LMFDB - L-function 40-48e20-1.1-c15e20-0-0

Dirichlet series

L(s) = 1

+ 1.13e6·9^-s + 3.18e8·13^-s + 2.76e11·25^-s − 1.81e12·37^-s + 4.33e13·49^-s + 4.61e13·61^-s + 8.32e13·73^-s + 1.90e13·81^-s − 1.21e15·97^-s + 2.92e13·109^-s + 3.62e14·117^-s − 4.11e16·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s − 4.83e17·169^-s + 173^-s + 179^-s + 181^-s + 191^-s + 193^-s + ⋯

L(s) = 1

+ 0.0791·9^-s + 1.40·13^-s + 9.06·25^-s − 3.13·37^-s + 9.12·49^-s + 1.87·61^-s + 0.881·73^-s + 0.0923·81^-s − 1.53·97^-s + 0.0153·109^-s + 0.111·117^-s − 9.84·121^-s − 9.43·169^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(16-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s+15/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$40$
Conductor:	$2^{80} \cdot 3^{20}$
Sign:	$1$
Analytic conductor:	$5.16308\times 10^{36}$
Root analytic conductor:	$8.27604$
Motivic weight:	$15$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(40,\ 2^{80} \cdot 3^{20} ,\ ( \ : [15/2]^{20} ),\ 1 )$

Particular Values

$L(8)$	$\approx$	$93.80119345$
$L(\frac12)$	$\approx$	$93.80119345$
$L(\frac{17}{2})$		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 $
	3	$ 1 - 378662 p T^{2} - 24323249011 p^{6} T^{4} - 14357614209201224 p^{11} T^{6} - 105434758079621632 p^{22} T^{8} + 9127615851432233984 p^{35} T^{10} - 105434758079621632 p^{52} T^{12} - 14357614209201224 p^{71} T^{14} - 24323249011 p^{96} T^{16} - 378662 p^{121} T^{18} + p^{150} T^{20} $
good	5	$ ( 1 - 138254313122 T^{2} + $$43\!\cdots\!97$$ p^{2} T^{4} - $$18\!\cdots\!56$$ p^{5} T^{6} + $$16\!\cdots\!78$$ p^{6} T^{8} - $$21\!\cdots\!08$$ p^{8} T^{10} + $$16\!\cdots\!78$$ p^{36} T^{12} - $$18\!\cdots\!56$$ p^{65} T^{14} + $$43\!\cdots\!97$$ p^{92} T^{16} - 138254313122 p^{120} T^{18} + p^{150} T^{20} )^{2} $
	7	$ ( 1 - 21655412425354 T^{2} + $$56\!\cdots\!45$$ p^{2} T^{4} - $$10\!\cdots\!20$$ p^{4} T^{6} + $$14\!\cdots\!42$$ p^{6} T^{8} - $$15\!\cdots\!76$$ p^{8} T^{10} + $$14\!\cdots\!42$$ p^{36} T^{12} - $$10\!\cdots\!20$$ p^{64} T^{14} + $$56\!\cdots\!45$$ p^{92} T^{16} - 21655412425354 p^{120} T^{18} + p^{150} T^{20} )^{2} $
	11	$ ( 1 + 20572367543138366 T^{2} + $$16\!\cdots\!95$$ p T^{4} + $$71\!\cdots\!40$$ p^{3} T^{6} + $$18\!\cdots\!54$$ p^{5} T^{8} + $$47\!\cdots\!28$$ p^{7} T^{10} + $$18\!\cdots\!54$$ p^{35} T^{12} + $$71\!\cdots\!40$$ p^{63} T^{14} + $$16\!\cdots\!95$$ p^{91} T^{16} + 20572367543138366 p^{120} T^{18} + p^{150} T^{20} )^{2} $
	13	$ ( 1 - 6130214 p T + 10510946863178181 p T^{2} - $$10\!\cdots\!88$$ p^{2} T^{3} + $$37\!\cdots\!34$$ p^{4} T^{4} - $$44\!\cdots\!24$$ p^{4} T^{5} + $$37\!\cdots\!34$$ p^{19} T^{6} - $$10\!\cdots\!88$$ p^{32} T^{7} + 10510946863178181 p^{46} T^{8} - 6130214 p^{61} T^{9} + p^{75} T^{10} )^{4} $
	17	$ ( 1 - 4270296280545531242 T^{2} - $$15\!\cdots\!19$$ T^{4} - $$55\!\cdots\!68$$ T^{6} + $$70\!\cdots\!06$$ p T^{8} - $$33\!\cdots\!44$$ T^{10} + $$70\!\cdots\!06$$ p^{31} T^{12} - $$55\!\cdots\!68$$ p^{60} T^{14} - $$15\!\cdots\!19$$ p^{90} T^{16} - 4270296280545531242 p^{120} T^{18} + p^{150} T^{20} )^{2} $
	19	$ ( 1 - 52703923602686533330 T^{2} + $$11\!\cdots\!65$$ T^{4} - $$19\!\cdots\!80$$ T^{6} + $$42\!\cdots\!70$$ T^{8} - $$81\!\cdots\!76$$ T^{10} + $$42\!\cdots\!70$$ p^{30} T^{12} - $$19\!\cdots\!80$$ p^{60} T^{14} + $$11\!\cdots\!65$$ p^{90} T^{16} - 52703923602686533330 p^{120} T^{18} + p^{150} T^{20} )^{2} $
	23	$ ( 1 + $$13\!\cdots\!90$$ T^{2} + $$10\!\cdots\!29$$ T^{4} + $$57\!\cdots\!20$$ T^{6} + $$99\!\cdots\!46$$ p T^{8} + $$69\!\cdots\!80$$ T^{10} + $$99\!\cdots\!46$$ p^{31} T^{12} + $$57\!\cdots\!20$$ p^{60} T^{14} + $$10\!\cdots\!29$$ p^{90} T^{16} + $$13\!\cdots\!90$$ p^{120} T^{18} + p^{150} T^{20} )^{2} $
	29	$ ( 1 - $$43\!\cdots\!30$$ T^{2} + $$97\!\cdots\!41$$ T^{4} - $$15\!\cdots\!20$$ T^{6} + $$17\!\cdots\!18$$ T^{8} - $$17\!\cdots\!80$$ T^{10} + $$17\!\cdots\!18$$ p^{30} T^{12} - $$15\!\cdots\!20$$ p^{60} T^{14} + $$97\!\cdots\!41$$ p^{90} T^{16} - $$43\!\cdots\!30$$ p^{120} T^{18} + p^{150} T^{20} )^{2} $
	31	$ ( 1 - $$43\!\cdots\!34$$ T^{2} + $$26\!\cdots\!05$$ T^{4} - $$81\!\cdots\!36$$ T^{6} + $$28\!\cdots\!10$$ T^{8} - $$63\!\cdots\!04$$ T^{10} + $$28\!\cdots\!10$$ p^{30} T^{12} - $$81\!\cdots\!36$$ p^{60} T^{14} + $$26\!\cdots\!05$$ p^{90} T^{16} - $$43\!\cdots\!34$$ p^{120} T^{18} + p^{150} T^{20} )^{2} $
	37	$ ( 1 + 452780009834 T + $$10\!\cdots\!77$$ T^{2} + $$21\!\cdots\!44$$ T^{3} + $$48\!\cdots\!70$$ T^{4} + $$56\!\cdots\!92$$ T^{5} + $$48\!\cdots\!70$$ p^{15} T^{6} + $$21\!\cdots\!44$$ p^{30} T^{7} + $$10\!\cdots\!77$$ p^{45} T^{8} + 452780009834 p^{60} T^{9} + p^{75} T^{10} )^{4} $
	41	$ ( 1 - $$84\!\cdots\!70$$ T^{2} + $$36\!\cdots\!41$$ T^{4} - $$10\!\cdots\!80$$ T^{6} + $$23\!\cdots\!18$$ T^{8} - $$41\!\cdots\!20$$ T^{10} + $$23\!\cdots\!18$$ p^{30} T^{12} - $$10\!\cdots\!80$$ p^{60} T^{14} + $$36\!\cdots\!41$$ p^{90} T^{16} - $$84\!\cdots\!70$$ p^{120} T^{18} + p^{150} T^{20} )^{2} $
	43	$ ( 1 - $$26\!\cdots\!42$$ T^{2} + $$33\!\cdots\!69$$ T^{4} - $$25\!\cdots\!92$$ T^{6} + $$13\!\cdots\!74$$ T^{8} - $$49\!\cdots\!16$$ T^{10} + $$13\!\cdots\!74$$ p^{30} T^{12} - $$25\!\cdots\!92$$ p^{60} T^{14} + $$33\!\cdots\!69$$ p^{90} T^{16} - $$26\!\cdots\!42$$ p^{120} T^{18} + p^{150} T^{20} )^{2} $
	47	$ ( 1 + $$44\!\cdots\!50$$ T^{2} + $$13\!\cdots\!09$$ T^{4} + $$28\!\cdots\!64$$ T^{6} + $$48\!\cdots\!26$$ T^{8} + $$64\!\cdots\!76$$ T^{10} + $$48\!\cdots\!26$$ p^{30} T^{12} + $$28\!\cdots\!64$$ p^{60} T^{14} + $$13\!\cdots\!09$$ p^{90} T^{16} + $$44\!\cdots\!50$$ p^{120} T^{18} + p^{150} T^{20} )^{2} $
	53	$ ( 1 - $$38\!\cdots\!30$$ T^{2} + $$69\!\cdots\!81$$ T^{4} - $$78\!\cdots\!36$$ T^{6} + $$66\!\cdots\!74$$ T^{8} - $$49\!\cdots\!28$$ T^{10} + $$66\!\cdots\!74$$ p^{30} T^{12} - $$78\!\cdots\!36$$ p^{60} T^{14} + $$69\!\cdots\!81$$ p^{90} T^{16} - $$38\!\cdots\!30$$ p^{120} T^{18} + p^{150} T^{20} )^{2} $
	59	$ ( 1 + $$14\!\cdots\!66$$ T^{2} + $$90\!\cdots\!69$$ T^{4} + $$39\!\cdots\!56$$ T^{6} + $$19\!\cdots\!82$$ T^{8} + $$87\!\cdots\!52$$ T^{10} + $$19\!\cdots\!82$$ p^{30} T^{12} + $$39\!\cdots\!56$$ p^{60} T^{14} + $$90\!\cdots\!69$$ p^{90} T^{16} + $$14\!\cdots\!66$$ p^{120} T^{18} + p^{150} T^{20} )^{2} $
	61	$ ( 1 - 11530211321758 T + $$21\!\cdots\!73$$ T^{2} - $$18\!\cdots\!12$$ T^{3} + $$21\!\cdots\!42$$ T^{4} - $$13\!\cdots\!28$$ T^{5} + $$21\!\cdots\!42$$ p^{15} T^{6} - $$18\!\cdots\!12$$ p^{30} T^{7} + $$21\!\cdots\!73$$ p^{45} T^{8} - 11530211321758 p^{60} T^{9} + p^{75} T^{10} )^{4} $
	67	$ ( 1 - $$14\!\cdots\!26$$ T^{2} + $$90\!\cdots\!37$$ T^{4} - $$35\!\cdots\!40$$ T^{6} + $$10\!\cdots\!42$$ T^{8} - $$27\!\cdots\!92$$ T^{10} + $$10\!\cdots\!42$$ p^{30} T^{12} - $$35\!\cdots\!40$$ p^{60} T^{14} + $$90\!\cdots\!37$$ p^{90} T^{16} - $$14\!\cdots\!26$$ p^{120} T^{18} + p^{150} T^{20} )^{2} $
	71	$ ( 1 + $$13\!\cdots\!30$$ T^{2} + $$14\!\cdots\!89$$ T^{4} + $$12\!\cdots\!52$$ T^{6} + $$96\!\cdots\!14$$ T^{8} + $$59\!\cdots\!44$$ T^{10} + $$96\!\cdots\!14$$ p^{30} T^{12} + $$12\!\cdots\!52$$ p^{60} T^{14} + $$14\!\cdots\!89$$ p^{90} T^{16} + $$13\!\cdots\!30$$ p^{120} T^{18} + p^{150} T^{20} )^{2} $
	73	$ ( 1 - 20805465951266 T + $$17\!\cdots\!49$$ T^{2} + $$12\!\cdots\!36$$ T^{3} + $$88\!\cdots\!42$$ T^{4} + $$22\!\cdots\!32$$ T^{5} + $$88\!\cdots\!42$$ p^{15} T^{6} + $$12\!\cdots\!36$$ p^{30} T^{7} + $$17\!\cdots\!49$$ p^{45} T^{8} - 20805465951266 p^{60} T^{9} + p^{75} T^{10} )^{4} $
	79	$ ( 1 - $$20\!\cdots\!98$$ T^{2} + $$20\!\cdots\!05$$ T^{4} - $$13\!\cdots\!92$$ T^{6} + $$60\!\cdots\!10$$ T^{8} - $$20\!\cdots\!88$$ T^{10} + $$60\!\cdots\!10$$ p^{30} T^{12} - $$13\!\cdots\!92$$ p^{60} T^{14} + $$20\!\cdots\!05$$ p^{90} T^{16} - $$20\!\cdots\!98$$ p^{120} T^{18} + p^{150} T^{20} )^{2} $
	83	$ ( 1 + $$23\!\cdots\!06$$ T^{2} + $$33\!\cdots\!53$$ T^{4} + $$35\!\cdots\!56$$ T^{6} + $$29\!\cdots\!86$$ T^{8} + $$19\!\cdots\!76$$ T^{10} + $$29\!\cdots\!86$$ p^{30} T^{12} + $$35\!\cdots\!56$$ p^{60} T^{14} + $$33\!\cdots\!53$$ p^{90} T^{16} + $$23\!\cdots\!06$$ p^{120} T^{18} + p^{150} T^{20} )^{2} $
	89	$ ( 1 - $$42\!\cdots\!54$$ T^{2} + $$11\!\cdots\!65$$ T^{4} - $$29\!\cdots\!52$$ T^{6} + $$65\!\cdots\!18$$ T^{8} - $$12\!\cdots\!20$$ T^{10} + $$65\!\cdots\!18$$ p^{30} T^{12} - $$29\!\cdots\!52$$ p^{60} T^{14} + $$11\!\cdots\!65$$ p^{90} T^{16} - $$42\!\cdots\!54$$ p^{120} T^{18} + p^{150} T^{20} )^{2} $
	97	$ ( 1 + 304759200448678 T + $$76\!\cdots\!45$$ T^{2} - $$89\!\cdots\!84$$ T^{3} + $$32\!\cdots\!10$$ T^{4} + $$80\!\cdots\!32$$ T^{5} + $$32\!\cdots\!10$$ p^{15} T^{6} - $$89\!\cdots\!84$$ p^{30} T^{7} + $$76\!\cdots\!45$$ p^{45} T^{8} + 304759200448678 p^{60} T^{9} + p^{75} T^{10} )^{4} $
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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

−1.77774351833024067432191409670, −1.58353467471525304544772752976, −1.58235029758526069445039976609, −1.39632082941924643982891371780, −1.33414774000410466123079146952, −1.32111682348918141508431975728, −1.30887820253897084596886392728, −1.26947229352260688674942927677, −1.25927342049615913448614292719, −1.21105031724364389540334475286, −1.12116488156340262404068440391, −1.05865967826431637179163361833, −1.01470470976767970581091056750, −0.886519218622078557861111683969, −0.854663281834177916005533288836, −0.76488759043661555700146120596, −0.66956634359084851514557744164, −0.54693144935443448005005170517, −0.49827951771738388981819056955, −0.45945452664045267022553213306, −0.37830269727882501684000064297, −0.23663777940600520967664125558, −0.17923430223793554657719929368, −0.16290506727124754984222736097, −0.13089242415006687903539717470, 0.13089242415006687903539717470, 0.16290506727124754984222736097, 0.17923430223793554657719929368, 0.23663777940600520967664125558, 0.37830269727882501684000064297, 0.45945452664045267022553213306, 0.49827951771738388981819056955, 0.54693144935443448005005170517, 0.66956634359084851514557744164, 0.76488759043661555700146120596, 0.854663281834177916005533288836, 0.886519218622078557861111683969, 1.01470470976767970581091056750, 1.05865967826431637179163361833, 1.12116488156340262404068440391, 1.21105031724364389540334475286, 1.25927342049615913448614292719, 1.26947229352260688674942927677, 1.30887820253897084596886392728, 1.32111682348918141508431975728, 1.33414774000410466123079146952, 1.39632082941924643982891371780, 1.58235029758526069445039976609, 1.58353467471525304544772752976, 1.77774351833024067432191409670

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 $\F_{q}$

\(L(8)\)	\(\approx\)	\(93.80119345\)
\(L(\frac12)\)	\(\approx\)	\(93.80119345\)
\(L(\frac{17}{2})\)		not available
\(L(1)\)		not available

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