LMFDB - L-function 20-38e10-1.1-c6e10-0-0

Dirichlet series

L(s) = 1

− 160·4^-s − 112·5^-s − 224·7^-s + 2.20e3·9^-s + 3.64e3·11^-s + 1.53e4·16^-s − 1.04e4·17^-s − 1.72e4·19^-s + 1.79e4·20^-s + 3.77e4·23^-s − 9.78e4·25^-s + 3.58e4·28^-s + 2.50e4·35^-s − 3.52e5·36^-s + 6.30e3·43^-s − 5.83e5·44^-s − 2.46e5·45^-s + 3.22e5·47^-s − 6.81e5·49^-s − 4.08e5·55^-s + 4.26e5·61^-s − 4.92e5·63^-s − 1.14e6·64^-s + 1.66e6·68^-s − 7.86e5·73^-s + 2.75e6·76^-s − 8.16e5·77^-s + ⋯

L(s) = 1

− 5/2·4^-s − 0.895·5^-s − 0.653·7^-s + 3.01·9^-s + 2.73·11^-s + 15/4·16^-s − 2.12·17^-s − 2.51·19^-s + 2.23·20^-s + 3.09·23^-s − 6.26·25^-s + 1.63·28^-s + 0.585·35^-s − 7.54·36^-s + 0.0793·43^-s − 6.84·44^-s − 2.70·45^-s + 3.10·47^-s − 5.78·49^-s − 2.45·55^-s + 1.87·61^-s − 1.97·63^-s − 4.37·64^-s + 5.30·68^-s − 2.02·73^-s + 6.28·76^-s − 1.78·77^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s+3)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$20$
Conductor:	$2^{10} \cdot 19^{10}$
Sign:	$1$
Analytic conductor:	$2.60696\times 10^{9}$
Root analytic conductor:	$2.95669$
Motivic weight:	$6$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(20,\ 2^{10} \cdot 19^{10} ,\ ( \ : [3]^{10} ),\ 1 )$

Particular Values

$L(\frac{7}{2})$	$\approx$	$0.06156130576$
$L(\frac12)$	$\approx$	$0.06156130576$
$L(4)$		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} )^{5} $
	19	$ 1 + 17230 T + 7817763 p T^{2} + 2124183392 p^{2} T^{3} + 27627228446 p^{4} T^{4} + 20756638956 p^{7} T^{5} + 27627228446 p^{10} T^{6} + 2124183392 p^{14} T^{7} + 7817763 p^{19} T^{8} + 17230 p^{24} T^{9} + p^{30} T^{10} $
good	3	$ 1 - 2200 T^{2} + 181270 p^{2} T^{4} - 57241582 p^{2} T^{6} + 2413949171 p^{5} T^{8} - 9031949427316 p^{4} T^{10} + 2413949171 p^{17} T^{12} - 57241582 p^{26} T^{14} + 181270 p^{38} T^{16} - 2200 p^{48} T^{18} + p^{60} T^{20} $
	5	$ ( 1 + 56 T + 2146 p^{2} T^{2} + 89502 p^{2} T^{3} + 2174881 p^{4} T^{4} + 13857148 p^{5} T^{5} + 2174881 p^{10} T^{6} + 89502 p^{14} T^{7} + 2146 p^{20} T^{8} + 56 p^{24} T^{9} + p^{30} T^{10} )^{2} $
	7	$ ( 1 + 16 p T + 359337 T^{2} + 34246958 T^{3} + 1268834993 p^{2} T^{4} + 5080318669710 T^{5} + 1268834993 p^{8} T^{6} + 34246958 p^{12} T^{7} + 359337 p^{18} T^{8} + 16 p^{25} T^{9} + p^{30} T^{10} )^{2} $
	11	$ ( 1 - 1822 T + 3854200 T^{2} - 5223454506 T^{3} + 11147752733431 T^{4} - 14402631038211088 T^{5} + 11147752733431 p^{6} T^{6} - 5223454506 p^{12} T^{7} + 3854200 p^{18} T^{8} - 1822 p^{24} T^{9} + p^{30} T^{10} )^{2} $
	13	$ 1 - 17666128 T^{2} + 11574881920638 p T^{4} - $$75\!\cdots\!02$$ T^{6} + $$23\!\cdots\!57$$ T^{8} - $$72\!\cdots\!08$$ T^{10} + $$23\!\cdots\!57$$ p^{12} T^{12} - $$75\!\cdots\!02$$ p^{24} T^{14} + 11574881920638 p^{37} T^{16} - 17666128 p^{48} T^{18} + p^{60} T^{20} $
	17	$ ( 1 + 5210 T + 51971035 T^{2} + 61430519544 T^{3} + 668715425929933 T^{4} - 1842471159207853078 T^{5} + 668715425929933 p^{6} T^{6} + 61430519544 p^{12} T^{7} + 51971035 p^{18} T^{8} + 5210 p^{24} T^{9} + p^{30} T^{10} )^{2} $
	23	$ ( 1 - 18856 T + 571101934 T^{2} - 9065914883226 T^{3} + 164009711653394737 T^{4} - $$18\!\cdots\!76$$ T^{5} + 164009711653394737 p^{6} T^{6} - 9065914883226 p^{12} T^{7} + 571101934 p^{18} T^{8} - 18856 p^{24} T^{9} + p^{30} T^{10} )^{2} $
	29	$ 1 - 2295730648 T^{2} + 3022757912409924870 T^{4} - $$26\!\cdots\!30$$ T^{6} + $$18\!\cdots\!13$$ T^{8} - $$11\!\cdots\!52$$ T^{10} + $$18\!\cdots\!13$$ p^{12} T^{12} - $$26\!\cdots\!30$$ p^{24} T^{14} + 3022757912409924870 p^{36} T^{16} - 2295730648 p^{48} T^{18} + p^{60} T^{20} $
	31	$ 1 - 2408322202 T^{2} + 3352416717263589885 T^{4} - $$38\!\cdots\!88$$ T^{6} + $$45\!\cdots\!50$$ T^{8} - $$46\!\cdots\!52$$ T^{10} + $$45\!\cdots\!50$$ p^{12} T^{12} - $$38\!\cdots\!88$$ p^{24} T^{14} + 3352416717263589885 p^{36} T^{16} - 2408322202 p^{48} T^{18} + p^{60} T^{20} $
	37	$ 1 + 2697991190 T^{2} + 11569645850506827021 T^{4} - $$95\!\cdots\!16$$ T^{6} - $$55\!\cdots\!38$$ T^{8} - $$35\!\cdots\!32$$ T^{10} - $$55\!\cdots\!38$$ p^{12} T^{12} - $$95\!\cdots\!16$$ p^{24} T^{14} + 11569645850506827021 p^{36} T^{16} + 2697991190 p^{48} T^{18} + p^{60} T^{20} $
	41	$ 1 - 13733644282 T^{2} + 81015381804592963677 T^{4} - $$19\!\cdots\!72$$ T^{6} - $$42\!\cdots\!54$$ T^{8} + $$45\!\cdots\!60$$ T^{10} - $$42\!\cdots\!54$$ p^{12} T^{12} - $$19\!\cdots\!72$$ p^{24} T^{14} + 81015381804592963677 p^{36} T^{16} - 13733644282 p^{48} T^{18} + p^{60} T^{20} $
	43	$ ( 1 - 3154 T + 21492915636 T^{2} - 397463731677654 T^{3} + $$21\!\cdots\!67$$ T^{4} - $$45\!\cdots\!84$$ T^{5} + $$21\!\cdots\!67$$ p^{6} T^{6} - 397463731677654 p^{12} T^{7} + 21492915636 p^{18} T^{8} - 3154 p^{24} T^{9} + p^{30} T^{10} )^{2} $
	47	$ ( 1 - 161110 T + 48913956136 T^{2} - 5932854681494826 T^{3} + $$98\!\cdots\!11$$ T^{4} - $$90\!\cdots\!28$$ T^{5} + $$98\!\cdots\!11$$ p^{6} T^{6} - 5932854681494826 p^{12} T^{7} + 48913956136 p^{18} T^{8} - 161110 p^{24} T^{9} + p^{30} T^{10} )^{2} $
	53	$ 1 - 128223896920 T^{2} + $$80\!\cdots\!82$$ T^{4} - $$33\!\cdots\!10$$ T^{6} + $$10\!\cdots\!21$$ T^{8} - $$26\!\cdots\!80$$ T^{10} + $$10\!\cdots\!21$$ p^{12} T^{12} - $$33\!\cdots\!10$$ p^{24} T^{14} + $$80\!\cdots\!82$$ p^{36} T^{16} - 128223896920 p^{48} T^{18} + p^{60} T^{20} $
	59	$ 1 - 227814608464 T^{2} + $$27\!\cdots\!38$$ T^{4} - $$22\!\cdots\!38$$ T^{6} + $$13\!\cdots\!09$$ T^{8} - $$65\!\cdots\!72$$ T^{10} + $$13\!\cdots\!09$$ p^{12} T^{12} - $$22\!\cdots\!38$$ p^{24} T^{14} + $$27\!\cdots\!38$$ p^{36} T^{16} - 227814608464 p^{48} T^{18} + p^{60} T^{20} $
	61	$ ( 1 - 213152 T + 149383123554 T^{2} - 43098614383216666 T^{3} + $$11\!\cdots\!33$$ T^{4} - $$31\!\cdots\!88$$ T^{5} + $$11\!\cdots\!33$$ p^{6} T^{6} - 43098614383216666 p^{12} T^{7} + 149383123554 p^{18} T^{8} - 213152 p^{24} T^{9} + p^{30} T^{10} )^{2} $
	67	$ 1 - 396287882632 T^{2} + $$73\!\cdots\!06$$ T^{4} - $$82\!\cdots\!98$$ T^{6} + $$68\!\cdots\!13$$ T^{8} - $$55\!\cdots\!12$$ T^{10} + $$68\!\cdots\!13$$ p^{12} T^{12} - $$82\!\cdots\!98$$ p^{24} T^{14} + $$73\!\cdots\!06$$ p^{36} T^{16} - 396287882632 p^{48} T^{18} + p^{60} T^{20} $
	71	$ 1 - 706078923010 T^{2} + $$26\!\cdots\!37$$ T^{4} - $$68\!\cdots\!92$$ T^{6} + $$12\!\cdots\!86$$ T^{8} - $$18\!\cdots\!64$$ T^{10} + $$12\!\cdots\!86$$ p^{12} T^{12} - $$68\!\cdots\!92$$ p^{24} T^{14} + $$26\!\cdots\!37$$ p^{36} T^{16} - 706078923010 p^{48} T^{18} + p^{60} T^{20} $
	73	$ ( 1 + 393038 T + 262845231891 T^{2} + 80521825338052188 T^{3} + $$48\!\cdots\!17$$ T^{4} + $$19\!\cdots\!18$$ T^{5} + $$48\!\cdots\!17$$ p^{6} T^{6} + 80521825338052188 p^{12} T^{7} + 262845231891 p^{18} T^{8} + 393038 p^{24} T^{9} + p^{30} T^{10} )^{2} $
	79	$ 1 - 1563370816666 T^{2} + $$12\!\cdots\!89$$ T^{4} - $$62\!\cdots\!48$$ T^{6} + $$23\!\cdots\!98$$ T^{8} - $$64\!\cdots\!48$$ T^{10} + $$23\!\cdots\!98$$ p^{12} T^{12} - $$62\!\cdots\!48$$ p^{24} T^{14} + $$12\!\cdots\!89$$ p^{36} T^{16} - 1563370816666 p^{48} T^{18} + p^{60} T^{20} $
	83	$ ( 1 + 50750 T + 830672751385 T^{2} + 45811815263701056 T^{3} + $$33\!\cdots\!98$$ T^{4} + $$24\!\cdots\!68$$ T^{5} + $$33\!\cdots\!98$$ p^{6} T^{6} + 45811815263701056 p^{12} T^{7} + 830672751385 p^{18} T^{8} + 50750 p^{24} T^{9} + p^{30} T^{10} )^{2} $
	89	$ 1 - 3396603770530 T^{2} + $$57\!\cdots\!37$$ T^{4} - $$61\!\cdots\!32$$ T^{6} + $$47\!\cdots\!86$$ T^{8} - $$27\!\cdots\!84$$ T^{10} + $$47\!\cdots\!86$$ p^{12} T^{12} - $$61\!\cdots\!32$$ p^{24} T^{14} + $$57\!\cdots\!37$$ p^{36} T^{16} - 3396603770530 p^{48} T^{18} + p^{60} T^{20} $
	97	$ 1 - 5659092169474 T^{2} + $$15\!\cdots\!21$$ T^{4} - $$28\!\cdots\!80$$ T^{6} + $$37\!\cdots\!34$$ T^{8} - $$35\!\cdots\!52$$ T^{10} + $$37\!\cdots\!34$$ p^{12} T^{12} - $$28\!\cdots\!80$$ p^{24} T^{14} + $$15\!\cdots\!21$$ p^{36} T^{16} - 5659092169474 p^{48} T^{18} + p^{60} T^{20} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−5.51385236615802927914568541628, −5.12486526502529185024049109391, −5.11314620554732559185490469663, −4.91331465708428092928447319842, −4.52800248332534957922935798640, −4.34254203975437924017092216848, −4.31739039182338991389965632526, −4.06428650327858334560016813089, −4.02479848563902741383354675949, −3.98386972499032034080511277307, −3.96878846235512310681459364185, −3.80495065365644384940558674838, −3.34342810804552054452169217127, −3.18887388143159736315617136492, −2.86686835526982583987332608739, −2.37719049816063630566633473771, −2.20031191778455532407894335979, −1.90564153685479980898852652225, −1.63320147756624966373125694287, −1.42248295170733489409069884644, −1.25344850822753627618803157842, −1.16626919971685974766930555358, −0.41397143803646312912193710737, −0.38400517740631515784228467933, −0.04982917952764402625666144158, 0.04982917952764402625666144158, 0.38400517740631515784228467933, 0.41397143803646312912193710737, 1.16626919971685974766930555358, 1.25344850822753627618803157842, 1.42248295170733489409069884644, 1.63320147756624966373125694287, 1.90564153685479980898852652225, 2.20031191778455532407894335979, 2.37719049816063630566633473771, 2.86686835526982583987332608739, 3.18887388143159736315617136492, 3.34342810804552054452169217127, 3.80495065365644384940558674838, 3.96878846235512310681459364185, 3.98386972499032034080511277307, 4.02479848563902741383354675949, 4.06428650327858334560016813089, 4.31739039182338991389965632526, 4.34254203975437924017092216848, 4.52800248332534957922935798640, 4.91331465708428092928447319842, 5.11314620554732559185490469663, 5.12486526502529185024049109391, 5.51385236615802927914568541628

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 $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(0.06156130576\)
\(L(\frac12)\)	\(\approx\)	\(0.06156130576\)
\(L(4)\)		not available
\(L(1)\)		not available

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