LMFDB - L-function 24-33e12-1.1-c6e12-0-0

Dirichlet series

L(s) = 1

+ 282·4^-s + 224·5^-s + 1.45e3·9^-s − 3.46e3·11^-s + 3.78e4·16^-s + 6.31e4·20^-s − 1.53e4·23^-s − 2.08e4·25^-s − 5.86e4·31^-s + 4.11e5·36^-s − 2.02e5·37^-s − 9.76e5·44^-s + 3.26e5·45^-s + 5.16e5·47^-s + 7.84e5·49^-s − 1.04e6·53^-s − 7.75e5·55^-s − 4.61e5·59^-s + 3.12e6·64^-s + 3.64e5·67^-s − 7.55e5·71^-s + 8.47e6·80^-s + 1.24e6·81^-s − 3.51e6·89^-s − 4.31e6·92^-s + 2.37e6·97^-s − 5.05e6·99^-s + ⋯

L(s) = 1

+ 4.40·4^-s + 1.79·5^-s + 2·9^-s − 2.60·11^-s + 9.23·16^-s + 7.89·20^-s − 1.25·23^-s − 1.33·25^-s − 1.96·31^-s + 8.81·36^-s − 3.99·37^-s − 11.4·44^-s + 3.58·45^-s + 4.97·47^-s + 6.67·49^-s − 7.00·53^-s − 4.66·55^-s − 2.24·59^-s + 11.9·64^-s + 1.21·67^-s − 2.10·71^-s + 16.5·80^-s + 7/3·81^-s − 4.98·89^-s − 5.54·92^-s + 2.59·97^-s − 5.20·99^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+3)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$24$
Conductor:	$3^{12} \cdot 11^{12}$
Sign:	$1$
Analytic conductor:	$3.66544\times 10^{10}$
Root analytic conductor:	$2.75531$
Motivic weight:	$6$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(24,\ 3^{12} \cdot 11^{12} ,\ ( \ : [3]^{12} ),\ 1 )$

Particular Values

$L(\frac{7}{2})$	$\approx$	$0.0001622276064$
$L(\frac12)$	$\approx$	$0.0001622276064$
$L(4)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ ( 1 - p^{5} T^{2} )^{6} $
	11	$ 1 + 3464 T + 739774 p T^{2} + 90237000 p^{2} T^{3} + 6476180021 p^{3} T^{4} - 8976173296 p^{5} T^{5} - 44209374932 p^{8} T^{6} - 8976173296 p^{11} T^{7} + 6476180021 p^{15} T^{8} + 90237000 p^{20} T^{9} + 739774 p^{25} T^{10} + 3464 p^{30} T^{11} + p^{36} T^{12} $
good	2	$ 1 - 141 p T^{2} + 41697 T^{4} - 1054777 p^{2} T^{6} + 85912053 p^{2} T^{8} - 24161997 p^{10} T^{10} + 50291025 p^{15} T^{12} - 24161997 p^{22} T^{14} + 85912053 p^{26} T^{16} - 1054777 p^{38} T^{18} + 41697 p^{48} T^{20} - 141 p^{61} T^{22} + p^{72} T^{24} $
	5	$ ( 1 - 112 T + 29234 T^{2} - 147048 p^{2} T^{3} + 24331963 p^{2} T^{4} - 656790328 p^{3} T^{5} + 19833344404 p^{4} T^{6} - 656790328 p^{9} T^{7} + 24331963 p^{14} T^{8} - 147048 p^{20} T^{9} + 29234 p^{24} T^{10} - 112 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	7	$ 1 - 784800 T^{2} + 46389854778 p T^{4} - 91703033397949408 T^{6} + $$19\!\cdots\!75$$ T^{8} - $$31\!\cdots\!84$$ T^{10} + $$41\!\cdots\!76$$ T^{12} - $$31\!\cdots\!84$$ p^{12} T^{14} + $$19\!\cdots\!75$$ p^{24} T^{16} - 91703033397949408 p^{36} T^{18} + 46389854778 p^{49} T^{20} - 784800 p^{60} T^{22} + p^{72} T^{24} $
	13	$ 1 - 11813004 T^{2} + 173649316619562 T^{4} - $$13\!\cdots\!40$$ T^{6} + $$11\!\cdots\!27$$ T^{8} - $$64\!\cdots\!96$$ T^{10} + $$35\!\cdots\!44$$ T^{12} - $$64\!\cdots\!96$$ p^{12} T^{14} + $$11\!\cdots\!27$$ p^{24} T^{16} - $$13\!\cdots\!40$$ p^{36} T^{18} + 173649316619562 p^{48} T^{20} - 11813004 p^{60} T^{22} + p^{72} T^{24} $
	17	$ 1 - 112969992 T^{2} + 7355558115683166 T^{4} - $$33\!\cdots\!68$$ T^{6} + $$74\!\cdots\!55$$ p T^{8} - $$39\!\cdots\!52$$ T^{10} + $$10\!\cdots\!48$$ T^{12} - $$39\!\cdots\!52$$ p^{12} T^{14} + $$74\!\cdots\!55$$ p^{25} T^{16} - $$33\!\cdots\!68$$ p^{36} T^{18} + 7355558115683166 p^{48} T^{20} - 112969992 p^{60} T^{22} + p^{72} T^{24} $
	19	$ 1 - 298936704 T^{2} + 47587972333287630 T^{4} - $$51\!\cdots\!12$$ T^{6} + $$42\!\cdots\!31$$ T^{8} - $$27\!\cdots\!04$$ T^{10} + $$14\!\cdots\!72$$ T^{12} - $$27\!\cdots\!04$$ p^{12} T^{14} + $$42\!\cdots\!31$$ p^{24} T^{16} - $$51\!\cdots\!12$$ p^{36} T^{18} + 47587972333287630 p^{48} T^{20} - 298936704 p^{60} T^{22} + p^{72} T^{24} $
	23	$ ( 1 + 7652 T + 314675762 T^{2} + 898155402636 T^{3} + 60250870240477171 T^{4} + $$30\!\cdots\!60$$ T^{5} + $$11\!\cdots\!60$$ T^{6} + $$30\!\cdots\!60$$ p^{6} T^{7} + 60250870240477171 p^{12} T^{8} + 898155402636 p^{18} T^{9} + 314675762 p^{24} T^{10} + 7652 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	29	$ 1 - 3552464472 T^{2} + 7069307690910966078 T^{4} - $$97\!\cdots\!24$$ T^{6} + $$34\!\cdots\!99$$ p T^{8} - $$83\!\cdots\!92$$ T^{10} + $$54\!\cdots\!60$$ T^{12} - $$83\!\cdots\!92$$ p^{12} T^{14} + $$34\!\cdots\!99$$ p^{25} T^{16} - $$97\!\cdots\!24$$ p^{36} T^{18} + 7069307690910966078 p^{48} T^{20} - 3552464472 p^{60} T^{22} + p^{72} T^{24} $
	31	$ ( 1 + 29304 T + 2909337822 T^{2} + 67282988073368 T^{3} + 4222373507783705727 T^{4} + $$85\!\cdots\!92$$ T^{5} + $$43\!\cdots\!24$$ T^{6} + $$85\!\cdots\!92$$ p^{6} T^{7} + 4222373507783705727 p^{12} T^{8} + 67282988073368 p^{18} T^{9} + 2909337822 p^{24} T^{10} + 29304 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	37	$ ( 1 + 101256 T + 8636966478 T^{2} + 484631270053544 T^{3} + 33956810947327452207 T^{4} + $$20\!\cdots\!48$$ T^{5} + $$32\!\cdots\!68$$ p T^{6} + $$20\!\cdots\!48$$ p^{6} T^{7} + 33956810947327452207 p^{12} T^{8} + 484631270053544 p^{18} T^{9} + 8636966478 p^{24} T^{10} + 101256 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	41	$ 1 - 18443226360 T^{2} + $$19\!\cdots\!26$$ T^{4} - $$14\!\cdots\!20$$ T^{6} + $$94\!\cdots\!19$$ T^{8} - $$52\!\cdots\!20$$ T^{10} + $$26\!\cdots\!44$$ T^{12} - $$52\!\cdots\!20$$ p^{12} T^{14} + $$94\!\cdots\!19$$ p^{24} T^{16} - $$14\!\cdots\!20$$ p^{36} T^{18} + $$19\!\cdots\!26$$ p^{48} T^{20} - 18443226360 p^{60} T^{22} + p^{72} T^{24} $
	43	$ 1 - 23801189952 T^{2} + $$39\!\cdots\!58$$ T^{4} - $$47\!\cdots\!28$$ T^{6} + $$45\!\cdots\!71$$ T^{8} - $$36\!\cdots\!92$$ T^{10} + $$24\!\cdots\!12$$ T^{12} - $$36\!\cdots\!92$$ p^{12} T^{14} + $$45\!\cdots\!71$$ p^{24} T^{16} - $$47\!\cdots\!28$$ p^{36} T^{18} + $$39\!\cdots\!58$$ p^{48} T^{20} - 23801189952 p^{60} T^{22} + p^{72} T^{24} $
	47	$ ( 1 - 258460 T + 80997697934 T^{2} - 13130966707228932 T^{3} + $$23\!\cdots\!79$$ T^{4} - $$27\!\cdots\!60$$ T^{5} + $$71\!\cdots\!08$$ p T^{6} - $$27\!\cdots\!60$$ p^{6} T^{7} + $$23\!\cdots\!79$$ p^{12} T^{8} - 13130966707228932 p^{18} T^{9} + 80997697934 p^{24} T^{10} - 258460 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	53	$ ( 1 + 9832 p T + 230179608818 T^{2} + 64512279082152192 T^{3} + $$15\!\cdots\!39$$ T^{4} + $$29\!\cdots\!76$$ T^{5} + $$50\!\cdots\!52$$ T^{6} + $$29\!\cdots\!76$$ p^{6} T^{7} + $$15\!\cdots\!39$$ p^{12} T^{8} + 64512279082152192 p^{18} T^{9} + 230179608818 p^{24} T^{10} + 9832 p^{31} T^{11} + p^{36} T^{12} )^{2} $
	59	$ ( 1 + 230504 T + 221620610954 T^{2} + 38998264192118184 T^{3} + $$20\!\cdots\!99$$ T^{4} + $$28\!\cdots\!68$$ T^{5} + $$11\!\cdots\!28$$ T^{6} + $$28\!\cdots\!68$$ p^{6} T^{7} + $$20\!\cdots\!99$$ p^{12} T^{8} + 38998264192118184 p^{18} T^{9} + 221620610954 p^{24} T^{10} + 230504 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	61	$ 1 - 344957455356 T^{2} + $$58\!\cdots\!66$$ T^{4} - $$63\!\cdots\!48$$ T^{6} + $$51\!\cdots\!59$$ T^{8} - $$32\!\cdots\!44$$ T^{10} + $$17\!\cdots\!88$$ T^{12} - $$32\!\cdots\!44$$ p^{12} T^{14} + $$51\!\cdots\!59$$ p^{24} T^{16} - $$63\!\cdots\!48$$ p^{36} T^{18} + $$58\!\cdots\!66$$ p^{48} T^{20} - 344957455356 p^{60} T^{22} + p^{72} T^{24} $
	67	$ ( 1 - 182376 T + 317875829886 T^{2} - 34532602304668456 T^{3} + $$46\!\cdots\!79$$ T^{4} - $$36\!\cdots\!72$$ T^{5} + $$48\!\cdots\!60$$ T^{6} - $$36\!\cdots\!72$$ p^{6} T^{7} + $$46\!\cdots\!79$$ p^{12} T^{8} - 34532602304668456 p^{18} T^{9} + 317875829886 p^{24} T^{10} - 182376 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	71	$ ( 1 + 377588 T + 6245146642 p T^{2} + 168724078391174988 T^{3} + $$10\!\cdots\!11$$ T^{4} + $$36\!\cdots\!64$$ T^{5} + $$16\!\cdots\!84$$ T^{6} + $$36\!\cdots\!64$$ p^{6} T^{7} + $$10\!\cdots\!11$$ p^{12} T^{8} + 168724078391174988 p^{18} T^{9} + 6245146642 p^{25} T^{10} + 377588 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	73	$ 1 - 1038551693052 T^{2} + $$52\!\cdots\!34$$ T^{4} - $$17\!\cdots\!24$$ T^{6} + $$44\!\cdots\!75$$ T^{8} - $$89\!\cdots\!52$$ T^{10} + $$14\!\cdots\!40$$ T^{12} - $$89\!\cdots\!52$$ p^{12} T^{14} + $$44\!\cdots\!75$$ p^{24} T^{16} - $$17\!\cdots\!24$$ p^{36} T^{18} + $$52\!\cdots\!34$$ p^{48} T^{20} - 1038551693052 p^{60} T^{22} + p^{72} T^{24} $
	79	$ 1 - 2060241143040 T^{2} + $$20\!\cdots\!02$$ T^{4} - $$12\!\cdots\!72$$ T^{6} + $$58\!\cdots\!51$$ T^{8} - $$20\!\cdots\!32$$ T^{10} + $$55\!\cdots\!96$$ T^{12} - $$20\!\cdots\!32$$ p^{12} T^{14} + $$58\!\cdots\!51$$ p^{24} T^{16} - $$12\!\cdots\!72$$ p^{36} T^{18} + $$20\!\cdots\!02$$ p^{48} T^{20} - 2060241143040 p^{60} T^{22} + p^{72} T^{24} $
	83	$ 1 - 1043972235372 T^{2} + $$84\!\cdots\!42$$ T^{4} - $$43\!\cdots\!80$$ T^{6} + $$20\!\cdots\!39$$ T^{8} - $$72\!\cdots\!88$$ T^{10} + $$25\!\cdots\!00$$ T^{12} - $$72\!\cdots\!88$$ p^{12} T^{14} + $$20\!\cdots\!39$$ p^{24} T^{16} - $$43\!\cdots\!80$$ p^{36} T^{18} + $$84\!\cdots\!42$$ p^{48} T^{20} - 1043972235372 p^{60} T^{22} + p^{72} T^{24} $
	89	$ ( 1 + 1756772 T + 2562798883442 T^{2} + 2577409891035591252 T^{3} + $$24\!\cdots\!83$$ T^{4} + $$19\!\cdots\!88$$ T^{5} + $$14\!\cdots\!88$$ T^{6} + $$19\!\cdots\!88$$ p^{6} T^{7} + $$24\!\cdots\!83$$ p^{12} T^{8} + 2577409891035591252 p^{18} T^{9} + 2562798883442 p^{24} T^{10} + 1756772 p^{30} T^{11} + p^{36} T^{12} )^{2} $
	97	$ ( 1 - 1185096 T + 3545530528134 T^{2} - 3370506166089168808 T^{3} + $$54\!\cdots\!91$$ T^{4} - $$43\!\cdots\!64$$ T^{5} + $$53\!\cdots\!96$$ T^{6} - $$43\!\cdots\!64$$ p^{6} T^{7} + $$54\!\cdots\!91$$ p^{12} T^{8} - 3370506166089168808 p^{18} T^{9} + 3545530528134 p^{24} T^{10} - 1185096 p^{30} T^{11} + p^{36} T^{12} )^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−5.06554293288893309691207413680, −4.95040704645644614494417329533, −4.57776221060929770589612168723, −4.49072428382574826542811934492, −4.19766028156401372681237985225, −4.18469403204910888853047962586, −3.94192123976492620428799955276, −3.79639424741695901015245695918, −3.57873691172918402812921768969, −3.22050055387751544524956502636, −3.02664307031210106553585887541, −2.90269473632836224807737029894, −2.80256667506157408310383699124, −2.58599372663401953198625448740, −2.38439482065377156074527019593, −2.07271372633331739032644253599, −1.93348304084515901400216595309, −1.89791084913676605870523943980, −1.80968805233278152121600546465, −1.77784599995074291590159499694, −1.47147528814225046857801491889, −1.21247151656150849529085846274, −0.68013833145883201412574428451, −0.50778356046003743427724412078, −0.00047192508862913396893149720, 0.00047192508862913396893149720, 0.50778356046003743427724412078, 0.68013833145883201412574428451, 1.21247151656150849529085846274, 1.47147528814225046857801491889, 1.77784599995074291590159499694, 1.80968805233278152121600546465, 1.89791084913676605870523943980, 1.93348304084515901400216595309, 2.07271372633331739032644253599, 2.38439482065377156074527019593, 2.58599372663401953198625448740, 2.80256667506157408310383699124, 2.90269473632836224807737029894, 3.02664307031210106553585887541, 3.22050055387751544524956502636, 3.57873691172918402812921768969, 3.79639424741695901015245695918, 3.94192123976492620428799955276, 4.18469403204910888853047962586, 4.19766028156401372681237985225, 4.49072428382574826542811934492, 4.57776221060929770589612168723, 4.95040704645644614494417329533, 5.06554293288893309691207413680

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 $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(0.0001622276064\)
\(L(\frac12)\)	\(\approx\)	\(0.0001622276064\)
\(L(4)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more