631768142934658, 4.96794541554377263893993248540, 5.12848509328607877220537272153, 5.31586157561232279469080273571, 5.41154980973924303426802446441

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 $\F_{q}$

\(L(\frac{5}{2})\)	\(\approx\)	\(0.06611522494\)
\(L(\frac12)\)	\(\approx\)	\(0.06611522494\)
\(L(3)\)		not available
\(L(1)\)		not available

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