LMFDB - L-function 40-117e20-1.1-c4e20-0-1

Dirichlet series

L(s) = 1

− 24·7^-s + 184·13^-s + 186·16^-s − 1.03e3·19^-s + 2.84e3·31^-s + 452·37^-s + 288·49^-s + 8.40e3·61^-s − 3.62e3·67^-s − 2.17e4·73^-s − 1.03e4·79^-s − 4.41e3·91^-s − 8.98e4·97^-s + 7.40e4·109^-s − 4.46e3·112^-s + 127^-s + 131^-s + 2.47e4·133^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 1.71e4·169^-s + 173^-s + ⋯

L(s) = 1

− 0.489·7^-s + 1.08·13^-s + 0.726·16^-s − 2.85·19^-s + 2.95·31^-s + 0.330·37^-s + 0.119·49^-s + 2.25·61^-s − 0.807·67^-s − 4.07·73^-s − 1.66·79^-s − 0.533·91^-s − 9.54·97^-s + 6.23·109^-s − 0.355·112^-s + 6.20e−5·127^-s + 5.82e−5·131^-s + 1.40·133^-s + 5.32e−5·137^-s + 5.17e−5·139^-s + 4.50e−5·149^-s + 4.38e−5·151^-s + 4.05e−5·157^-s + 3.76e−5·163^-s + 3.58e−5·167^-s + 0.600·169^-s + 3.34e−5·173^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$40$
Conductor:	$3^{40} \cdot 13^{20}$
Sign:	$1$
Analytic conductor:	$4.48336\times 10^{21}$
Root analytic conductor:	$3.47768$
Motivic weight:	$4$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(40,\ 3^{40} \cdot 13^{20} ,\ ( \ : [2]^{20} ),\ 1 )$

Particular Values

$L(\frac{5}{2})$	$\approx$	$12.75393758$
$L(\frac12)$	$\approx$	$12.75393758$
$L(3)$		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 $
	13	$ ( 1 - 92 T + 317 p T^{2} + 8912 p T^{3} + 964070 p^{2} T^{4} - 591688 p^{5} T^{5} + 964070 p^{6} T^{6} + 8912 p^{9} T^{7} + 317 p^{13} T^{8} - 92 p^{16} T^{9} + p^{20} T^{10} )^{2} $
good	2	$ 1 - 93 p T^{4} + 81505 T^{8} - 6267917 p^{2} T^{12} + 544818835 p^{4} T^{16} - 13157598215 p^{6} T^{20} + 544818835 p^{20} T^{24} - 6267917 p^{34} T^{28} + 81505 p^{48} T^{32} - 93 p^{65} T^{36} + p^{80} T^{40} $
	5	$ 1 - 615642 T^{4} + 208961610829 T^{8} - 69095118676607864 T^{12} + $$77\!\cdots\!54$$ p T^{16} - $$23\!\cdots\!24$$ T^{20} + $$77\!\cdots\!54$$ p^{17} T^{24} - 69095118676607864 p^{32} T^{28} + 208961610829 p^{48} T^{32} - 615642 p^{64} T^{36} + p^{80} T^{40} $
	7	$ ( 1 + 12 T + 72 T^{2} - 149276 T^{3} - 5057999 T^{4} + 56969928 T^{5} + 12189477152 T^{6} + 165015912904 p T^{7} + 32976411773902 T^{8} - 2184258717276464 T^{9} - 137738664564139600 T^{10} - 2184258717276464 p^{4} T^{11} + 32976411773902 p^{8} T^{12} + 165015912904 p^{13} T^{13} + 12189477152 p^{16} T^{14} + 56969928 p^{20} T^{15} - 5057999 p^{24} T^{16} - 149276 p^{28} T^{17} + 72 p^{32} T^{18} + 12 p^{36} T^{19} + p^{40} T^{20} )^{2} $
	11	$ 1 - 23851866 p T^{4} + 35033948562758077 T^{8} + $$66\!\cdots\!28$$ T^{12} + $$17\!\cdots\!26$$ T^{16} - $$95\!\cdots\!96$$ T^{20} + $$17\!\cdots\!26$$ p^{16} T^{24} + $$66\!\cdots\!28$$ p^{32} T^{28} + 35033948562758077 p^{48} T^{32} - 23851866 p^{65} T^{36} + p^{80} T^{40} $
	17	$ ( 1 - 226522 T^{2} + 22203479853 T^{4} - 1794867031058232 T^{6} + $$22\!\cdots\!34$$ T^{8} - $$24\!\cdots\!08$$ T^{10} + $$22\!\cdots\!34$$ p^{8} T^{12} - 1794867031058232 p^{16} T^{14} + 22203479853 p^{24} T^{16} - 226522 p^{32} T^{18} + p^{40} T^{20} )^{2} $
	19	$ ( 1 + 516 T + 133128 T^{2} + 60343756 T^{3} + 6209253457 T^{4} - 3606145512552 T^{5} - 866712134606560 T^{6} - 426170978368170200 T^{7} - $$13\!\cdots\!34$$ T^{8} - $$15\!\cdots\!32$$ T^{9} - $$38\!\cdots\!88$$ T^{10} - $$15\!\cdots\!32$$ p^{4} T^{11} - $$13\!\cdots\!34$$ p^{8} T^{12} - 426170978368170200 p^{12} T^{13} - 866712134606560 p^{16} T^{14} - 3606145512552 p^{20} T^{15} + 6209253457 p^{24} T^{16} + 60343756 p^{28} T^{17} + 133128 p^{32} T^{18} + 516 p^{36} T^{19} + p^{40} T^{20} )^{2} $
	23	$ ( 1 - 1764706 T^{2} + 1452318022845 T^{4} - 760738245306207096 T^{6} + $$12\!\cdots\!38$$ p T^{8} - $$90\!\cdots\!76$$ T^{10} + $$12\!\cdots\!38$$ p^{9} T^{12} - 760738245306207096 p^{16} T^{14} + 1452318022845 p^{24} T^{16} - 1764706 p^{32} T^{18} + p^{40} T^{20} )^{2} $
	29	$ ( 1 + 3460710 T^{2} + 6693089241517 T^{4} + 8924811199889379976 T^{6} + $$90\!\cdots\!26$$ T^{8} + $$71\!\cdots\!16$$ T^{10} + $$90\!\cdots\!26$$ p^{8} T^{12} + 8924811199889379976 p^{16} T^{14} + 6693089241517 p^{24} T^{16} + 3460710 p^{32} T^{18} + p^{40} T^{20} )^{2} $
	31	$ ( 1 - 1420 T + 1008200 T^{2} - 1698812324 T^{3} + 2188589515057 T^{4} - 807320205043336 T^{5} + 382840398168210208 T^{6} - $$46\!\cdots\!40$$ T^{7} - $$54\!\cdots\!14$$ T^{8} + $$10\!\cdots\!12$$ T^{9} - $$68\!\cdots\!88$$ T^{10} + $$10\!\cdots\!12$$ p^{4} T^{11} - $$54\!\cdots\!14$$ p^{8} T^{12} - $$46\!\cdots\!40$$ p^{12} T^{13} + 382840398168210208 p^{16} T^{14} - 807320205043336 p^{20} T^{15} + 2188589515057 p^{24} T^{16} - 1698812324 p^{28} T^{17} + 1008200 p^{32} T^{18} - 1420 p^{36} T^{19} + p^{40} T^{20} )^{2} $
	37	$ ( 1 - 226 T + 25538 T^{2} - 2603391938 T^{3} + 2126040092509 T^{4} + 4742577048717704 T^{5} + 2262707566529001976 T^{6} - $$52\!\cdots\!20$$ T^{7} - $$17\!\cdots\!66$$ T^{8} + $$24\!\cdots\!32$$ T^{9} + $$14\!\cdots\!60$$ T^{10} + $$24\!\cdots\!32$$ p^{4} T^{11} - $$17\!\cdots\!66$$ p^{8} T^{12} - $$52\!\cdots\!20$$ p^{12} T^{13} + 2262707566529001976 p^{16} T^{14} + 4742577048717704 p^{20} T^{15} + 2126040092509 p^{24} T^{16} - 2603391938 p^{28} T^{17} + 25538 p^{32} T^{18} - 226 p^{36} T^{19} + p^{40} T^{20} )^{2} $
	41	$ 1 + 2137885117158 T^{4} - $$88\!\cdots\!47$$ T^{8} - $$78\!\cdots\!96$$ T^{12} + $$21\!\cdots\!62$$ T^{16} + $$82\!\cdots\!48$$ T^{20} + $$21\!\cdots\!62$$ p^{16} T^{24} - $$78\!\cdots\!96$$ p^{32} T^{28} - $$88\!\cdots\!47$$ p^{48} T^{32} + 2137885117158 p^{64} T^{36} + p^{80} T^{40} $
	43	$ ( 1 - 23865226 T^{2} + 280677245179149 T^{4} - $$21\!\cdots\!24$$ T^{6} + $$11\!\cdots\!30$$ T^{8} - $$44\!\cdots\!60$$ T^{10} + $$11\!\cdots\!30$$ p^{8} T^{12} - $$21\!\cdots\!24$$ p^{16} T^{14} + 280677245179149 p^{24} T^{16} - 23865226 p^{32} T^{18} + p^{40} T^{20} )^{2} $
	47	$ 1 - 39156456358302 T^{4} + $$15\!\cdots\!69$$ T^{8} - $$62\!\cdots\!04$$ T^{12} + $$39\!\cdots\!78$$ p T^{16} - $$41\!\cdots\!36$$ T^{20} + $$39\!\cdots\!78$$ p^{17} T^{24} - $$62\!\cdots\!04$$ p^{32} T^{28} + $$15\!\cdots\!69$$ p^{48} T^{32} - 39156456358302 p^{64} T^{36} + p^{80} T^{40} $
	53	$ ( 1 + 34865706 T^{2} + 614155494714637 T^{4} + $$76\!\cdots\!60$$ T^{6} + $$76\!\cdots\!14$$ T^{8} + $$64\!\cdots\!32$$ T^{10} + $$76\!\cdots\!14$$ p^{8} T^{12} + $$76\!\cdots\!60$$ p^{16} T^{14} + 614155494714637 p^{24} T^{16} + 34865706 p^{32} T^{18} + p^{40} T^{20} )^{2} $
	59	$ 1 - 428757914085342 T^{4} + $$10\!\cdots\!05$$ T^{8} - $$14\!\cdots\!24$$ T^{12} + $$12\!\cdots\!14$$ T^{16} - $$91\!\cdots\!28$$ T^{20} + $$12\!\cdots\!14$$ p^{16} T^{24} - $$14\!\cdots\!24$$ p^{32} T^{28} + $$10\!\cdots\!05$$ p^{48} T^{32} - 428757914085342 p^{64} T^{36} + p^{80} T^{40} $
	61	$ ( 1 - 2102 T + 35645621 T^{2} - 107984499544 T^{3} + 844980337729706 T^{4} - 1771987045502293444 T^{5} + 844980337729706 p^{4} T^{6} - 107984499544 p^{8} T^{7} + 35645621 p^{12} T^{8} - 2102 p^{16} T^{9} + p^{20} T^{10} )^{4} $
	67	$ ( 1 + 1812 T + 1641672 T^{2} - 140775782852 T^{3} - 148981727955119 T^{4} + 2804169560311225656 T^{5} + $$15\!\cdots\!92$$ T^{6} + $$10\!\cdots\!04$$ T^{7} - $$10\!\cdots\!02$$ T^{8} - $$23\!\cdots\!80$$ T^{9} + $$44\!\cdots\!92$$ T^{10} - $$23\!\cdots\!80$$ p^{4} T^{11} - $$10\!\cdots\!02$$ p^{8} T^{12} + $$10\!\cdots\!04$$ p^{12} T^{13} + $$15\!\cdots\!92$$ p^{16} T^{14} + 2804169560311225656 p^{20} T^{15} - 148981727955119 p^{24} T^{16} - 140775782852 p^{28} T^{17} + 1641672 p^{32} T^{18} + 1812 p^{36} T^{19} + p^{40} T^{20} )^{2} $
	71	$ 1 + 368598935837154 T^{4} + $$61\!\cdots\!33$$ T^{8} + $$38\!\cdots\!24$$ T^{12} + $$29\!\cdots\!82$$ T^{16} + $$29\!\cdots\!16$$ T^{20} + $$29\!\cdots\!82$$ p^{16} T^{24} + $$38\!\cdots\!24$$ p^{32} T^{28} + $$61\!\cdots\!33$$ p^{48} T^{32} + 368598935837154 p^{64} T^{36} + p^{80} T^{40} $
	73	$ ( 1 + 10866 T + 59034978 T^{2} + 386404953010 T^{3} + 1329607557863485 T^{4} - 812923667910635976 T^{5} - $$12\!\cdots\!96$$ T^{6} - $$93\!\cdots\!92$$ T^{7} + $$37\!\cdots\!58$$ T^{8} + $$72\!\cdots\!60$$ T^{9} + $$37\!\cdots\!64$$ T^{10} + $$72\!\cdots\!60$$ p^{4} T^{11} + $$37\!\cdots\!58$$ p^{8} T^{12} - $$93\!\cdots\!92$$ p^{12} T^{13} - $$12\!\cdots\!96$$ p^{16} T^{14} - 812923667910635976 p^{20} T^{15} + 1329607557863485 p^{24} T^{16} + 386404953010 p^{28} T^{17} + 59034978 p^{32} T^{18} + 10866 p^{36} T^{19} + p^{40} T^{20} )^{2} $
	79	$ ( 1 + 2596 T + 63922325 T^{2} - 135507983536 T^{3} + 2076994237683578 T^{4} - 10654532650746409384 T^{5} + 2076994237683578 p^{4} T^{6} - 135507983536 p^{8} T^{7} + 63922325 p^{12} T^{8} + 2596 p^{16} T^{9} + p^{20} T^{10} )^{4} $
	83	$ 1 - 2477306882093406 T^{4} + $$93\!\cdots\!33$$ T^{8} + $$88\!\cdots\!80$$ T^{12} - $$33\!\cdots\!78$$ T^{16} + $$21\!\cdots\!64$$ T^{20} - $$33\!\cdots\!78$$ p^{16} T^{24} + $$88\!\cdots\!80$$ p^{32} T^{28} + $$93\!\cdots\!33$$ p^{48} T^{32} - 2477306882093406 p^{64} T^{36} + p^{80} T^{40} $
	89	$ 1 + 1001300595451878 T^{4} - $$17\!\cdots\!87$$ T^{8} + $$24\!\cdots\!16$$ T^{12} + $$25\!\cdots\!90$$ T^{16} - $$19\!\cdots\!80$$ T^{20} + $$25\!\cdots\!90$$ p^{16} T^{24} + $$24\!\cdots\!16$$ p^{32} T^{28} - $$17\!\cdots\!87$$ p^{48} T^{32} + 1001300595451878 p^{64} T^{36} + p^{80} T^{40} $
	97	$ ( 1 + 44906 T + 1008274418 T^{2} + 16304202269002 T^{3} + 212618341421644093 T^{4} + $$23\!\cdots\!40$$ T^{5} + $$24\!\cdots\!68$$ T^{6} + $$24\!\cdots\!48$$ T^{7} + $$27\!\cdots\!06$$ T^{8} + $$28\!\cdots\!64$$ T^{9} + $$28\!\cdots\!48$$ T^{10} + $$28\!\cdots\!64$$ p^{4} T^{11} + $$27\!\cdots\!06$$ p^{8} T^{12} + $$24\!\cdots\!48$$ p^{12} T^{13} + $$24\!\cdots\!68$$ p^{16} T^{14} + $$23\!\cdots\!40$$ p^{20} T^{15} + 212618341421644093 p^{24} T^{16} + 16304202269002 p^{28} T^{17} + 1008274418 p^{32} T^{18} + 44906 p^{36} T^{19} + p^{40} T^{20} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−2.59471019454277398550332287882, −2.58486676702237049715387399984, −2.50498076232447057243580009584, −2.35216941477433349144338542412, −2.33091188578851148290161644404, −2.30171361070833522710915438942, −2.17432040164799402601090448049, −1.96643646188212369632565211454, −1.82403364562149572155366991928, −1.82053402429593218854323197149, −1.74210972008003627546533159529, −1.47826278660584535140429694681, −1.32907598237620676970932270967, −1.32668848843240324041502087594, −1.31111475513047009915235540778, −1.25262630155003519910624023630, −1.12962816173785792185127218874, −1.07672054339819849017378389201, −0.67878243860015004873584697888, −0.57304248762318112211908810548, −0.50372604908056506411279019687, −0.40650273945451378111267861317, −0.39637579951268511824290086870, −0.28941962285927075724413315529, −0.11602454275521653692031729563, 0.11602454275521653692031729563, 0.28941962285927075724413315529, 0.39637579951268511824290086870, 0.40650273945451378111267861317, 0.50372604908056506411279019687, 0.57304248762318112211908810548, 0.67878243860015004873584697888, 1.07672054339819849017378389201, 1.12962816173785792185127218874, 1.25262630155003519910624023630, 1.31111475513047009915235540778, 1.32668848843240324041502087594, 1.32907598237620676970932270967, 1.47826278660584535140429694681, 1.74210972008003627546533159529, 1.82053402429593218854323197149, 1.82403364562149572155366991928, 1.96643646188212369632565211454, 2.17432040164799402601090448049, 2.30171361070833522710915438942, 2.33091188578851148290161644404, 2.35216941477433349144338542412, 2.50498076232447057243580009584, 2.58486676702237049715387399984, 2.59471019454277398550332287882

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 $\F_{q}$

\(L(\frac{5}{2})\)	\(\approx\)	\(12.75393758\)
\(L(\frac12)\)	\(\approx\)	\(12.75393758\)
\(L(3)\)		not available
\(L(1)\)		not available

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