LMFDB - L-function 12-126e6-1.1-c11e6-0-1

Dirichlet series

L(s) = 1

− 96·2^-s + 3.07e3·4^-s − 1.33e3·5^-s − 8.35e4·7^-s + 6.55e4·8^-s + 1.27e5·10^-s + 3.56e5·11^-s − 1.95e6·13^-s + 8.02e6·14^-s − 9.43e6·16^-s − 1.73e6·17^-s − 5.28e6·19^-s − 4.08e6·20^-s − 3.42e7·22^-s + 5.67e6·23^-s + 9.91e7·25^-s + 1.87e8·26^-s − 2.56e8·28^-s − 7.48e6·29^-s + 1.45e8·31^-s + 3.01e8·32^-s + 1.66e8·34^-s + 1.11e8·35^-s + 2.66e8·37^-s + 5.07e8·38^-s − 8.72e7·40^-s − 9.85e8·41^-s + ⋯

L(s) = 1

− 2.12·2^-s + 3/2·4^-s − 0.190·5^-s − 1.87·7^-s + 0.707·8^-s + 0.404·10^-s + 0.667·11^-s − 1.45·13^-s + 3.98·14^-s − 9/4·16^-s − 0.295·17^-s − 0.490·19^-s − 0.285·20^-s − 1.41·22^-s + 0.183·23^-s + 2.03·25^-s + 3.09·26^-s − 2.81·28^-s − 0.0677·29^-s + 0.910·31^-s + 1.59·32^-s + 0.627·34^-s + 0.357·35^-s + 0.630·37^-s + 1.03·38^-s − 0.134·40^-s − 1.32·41^-s + ⋯

Functional equation

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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$12$
Conductor:	$2^{6} \cdot 3^{12} \cdot 7^{6}$
Sign:	$1$
Analytic conductor:	$8.23293\times 10^{11}$
Root analytic conductor:	$9.83927$
Motivic weight:	$11$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [11/2]^{6} ),\ 1 )$

Particular Values

$L(6)$	$\approx$	$3.584065306$
$L(\frac12)$	$\approx$	$3.584065306$
$L(\frac{13}{2})$		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^{5} T + p^{10} T^{2} )^{3} $
	3	$ 1 $
	7	$ 1 + 1705 p^{2} T + 85040 p^{5} T^{2} - 98135 p^{10} T^{3} + 85040 p^{16} T^{4} + 1705 p^{24} T^{5} + p^{33} T^{6} $
good	5	$ 1 + 1331 T - 97399154 T^{2} + 27352063513 p T^{3} + 199827246042932 p^{2} T^{4} - 79625073059973149 p^{3} T^{5} - $$40\!\cdots\!76$$ p^{4} T^{6} - 79625073059973149 p^{14} T^{7} + 199827246042932 p^{24} T^{8} + 27352063513 p^{34} T^{9} - 97399154 p^{44} T^{10} + 1331 p^{55} T^{11} + p^{66} T^{12} $
	11	$ 1 - 356381 T + 302945706784 T^{2} - 232553755683207713 T^{3} + $$13\!\cdots\!60$$ T^{4} - $$90\!\cdots\!61$$ T^{5} + $$49\!\cdots\!06$$ T^{6} - $$90\!\cdots\!61$$ p^{11} T^{7} + $$13\!\cdots\!60$$ p^{22} T^{8} - 232553755683207713 p^{33} T^{9} + 302945706784 p^{44} T^{10} - 356381 p^{55} T^{11} + p^{66} T^{12} $
	13	$ ( 1 + 977174 T + 3778834995400 T^{2} + 1997082454585484716 T^{3} + 3778834995400 p^{11} T^{4} + 977174 p^{22} T^{5} + p^{33} T^{6} )^{2} $
	17	$ 1 + 1732024 T - 75735751052963 T^{2} - $$15\!\cdots\!32$$ T^{3} + $$32\!\cdots\!18$$ T^{4} + $$23\!\cdots\!08$$ p T^{5} - $$11\!\cdots\!27$$ T^{6} + $$23\!\cdots\!08$$ p^{12} T^{7} + $$32\!\cdots\!18$$ p^{22} T^{8} - $$15\!\cdots\!32$$ p^{33} T^{9} - 75735751052963 p^{44} T^{10} + 1732024 p^{55} T^{11} + p^{66} T^{12} $
	19	$ 1 + 5289402 T - 318718494990930 T^{2} - $$58\!\cdots\!24$$ T^{3} + $$73\!\cdots\!18$$ T^{4} + $$63\!\cdots\!02$$ T^{5} - $$97\!\cdots\!74$$ T^{6} + $$63\!\cdots\!02$$ p^{11} T^{7} + $$73\!\cdots\!18$$ p^{22} T^{8} - $$58\!\cdots\!24$$ p^{33} T^{9} - 318718494990930 p^{44} T^{10} + 5289402 p^{55} T^{11} + p^{66} T^{12} $
	23	$ 1 - 246904 p T - 446877689003429 T^{2} - $$25\!\cdots\!40$$ T^{3} - $$16\!\cdots\!98$$ T^{4} + $$54\!\cdots\!12$$ T^{5} + $$16\!\cdots\!51$$ T^{6} + $$54\!\cdots\!12$$ p^{11} T^{7} - $$16\!\cdots\!98$$ p^{22} T^{8} - $$25\!\cdots\!40$$ p^{33} T^{9} - 446877689003429 p^{44} T^{10} - 246904 p^{56} T^{11} + p^{66} T^{12} $
	29	$ ( 1 + 3740911 T + 479801432736123 p T^{2} - $$18\!\cdots\!62$$ T^{3} + 479801432736123 p^{12} T^{4} + 3740911 p^{22} T^{5} + p^{33} T^{6} )^{2} $
	31	$ 1 - 145122587 T - 50173077006188295 T^{2} + $$28\!\cdots\!90$$ T^{3} + $$23\!\cdots\!39$$ T^{4} - $$39\!\cdots\!47$$ T^{5} - $$65\!\cdots\!62$$ T^{6} - $$39\!\cdots\!47$$ p^{11} T^{7} + $$23\!\cdots\!39$$ p^{22} T^{8} + $$28\!\cdots\!90$$ p^{33} T^{9} - 50173077006188295 p^{44} T^{10} - 145122587 p^{55} T^{11} + p^{66} T^{12} $
	37	$ 1 - 266146806 T - 481335091338530436 T^{2} + $$42\!\cdots\!40$$ T^{3} + $$18\!\cdots\!32$$ T^{4} - $$22\!\cdots\!02$$ p T^{5} - $$36\!\cdots\!86$$ T^{6} - $$22\!\cdots\!02$$ p^{12} T^{7} + $$18\!\cdots\!32$$ p^{22} T^{8} + $$42\!\cdots\!40$$ p^{33} T^{9} - 481335091338530436 p^{44} T^{10} - 266146806 p^{55} T^{11} + p^{66} T^{12} $
	41	$ ( 1 + 492748620 T + 1569920190915123807 T^{2} + $$52\!\cdots\!60$$ T^{3} + 1569920190915123807 p^{11} T^{4} + 492748620 p^{22} T^{5} + p^{33} T^{6} )^{2} $
	43	$ ( 1 - 856196724 T + 2765765613777350466 T^{2} - $$15\!\cdots\!86$$ T^{3} + 2765765613777350466 p^{11} T^{4} - 856196724 p^{22} T^{5} + p^{33} T^{6} )^{2} $
	47	$ 1 - 1523448978 T - 2839164586319750469 T^{2} + $$65\!\cdots\!66$$ T^{3} + $$73\!\cdots\!50$$ T^{4} + $$75\!\cdots\!02$$ T^{5} - $$31\!\cdots\!29$$ T^{6} + $$75\!\cdots\!02$$ p^{11} T^{7} + $$73\!\cdots\!50$$ p^{22} T^{8} + $$65\!\cdots\!66$$ p^{33} T^{9} - 2839164586319750469 p^{44} T^{10} - 1523448978 p^{55} T^{11} + p^{66} T^{12} $
	53	$ 1 + 5590753641 T - 1628536001919688050 T^{2} - $$22\!\cdots\!73$$ T^{3} + $$20\!\cdots\!52$$ T^{4} + $$46\!\cdots\!77$$ T^{5} - $$50\!\cdots\!16$$ T^{6} + $$46\!\cdots\!77$$ p^{11} T^{7} + $$20\!\cdots\!52$$ p^{22} T^{8} - $$22\!\cdots\!73$$ p^{33} T^{9} - 1628536001919688050 p^{44} T^{10} + 5590753641 p^{55} T^{11} + p^{66} T^{12} $
	59	$ 1 + 2495065619 T - 9506680356468604892 T^{2} + $$26\!\cdots\!95$$ T^{3} + $$53\!\cdots\!68$$ T^{4} - $$18\!\cdots\!69$$ T^{5} + $$54\!\cdots\!22$$ T^{6} - $$18\!\cdots\!69$$ p^{11} T^{7} + $$53\!\cdots\!68$$ p^{22} T^{8} + $$26\!\cdots\!95$$ p^{33} T^{9} - 9506680356468604892 p^{44} T^{10} + 2495065619 p^{55} T^{11} + p^{66} T^{12} $
	61	$ 1 + 5120890846 T - 17867627105650130871 T^{2} + $$30\!\cdots\!78$$ T^{3} + $$23\!\cdots\!10$$ T^{4} - $$11\!\cdots\!34$$ T^{5} + $$32\!\cdots\!61$$ T^{6} - $$11\!\cdots\!34$$ p^{11} T^{7} + $$23\!\cdots\!10$$ p^{22} T^{8} + $$30\!\cdots\!78$$ p^{33} T^{9} - 17867627105650130871 p^{44} T^{10} + 5120890846 p^{55} T^{11} + p^{66} T^{12} $
	67	$ 1 - 16130531092 T - $$17\!\cdots\!22$$ T^{2} + $$10\!\cdots\!72$$ T^{3} + $$72\!\cdots\!94$$ T^{4} - $$27\!\cdots\!08$$ T^{5} - $$73\!\cdots\!26$$ T^{6} - $$27\!\cdots\!08$$ p^{11} T^{7} + $$72\!\cdots\!94$$ p^{22} T^{8} + $$10\!\cdots\!72$$ p^{33} T^{9} - $$17\!\cdots\!22$$ p^{44} T^{10} - 16130531092 p^{55} T^{11} + p^{66} T^{12} $
	71	$ ( 1 - 51069173614 T + $$15\!\cdots\!73$$ T^{2} - $$27\!\cdots\!60$$ T^{3} + $$15\!\cdots\!73$$ p^{11} T^{4} - 51069173614 p^{22} T^{5} + p^{33} T^{6} )^{2} $
	73	$ 1 - 9815735116 T - 57368224540646980600 T^{2} + $$17\!\cdots\!48$$ T^{3} - $$11\!\cdots\!28$$ T^{4} - $$14\!\cdots\!12$$ T^{5} + $$13\!\cdots\!54$$ T^{6} - $$14\!\cdots\!12$$ p^{11} T^{7} - $$11\!\cdots\!28$$ p^{22} T^{8} + $$17\!\cdots\!48$$ p^{33} T^{9} - 57368224540646980600 p^{44} T^{10} - 9815735116 p^{55} T^{11} + p^{66} T^{12} $
	79	$ 1 + 2043182857 T - $$43\!\cdots\!27$$ T^{2} + $$52\!\cdots\!34$$ T^{3} - $$82\!\cdots\!21$$ T^{4} - $$11\!\cdots\!95$$ T^{5} + $$13\!\cdots\!66$$ T^{6} - $$11\!\cdots\!95$$ p^{11} T^{7} - $$82\!\cdots\!21$$ p^{22} T^{8} + $$52\!\cdots\!34$$ p^{33} T^{9} - $$43\!\cdots\!27$$ p^{44} T^{10} + 2043182857 p^{55} T^{11} + p^{66} T^{12} $
	83	$ ( 1 - 72650826067 T + $$34\!\cdots\!41$$ T^{2} - $$13\!\cdots\!78$$ T^{3} + $$34\!\cdots\!41$$ p^{11} T^{4} - 72650826067 p^{22} T^{5} + p^{33} T^{6} )^{2} $
	89	$ 1 + 62019734142 T + $$66\!\cdots\!21$$ T^{2} - $$13\!\cdots\!66$$ T^{3} - $$90\!\cdots\!90$$ T^{4} - $$88\!\cdots\!38$$ T^{5} + $$12\!\cdots\!29$$ T^{6} - $$88\!\cdots\!38$$ p^{11} T^{7} - $$90\!\cdots\!90$$ p^{22} T^{8} - $$13\!\cdots\!66$$ p^{33} T^{9} + $$66\!\cdots\!21$$ p^{44} T^{10} + 62019734142 p^{55} T^{11} + p^{66} T^{12} $
	97	$ ( 1 + 182732589085 T + $$24\!\cdots\!91$$ T^{2} + $$21\!\cdots\!42$$ T^{3} + $$24\!\cdots\!91$$ p^{11} T^{4} + 182732589085 p^{22} T^{5} + p^{33} 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

−5.19473005044561178106288050044, −5.19461320495874066152784834506, −4.97951345053152997030459876199, −4.93521730831081783994929756872, −4.28738224340966732317438199548, −4.26224988496265714754689541079, −4.07068212043657723763081997531, −3.95483430408905952125746674443, −3.62120509926440440587711571325, −3.47013530328623990141453628633, −3.07297724618886046517420962569, −2.98379149565813523980374528683, −2.67304991053359531696321580068, −2.51186586645026125203909081130, −2.44434815032180263947847977348, −2.06214598531968854435161138334, −1.83561716573347024481209954554, −1.64847518171571536087919641786, −1.31478490369659245847932833912, −1.12028841487385658333271173606, −0.69516237296576049730407714772, −0.57702757548388586049471552303, −0.50908587893287797601056544384, −0.49381450396452801618437969878, −0.36521817238513712435174446930, 0.36521817238513712435174446930, 0.49381450396452801618437969878, 0.50908587893287797601056544384, 0.57702757548388586049471552303, 0.69516237296576049730407714772, 1.12028841487385658333271173606, 1.31478490369659245847932833912, 1.64847518171571536087919641786, 1.83561716573347024481209954554, 2.06214598531968854435161138334, 2.44434815032180263947847977348, 2.51186586645026125203909081130, 2.67304991053359531696321580068, 2.98379149565813523980374528683, 3.07297724618886046517420962569, 3.47013530328623990141453628633, 3.62120509926440440587711571325, 3.95483430408905952125746674443, 4.07068212043657723763081997531, 4.26224988496265714754689541079, 4.28738224340966732317438199548, 4.93521730831081783994929756872, 4.97951345053152997030459876199, 5.19461320495874066152784834506, 5.19473005044561178106288050044

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 $\F_{q}$

\(L(6)\)	\(\approx\)	\(3.584065306\)
\(L(\frac12)\)	\(\approx\)	\(3.584065306\)
\(L(\frac{13}{2})\)		not available
\(L(1)\)		not available

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