LMFDB - L-function 20-234e10-1.1-c7e10-0-1

Dirichlet series

L(s) = 1

− 320·4^-s + 3.43e3·13^-s + 6.14e4·16^-s + 6.91e3·17^-s − 9.41e4·23^-s + 2.25e5·25^-s + 1.31e5·29^-s + 2.74e6·43^-s + 2.57e6·49^-s − 1.09e6·52^-s + 3.60e6·53^-s + 1.17e7·61^-s − 9.17e6·64^-s − 2.21e6·68^-s − 2.84e7·79^-s + 3.01e7·92^-s − 7.20e7·100^-s + 6.68e7·101^-s + 3.96e7·103^-s − 3.23e7·107^-s − 1.26e7·113^-s − 4.20e7·116^-s + 1.20e8·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + ⋯

L(s) = 1

− 5/2·4^-s + 0.433·13^-s + 15/4·16^-s + 0.341·17^-s − 1.61·23^-s + 2.88·25^-s + 0.999·29^-s + 5.25·43^-s + 3.12·49^-s − 1.08·52^-s + 3.32·53^-s + 6.61·61^-s − 4.37·64^-s − 0.853·68^-s − 6.49·79^-s + 4.03·92^-s − 7.20·100^-s + 6.45·101^-s + 3.57·103^-s − 2.55·107^-s − 0.823·113^-s − 2.49·116^-s + 6.19·121^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$20$
Conductor:	$2^{10} \cdot 3^{20} \cdot 13^{10}$
Sign:	$1$
Analytic conductor:	$4.35572\times 10^{18}$
Root analytic conductor:	$8.54974$
Motivic weight:	$7$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(20,\ 2^{10} \cdot 3^{20} \cdot 13^{10} ,\ ( \ : [7/2]^{10} ),\ 1 )$

Particular Values

$L(4)$	$\approx$	$20.60625362$
$L(\frac12)$	$\approx$	$20.60625362$
$L(\frac{9}{2})$		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 + p^{6} T^{2} )^{5} $
	3	$ 1 $
	13	$ 1 - 264 p T - 22993699 T^{2} - 4924787536 p T^{3} - 968688324290 p^{3} T^{4} + 6063923562544 p^{6} T^{5} - 968688324290 p^{10} T^{6} - 4924787536 p^{15} T^{7} - 22993699 p^{21} T^{8} - 264 p^{29} T^{9} + p^{35} T^{10} $
good	5	$ 1 - 225231 T^{2} + 32288663936 T^{4} - 152778921987493 p^{2} T^{6} + 602856476164317307 p^{4} T^{8} - $$20\!\cdots\!44$$ p^{6} T^{10} + 602856476164317307 p^{18} T^{12} - 152778921987493 p^{30} T^{14} + 32288663936 p^{42} T^{16} - 225231 p^{56} T^{18} + p^{70} T^{20} $
	7	$ 1 - 2575763 T^{2} + 671623904160 p T^{4} - 5365720625267088049 T^{6} + $$53\!\cdots\!99$$ T^{8} - $$43\!\cdots\!96$$ T^{10} + $$53\!\cdots\!99$$ p^{14} T^{12} - 5365720625267088049 p^{28} T^{14} + 671623904160 p^{43} T^{16} - 2575763 p^{56} T^{18} + p^{70} T^{20} $
	11	$ 1 - 120702854 T^{2} + 6794589484941669 T^{4} - $$24\!\cdots\!20$$ T^{6} + $$61\!\cdots\!74$$ T^{8} - $$13\!\cdots\!40$$ T^{10} + $$61\!\cdots\!74$$ p^{14} T^{12} - $$24\!\cdots\!20$$ p^{28} T^{14} + 6794589484941669 p^{42} T^{16} - 120702854 p^{56} T^{18} + p^{70} T^{20} $
	17	$ ( 1 - 3459 T + 859860888 T^{2} - 7710109622385 T^{3} + 576376045055414203 T^{4} - $$15\!\cdots\!16$$ p T^{5} + 576376045055414203 p^{7} T^{6} - 7710109622385 p^{14} T^{7} + 859860888 p^{21} T^{8} - 3459 p^{28} T^{9} + p^{35} T^{10} )^{2} $
	19	$ 1 - 6136392106 T^{2} + 17676334383504800229 T^{4} - $$32\!\cdots\!00$$ T^{6} + $$41\!\cdots\!86$$ T^{8} - $$41\!\cdots\!20$$ T^{10} + $$41\!\cdots\!86$$ p^{14} T^{12} - $$32\!\cdots\!00$$ p^{28} T^{14} + 17676334383504800229 p^{42} T^{16} - 6136392106 p^{56} T^{18} + p^{70} T^{20} $
	23	$ ( 1 + 47082 T + 6606601023 T^{2} + 217461172664112 T^{3} + 31436770794169134934 T^{4} + $$12\!\cdots\!24$$ T^{5} + 31436770794169134934 p^{7} T^{6} + 217461172664112 p^{14} T^{7} + 6606601023 p^{21} T^{8} + 47082 p^{28} T^{9} + p^{35} T^{10} )^{2} $
	29	$ ( 1 - 65652 T + 26235231693 T^{2} - 1129309776900624 T^{3} + $$74\!\cdots\!82$$ T^{4} - $$44\!\cdots\!88$$ T^{5} + $$74\!\cdots\!82$$ p^{7} T^{6} - 1129309776900624 p^{14} T^{7} + 26235231693 p^{21} T^{8} - 65652 p^{28} T^{9} + p^{35} T^{10} )^{2} $
	31	$ 1 - 185089377182 T^{2} + $$17\!\cdots\!25$$ T^{4} - $$10\!\cdots\!88$$ T^{6} + $$44\!\cdots\!70$$ T^{8} - $$14\!\cdots\!72$$ T^{10} + $$44\!\cdots\!70$$ p^{14} T^{12} - $$10\!\cdots\!88$$ p^{28} T^{14} + $$17\!\cdots\!25$$ p^{42} T^{16} - 185089377182 p^{56} T^{18} + p^{70} T^{20} $
	37	$ 1 - 343402102127 T^{2} + $$55\!\cdots\!88$$ T^{4} - $$57\!\cdots\!09$$ T^{6} + $$45\!\cdots\!87$$ T^{8} - $$35\!\cdots\!48$$ T^{10} + $$45\!\cdots\!87$$ p^{14} T^{12} - $$57\!\cdots\!09$$ p^{28} T^{14} + $$55\!\cdots\!88$$ p^{42} T^{16} - 343402102127 p^{56} T^{18} + p^{70} T^{20} $
	41	$ 1 + 112502588934 T^{2} + $$77\!\cdots\!45$$ T^{4} + $$13\!\cdots\!16$$ T^{6} + $$47\!\cdots\!78$$ T^{8} + $$60\!\cdots\!92$$ T^{10} + $$47\!\cdots\!78$$ p^{14} T^{12} + $$13\!\cdots\!16$$ p^{28} T^{14} + $$77\!\cdots\!45$$ p^{42} T^{16} + 112502588934 p^{56} T^{18} + p^{70} T^{20} $
	43	$ ( 1 - 1370387 T + 1564308442782 T^{2} - 1225930881362215225 T^{3} + $$83\!\cdots\!73$$ T^{4} - $$45\!\cdots\!76$$ T^{5} + $$83\!\cdots\!73$$ p^{7} T^{6} - 1225930881362215225 p^{14} T^{7} + 1564308442782 p^{21} T^{8} - 1370387 p^{28} T^{9} + p^{35} T^{10} )^{2} $
	47	$ 1 - 3093842497907 T^{2} + $$44\!\cdots\!04$$ T^{4} - $$39\!\cdots\!05$$ T^{6} + $$25\!\cdots\!19$$ T^{8} - $$13\!\cdots\!36$$ T^{10} + $$25\!\cdots\!19$$ p^{14} T^{12} - $$39\!\cdots\!05$$ p^{28} T^{14} + $$44\!\cdots\!04$$ p^{42} T^{16} - 3093842497907 p^{56} T^{18} + p^{70} T^{20} $
	53	$ ( 1 - 1802958 T + 5639334494953 T^{2} - 7115810756500449768 T^{3} + $$12\!\cdots\!62$$ T^{4} - $$11\!\cdots\!00$$ T^{5} + $$12\!\cdots\!62$$ p^{7} T^{6} - 7115810756500449768 p^{14} T^{7} + 5639334494953 p^{21} T^{8} - 1802958 p^{28} T^{9} + p^{35} T^{10} )^{2} $
	59	$ 1 - 13883635821978 T^{2} + $$10\!\cdots\!93$$ T^{4} - $$49\!\cdots\!40$$ T^{6} + $$17\!\cdots\!42$$ T^{8} - $$50\!\cdots\!76$$ T^{10} + $$17\!\cdots\!42$$ p^{14} T^{12} - $$49\!\cdots\!40$$ p^{28} T^{14} + $$10\!\cdots\!93$$ p^{42} T^{16} - 13883635821978 p^{56} T^{18} + p^{70} T^{20} $
	61	$ ( 1 - 5863828 T + 26869621220877 T^{2} - 80701347370664552240 T^{3} + $$20\!\cdots\!46$$ T^{4} - $$39\!\cdots\!52$$ T^{5} + $$20\!\cdots\!46$$ p^{7} T^{6} - 80701347370664552240 p^{14} T^{7} + 26869621220877 p^{21} T^{8} - 5863828 p^{28} T^{9} + p^{35} T^{10} )^{2} $
	67	$ 1 - 27541658882918 T^{2} + $$40\!\cdots\!49$$ T^{4} - $$42\!\cdots\!60$$ T^{6} + $$34\!\cdots\!94$$ T^{8} - $$23\!\cdots\!64$$ T^{10} + $$34\!\cdots\!94$$ p^{14} T^{12} - $$42\!\cdots\!60$$ p^{28} T^{14} + $$40\!\cdots\!49$$ p^{42} T^{16} - 27541658882918 p^{56} T^{18} + p^{70} T^{20} $
	71	$ 1 - 33623256709835 T^{2} + $$58\!\cdots\!20$$ T^{4} - $$61\!\cdots\!45$$ T^{6} + $$46\!\cdots\!35$$ T^{8} - $$33\!\cdots\!52$$ T^{10} + $$46\!\cdots\!35$$ p^{14} T^{12} - $$61\!\cdots\!45$$ p^{28} T^{14} + $$58\!\cdots\!20$$ p^{42} T^{16} - 33623256709835 p^{56} T^{18} + p^{70} T^{20} $
	73	$ 1 - 95909133451406 T^{2} + $$42\!\cdots\!41$$ T^{4} - $$11\!\cdots\!32$$ T^{6} + $$21\!\cdots\!78$$ T^{8} - $$27\!\cdots\!84$$ T^{10} + $$21\!\cdots\!78$$ p^{14} T^{12} - $$11\!\cdots\!32$$ p^{28} T^{14} + $$42\!\cdots\!41$$ p^{42} T^{16} - 95909133451406 p^{56} T^{18} + p^{70} T^{20} $
	79	$ ( 1 + 14224614 T + 123964613166215 T^{2} + $$65\!\cdots\!56$$ T^{3} + $$28\!\cdots\!14$$ T^{4} + $$10\!\cdots\!80$$ T^{5} + $$28\!\cdots\!14$$ p^{7} T^{6} + $$65\!\cdots\!56$$ p^{14} T^{7} + 123964613166215 p^{21} T^{8} + 14224614 p^{28} T^{9} + p^{35} T^{10} )^{2} $
	83	$ 1 - 212120486293674 T^{2} + $$21\!\cdots\!45$$ T^{4} - $$13\!\cdots\!96$$ T^{6} + $$57\!\cdots\!74$$ T^{8} - $$18\!\cdots\!20$$ T^{10} + $$57\!\cdots\!74$$ p^{14} T^{12} - $$13\!\cdots\!96$$ p^{28} T^{14} + $$21\!\cdots\!45$$ p^{42} T^{16} - 212120486293674 p^{56} T^{18} + p^{70} T^{20} $
	89	$ 1 - 298599996412142 T^{2} + $$41\!\cdots\!89$$ T^{4} - $$36\!\cdots\!56$$ T^{6} + $$22\!\cdots\!06$$ T^{8} - $$11\!\cdots\!56$$ T^{10} + $$22\!\cdots\!06$$ p^{14} T^{12} - $$36\!\cdots\!56$$ p^{28} T^{14} + $$41\!\cdots\!89$$ p^{42} T^{16} - 298599996412142 p^{56} T^{18} + p^{70} T^{20} $
	97	$ 1 - 508032728860570 T^{2} + $$12\!\cdots\!41$$ T^{4} - $$21\!\cdots\!20$$ T^{6} + $$26\!\cdots\!86$$ T^{8} - $$24\!\cdots\!40$$ T^{10} + $$26\!\cdots\!86$$ p^{14} T^{12} - $$21\!\cdots\!20$$ p^{28} T^{14} + $$12\!\cdots\!41$$ p^{42} T^{16} - 508032728860570 p^{56} T^{18} + p^{70} 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

−3.29414867743035764599868339449, −3.23801599308529551063895885985, −3.19533100369994420008028669067, −3.08587900193629026482591796945, −3.02481915747946150820317717286, −2.79334269628202235869531982837, −2.50845744111743333968230146962, −2.49456389513205205612119693509, −2.29977360204920683462241585336, −2.15992412938200714332180062184, −2.07963697575632900463823860263, −2.07171134505883015576027035832, −1.89632325012941739919426921886, −1.67751990599398960519057929798, −1.40528054018674921120832457116, −1.15276632558209145796599753865, −0.909868413131828452358452574068, −0.901690103486054421450845751483, −0.871277228110544982994515668925, −0.793751423629071173132837265747, −0.77485044444179829507354481768, −0.60747449243357315230173779986, −0.57037342486153059289766469848, −0.27702381932856359080162017263, −0.16486403679890947566617280619, 0.16486403679890947566617280619, 0.27702381932856359080162017263, 0.57037342486153059289766469848, 0.60747449243357315230173779986, 0.77485044444179829507354481768, 0.793751423629071173132837265747, 0.871277228110544982994515668925, 0.901690103486054421450845751483, 0.909868413131828452358452574068, 1.15276632558209145796599753865, 1.40528054018674921120832457116, 1.67751990599398960519057929798, 1.89632325012941739919426921886, 2.07171134505883015576027035832, 2.07963697575632900463823860263, 2.15992412938200714332180062184, 2.29977360204920683462241585336, 2.49456389513205205612119693509, 2.50845744111743333968230146962, 2.79334269628202235869531982837, 3.02481915747946150820317717286, 3.08587900193629026482591796945, 3.19533100369994420008028669067, 3.23801599308529551063895885985, 3.29414867743035764599868339449

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 $\F_{q}$
Belyi maps

\(L(4)\)	\(\approx\)	\(20.60625362\)
\(L(\frac12)\)	\(\approx\)	\(20.60625362\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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