LMFDB - L-function 12-126e6-1.1-c9e6-0-1

Dirichlet series

L(s) = 1

− 48·2^-s + 768·4^-s − 361·5^-s + 1.25e4·7^-s + 8.19e3·8^-s + 1.73e4·10^-s − 3.77e4·11^-s − 4.41e5·13^-s − 6.00e5·14^-s − 5.89e5·16^-s + 7.81e5·17^-s − 6.20e5·19^-s − 2.77e5·20^-s + 1.81e6·22^-s − 1.20e6·23^-s + 9.84e5·25^-s + 2.11e7·26^-s + 9.60e6·28^-s + 1.05e7·29^-s + 1.28e7·31^-s + 9.43e6·32^-s − 3.75e7·34^-s − 4.51e6·35^-s + 8.01e6·37^-s + 2.97e7·38^-s − 2.95e6·40^-s − 1.30e7·41^-s + ⋯

L(s) = 1

− 2.12·2^-s + 3/2·4^-s − 0.258·5^-s + 1.96·7^-s + 0.707·8^-s + 0.547·10^-s − 0.778·11^-s − 4.28·13^-s − 4.17·14^-s − 9/4·16^-s + 2.27·17^-s − 1.09·19^-s − 0.387·20^-s + 1.65·22^-s − 0.897·23^-s + 0.503·25^-s + 9.08·26^-s + 2.95·28^-s + 2.77·29^-s + 2.49·31^-s + 1.59·32^-s − 4.81·34^-s − 0.508·35^-s + 0.703·37^-s + 2.31·38^-s − 0.182·40^-s − 0.718·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$12$
Conductor:	$2^{6} \cdot 3^{12} \cdot 7^{6}$
Sign:	$1$
Analytic conductor:	$7.46875\times 10^{10}$
Root analytic conductor:	$8.05571$
Motivic weight:	$9$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [9/2]^{6} ),\ 1 )$

Particular Values

$L(5)$	$\approx$	$0.5190785532$
$L(\frac12)$	$\approx$	$0.5190785532$
$L(\frac{11}{2})$		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^{4} T + p^{8} T^{2} )^{3} $
	3	$ 1 $
	7	$ 1 - 1787 p T + 363572 p^{3} T^{2} - 1169207 p^{7} T^{3} + 363572 p^{12} T^{4} - 1787 p^{19} T^{5} + p^{27} T^{6} $
good	5	$ 1 + 361 T - 854054 T^{2} + 3346723 p^{4} T^{3} - 1024325536 p^{4} T^{4} - 88639787 p^{10} T^{5} + 33958862514976 p^{8} T^{6} - 88639787 p^{19} T^{7} - 1024325536 p^{22} T^{8} + 3346723 p^{31} T^{9} - 854054 p^{36} T^{10} + 361 p^{45} T^{11} + p^{54} T^{12} $
	11	$ 1 + 37799 T - 1510556936 T^{2} - 80478148006045 T^{3} - 1911951640485624844 T^{4} - $$32\!\cdots\!49$$ T^{5} + $$70\!\cdots\!98$$ T^{6} - $$32\!\cdots\!49$$ p^{9} T^{7} - 1911951640485624844 p^{18} T^{8} - 80478148006045 p^{27} T^{9} - 1510556936 p^{36} T^{10} + 37799 p^{45} T^{11} + p^{54} T^{12} $
	13	$ ( 1 + 220586 T + 3402528616 p T^{2} + 4732599775105960 T^{3} + 3402528616 p^{10} T^{4} + 220586 p^{18} T^{5} + p^{27} T^{6} )^{2} $
	17	$ 1 - 781816 T + 275472832141 T^{2} - 25404569111963272 T^{3} - $$19\!\cdots\!74$$ T^{4} + $$11\!\cdots\!56$$ T^{5} - $$45\!\cdots\!27$$ T^{6} + $$11\!\cdots\!56$$ p^{9} T^{7} - $$19\!\cdots\!74$$ p^{18} T^{8} - 25404569111963272 p^{27} T^{9} + 275472832141 p^{36} T^{10} - 781816 p^{45} T^{11} + p^{54} T^{12} $
	19	$ 1 + 620154 T - 34603144686 T^{2} + 147018356026129220 T^{3} + $$54\!\cdots\!70$$ T^{4} - $$79\!\cdots\!62$$ T^{5} - $$26\!\cdots\!58$$ T^{6} - $$79\!\cdots\!62$$ p^{9} T^{7} + $$54\!\cdots\!70$$ p^{18} T^{8} + 147018356026129220 p^{27} T^{9} - 34603144686 p^{36} T^{10} + 620154 p^{45} T^{11} + p^{54} T^{12} $
	23	$ 1 + 1204784 T - 3580841239685 T^{2} - 2612212770876712528 T^{3} + $$11\!\cdots\!74$$ T^{4} + $$38\!\cdots\!28$$ T^{5} - $$19\!\cdots\!77$$ T^{6} + $$38\!\cdots\!28$$ p^{9} T^{7} + $$11\!\cdots\!74$$ p^{18} T^{8} - 2612212770876712528 p^{27} T^{9} - 3580841239685 p^{36} T^{10} + 1204784 p^{45} T^{11} + p^{54} T^{12} $
	29	$ ( 1 - 5280535 T + 19018715255919 T^{2} - 72862120772710617958 T^{3} + 19018715255919 p^{9} T^{4} - 5280535 p^{18} T^{5} + p^{27} T^{6} )^{2} $
	31	$ 1 - 12838457 T + 51083476203621 T^{2} - $$18\!\cdots\!58$$ T^{3} + $$17\!\cdots\!63$$ T^{4} - $$44\!\cdots\!37$$ T^{5} - $$12\!\cdots\!90$$ T^{6} - $$44\!\cdots\!37$$ p^{9} T^{7} + $$17\!\cdots\!63$$ p^{18} T^{8} - $$18\!\cdots\!58$$ p^{27} T^{9} + 51083476203621 p^{36} T^{10} - 12838457 p^{45} T^{11} + p^{54} T^{12} $
	37	$ 1 - 8014770 T - 308120275501164 T^{2} + $$87\!\cdots\!72$$ T^{3} + $$74\!\cdots\!48$$ T^{4} - $$78\!\cdots\!74$$ T^{5} - $$10\!\cdots\!50$$ T^{6} - $$78\!\cdots\!74$$ p^{9} T^{7} + $$74\!\cdots\!48$$ p^{18} T^{8} + $$87\!\cdots\!72$$ p^{27} T^{9} - 308120275501164 p^{36} T^{10} - 8014770 p^{45} T^{11} + p^{54} T^{12} $
	41	$ ( 1 + 6503028 T + 226656650449023 T^{2} - $$51\!\cdots\!08$$ T^{3} + 226656650449023 p^{9} T^{4} + 6503028 p^{18} T^{5} + p^{27} T^{6} )^{2} $
	43	$ ( 1 + 22543488 T + 1418023854308070 T^{2} + $$20\!\cdots\!26$$ T^{3} + 1418023854308070 p^{9} T^{4} + 22543488 p^{18} T^{5} + p^{27} T^{6} )^{2} $
	47	$ 1 - 21473082 T - 1131307260614133 T^{2} - $$51\!\cdots\!34$$ p T^{3} + $$64\!\cdots\!14$$ T^{4} + $$41\!\cdots\!02$$ T^{5} - $$58\!\cdots\!49$$ T^{6} + $$41\!\cdots\!02$$ p^{9} T^{7} + $$64\!\cdots\!14$$ p^{18} T^{8} - $$51\!\cdots\!34$$ p^{28} T^{9} - 1131307260614133 p^{36} T^{10} - 21473082 p^{45} T^{11} + p^{54} T^{12} $
	53	$ 1 + 13685715 T - 9491736819770142 T^{2} - $$92\!\cdots\!31$$ p T^{3} + $$60\!\cdots\!68$$ T^{4} + $$15\!\cdots\!63$$ T^{5} - $$23\!\cdots\!12$$ T^{6} + $$15\!\cdots\!63$$ p^{9} T^{7} + $$60\!\cdots\!68$$ p^{18} T^{8} - $$92\!\cdots\!31$$ p^{28} T^{9} - 9491736819770142 p^{36} T^{10} + 13685715 p^{45} T^{11} + p^{54} T^{12} $
	59	$ 1 - 92528141 T - 1442613099568880 T^{2} + $$34\!\cdots\!03$$ T^{3} - $$45\!\cdots\!40$$ T^{4} + $$27\!\cdots\!95$$ T^{5} + $$19\!\cdots\!22$$ T^{6} + $$27\!\cdots\!95$$ p^{9} T^{7} - $$45\!\cdots\!40$$ p^{18} T^{8} + $$34\!\cdots\!03$$ p^{27} T^{9} - 1442613099568880 p^{36} T^{10} - 92528141 p^{45} T^{11} + p^{54} T^{12} $
	61	$ 1 + 7516114 T - 29355354716966031 T^{2} - $$30\!\cdots\!38$$ T^{3} + $$51\!\cdots\!10$$ T^{4} + $$36\!\cdots\!54$$ T^{5} - $$68\!\cdots\!59$$ T^{6} + $$36\!\cdots\!54$$ p^{9} T^{7} + $$51\!\cdots\!10$$ p^{18} T^{8} - $$30\!\cdots\!38$$ p^{27} T^{9} - 29355354716966031 p^{36} T^{10} + 7516114 p^{45} T^{11} + p^{54} T^{12} $
	67	$ 1 + 137325404 T - 12037795994200774 T^{2} + $$23\!\cdots\!28$$ T^{3} - $$16\!\cdots\!94$$ T^{4} - $$99\!\cdots\!64$$ T^{5} + $$12\!\cdots\!58$$ T^{6} - $$99\!\cdots\!64$$ p^{9} T^{7} - $$16\!\cdots\!94$$ p^{18} T^{8} + $$23\!\cdots\!28$$ p^{27} T^{9} - 12037795994200774 p^{36} T^{10} + 137325404 p^{45} T^{11} + p^{54} T^{12} $
	71	$ ( 1 + 8190346 T + 64452383199706977 T^{2} - $$47\!\cdots\!84$$ T^{3} + 64452383199706977 p^{9} T^{4} + 8190346 p^{18} T^{5} + p^{27} T^{6} )^{2} $
	73	$ 1 - 394016164 T + 30400712426791688 T^{2} - $$60\!\cdots\!52$$ T^{3} - $$13\!\cdots\!60$$ T^{4} + $$13\!\cdots\!56$$ T^{5} - $$45\!\cdots\!62$$ T^{6} + $$13\!\cdots\!56$$ p^{9} T^{7} - $$13\!\cdots\!60$$ p^{18} T^{8} - $$60\!\cdots\!52$$ p^{27} T^{9} + 30400712426791688 p^{36} T^{10} - 394016164 p^{45} T^{11} + p^{54} T^{12} $
	79	$ 1 - 35090117 T - 357321337837001559 T^{2} + $$42\!\cdots\!70$$ T^{3} + $$85\!\cdots\!11$$ T^{4} - $$50\!\cdots\!93$$ T^{5} - $$11\!\cdots\!38$$ T^{6} - $$50\!\cdots\!93$$ p^{9} T^{7} + $$85\!\cdots\!11$$ p^{18} T^{8} + $$42\!\cdots\!70$$ p^{27} T^{9} - 357321337837001559 p^{36} T^{10} - 35090117 p^{45} T^{11} + p^{54} T^{12} $
	83	$ ( 1 + 956948005 T + 803075196332581041 T^{2} + $$37\!\cdots\!58$$ T^{3} + 803075196332581041 p^{9} T^{4} + 956948005 p^{18} T^{5} + p^{27} T^{6} )^{2} $
	89	$ 1 - 855499686 T - 41037443393549415 T^{2} - $$46\!\cdots\!10$$ p T^{3} + $$45\!\cdots\!14$$ T^{4} + $$10\!\cdots\!54$$ T^{5} - $$10\!\cdots\!23$$ T^{6} + $$10\!\cdots\!54$$ p^{9} T^{7} + $$45\!\cdots\!14$$ p^{18} T^{8} - $$46\!\cdots\!10$$ p^{28} T^{9} - 41037443393549415 p^{36} T^{10} - 855499686 p^{45} T^{11} + p^{54} T^{12} $
	97	$ ( 1 - 2010569795 T + 2908356430304559059 T^{2} - $$28\!\cdots\!26$$ T^{3} + 2908356430304559059 p^{9} T^{4} - 2010569795 p^{18} T^{5} + p^{27} T^{6} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−5.53959162294807314622062816254, −5.20434074915253060679432914809, −5.05591568777402917736613754455, −5.00433891928677611276875742198, −4.75851657023038262284010513438, −4.69675746038532130894388426664, −4.61030281936384480226191078105, −4.40747424117005822624357346725, −3.94344885791375914522098771326, −3.82855066749489549003060882137, −3.33038758014725548937151344968, −2.97370011849766159356313546011, −2.85963202174100004275801762510, −2.84123064366323768782240212439, −2.34681204870527995328068234946, −2.29870301723127055297071683820, −1.88209804986761893724255125947, −1.81060792626966657117411262822, −1.68628687343581168156432072797, −1.15172516545358021035981438482, −0.955252321243991270987312943077, −0.78521725592809017635220957305, −0.65795548866868697514187064235, −0.35249984516060227454269050564, −0.12555645863656350173035535133, 0.12555645863656350173035535133, 0.35249984516060227454269050564, 0.65795548866868697514187064235, 0.78521725592809017635220957305, 0.955252321243991270987312943077, 1.15172516545358021035981438482, 1.68628687343581168156432072797, 1.81060792626966657117411262822, 1.88209804986761893724255125947, 2.29870301723127055297071683820, 2.34681204870527995328068234946, 2.84123064366323768782240212439, 2.85963202174100004275801762510, 2.97370011849766159356313546011, 3.33038758014725548937151344968, 3.82855066749489549003060882137, 3.94344885791375914522098771326, 4.40747424117005822624357346725, 4.61030281936384480226191078105, 4.69675746038532130894388426664, 4.75851657023038262284010513438, 5.00433891928677611276875742198, 5.05591568777402917736613754455, 5.20434074915253060679432914809, 5.53959162294807314622062816254

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(0.5190785532\)
\(L(\frac12)\)	\(\approx\)	\(0.5190785532\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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