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

Dirichlet series

L(s) = 1

− 48·2^-s + 768·4^-s + 1.08e3·5^-s − 6.79e3·7^-s + 8.19e3·8^-s − 5.20e4·10^-s − 2.55e3·11^-s + 3.61e4·13^-s + 3.26e5·14^-s − 5.89e5·16^-s − 2.07e4·17^-s + 1.22e6·19^-s + 8.33e5·20^-s + 1.22e5·22^-s + 1.96e6·23^-s + 3.08e6·25^-s − 1.73e6·26^-s − 5.21e6·28^-s + 5.21e6·29^-s − 9.37e6·31^-s + 9.43e6·32^-s + 9.96e5·34^-s − 7.37e6·35^-s − 2.58e7·37^-s − 5.85e7·38^-s + 8.88e6·40^-s − 4.18e5·41^-s + ⋯

L(s) = 1

− 2.12·2^-s + 3/2·4^-s + 0.776·5^-s − 1.06·7^-s + 0.707·8^-s − 1.64·10^-s − 0.0526·11^-s + 0.350·13^-s + 2.26·14^-s − 9/4·16^-s − 0.0602·17^-s + 2.14·19^-s + 1.16·20^-s + 0.111·22^-s + 1.46·23^-s + 1.58·25^-s − 0.744·26^-s − 1.60·28^-s + 1.36·29^-s − 1.82·31^-s + 1.59·32^-s + 0.127·34^-s − 0.830·35^-s − 2.26·37^-s − 4.55·38^-s + 0.548·40^-s − 0.0231·41^-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

$L(5)$	$\approx$	$0.03413597216$
$L(\frac12)$	$\approx$	$0.03413597216$
$L(\frac{11}{2})$		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 + 6796 T - 635515 p^{2} T^{2} - 25759976 p^{5} T^{3} - 635515 p^{11} T^{4} + 6796 p^{18} T^{5} + p^{27} T^{6} $
good	5	$ 1 - 217 p T - 1909289 T^{2} + 1197997304 p T^{3} - 1568107392979 T^{4} - 1093936850140519 p T^{5} + 544294970253925366 p^{2} T^{6} - 1093936850140519 p^{10} T^{7} - 1568107392979 p^{18} T^{8} + 1197997304 p^{28} T^{9} - 1909289 p^{36} T^{10} - 217 p^{46} T^{11} + p^{54} T^{12} $
	11	$ 1 + 2555 T - 2975965331 T^{2} - 127913425295698 T^{3} + 1689558999970696595 T^{4} + $$18\!\cdots\!79$$ T^{5} + $$98\!\cdots\!62$$ T^{6} + $$18\!\cdots\!79$$ p^{9} T^{7} + 1689558999970696595 p^{18} T^{8} - 127913425295698 p^{27} T^{9} - 2975965331 p^{36} T^{10} + 2555 p^{45} T^{11} + p^{54} T^{12} $
	13	$ ( 1 - 1390 p T + 21306979843 T^{2} - 14271766002020 T^{3} + 21306979843 p^{9} T^{4} - 1390 p^{19} T^{5} + p^{27} T^{6} )^{2} $
	17	$ 1 + 20759 T - 247431255533 T^{2} - 11560274507094196 T^{3} + $$31\!\cdots\!89$$ T^{4} + $$11\!\cdots\!41$$ T^{5} - $$38\!\cdots\!74$$ T^{6} + $$11\!\cdots\!41$$ p^{9} T^{7} + $$31\!\cdots\!89$$ p^{18} T^{8} - 11560274507094196 p^{27} T^{9} - 247431255533 p^{36} T^{10} + 20759 p^{45} T^{11} + p^{54} T^{12} $
	19	$ 1 - 1220649 T + 202592460021 T^{2} - 34374843892736986 T^{3} + $$30\!\cdots\!47$$ T^{4} - $$10\!\cdots\!09$$ T^{5} - $$32\!\cdots\!74$$ T^{6} - $$10\!\cdots\!09$$ p^{9} T^{7} + $$30\!\cdots\!47$$ p^{18} T^{8} - 34374843892736986 p^{27} T^{9} + 202592460021 p^{36} T^{10} - 1220649 p^{45} T^{11} + p^{54} T^{12} $
	23	$ 1 - 1960903 T + 3690258693265 T^{2} - 5968087950532066114 T^{3} + $$52\!\cdots\!87$$ T^{4} - $$48\!\cdots\!71$$ T^{5} + $$71\!\cdots\!86$$ T^{6} - $$48\!\cdots\!71$$ p^{9} T^{7} + $$52\!\cdots\!87$$ p^{18} T^{8} - 5968087950532066114 p^{27} T^{9} + 3690258693265 p^{36} T^{10} - 1960903 p^{45} T^{11} + p^{54} T^{12} $
	29	$ ( 1 - 2606146 T + 38672432258067 T^{2} - 74479453247155062316 T^{3} + 38672432258067 p^{9} T^{4} - 2606146 p^{18} T^{5} + p^{27} T^{6} )^{2} $
	31	$ 1 + 9377989 T + 21939918309201 T^{2} + 21641575053019249710 T^{3} - $$61\!\cdots\!35$$ p T^{4} - $$64\!\cdots\!11$$ T^{5} - $$50\!\cdots\!14$$ T^{6} - $$64\!\cdots\!11$$ p^{9} T^{7} - $$61\!\cdots\!35$$ p^{19} T^{8} + 21641575053019249710 p^{27} T^{9} + 21939918309201 p^{36} T^{10} + 9377989 p^{45} T^{11} + p^{54} T^{12} $
	37	$ 1 + 25814913 T + 77628810320679 T^{2} + $$82\!\cdots\!92$$ T^{3} + $$84\!\cdots\!13$$ T^{4} + $$71\!\cdots\!07$$ T^{5} - $$23\!\cdots\!14$$ T^{6} + $$71\!\cdots\!07$$ p^{9} T^{7} + $$84\!\cdots\!13$$ p^{18} T^{8} + $$82\!\cdots\!92$$ p^{27} T^{9} + 77628810320679 p^{36} T^{10} + 25814913 p^{45} T^{11} + p^{54} T^{12} $
	41	$ ( 1 + 209250 T + 750185439280791 T^{2} - $$10\!\cdots\!24$$ T^{3} + 750185439280791 p^{9} T^{4} + 209250 p^{18} T^{5} + p^{27} T^{6} )^{2} $
	43	$ ( 1 - 2944308 T + 1450396321305105 T^{2} - $$28\!\cdots\!96$$ T^{3} + 1450396321305105 p^{9} T^{4} - 2944308 p^{18} T^{5} + p^{27} T^{6} )^{2} $
	47	$ 1 + 48391269 T - 150981519635583 T^{2} - $$25\!\cdots\!86$$ T^{3} + $$44\!\cdots\!79$$ T^{4} - $$55\!\cdots\!99$$ T^{5} - $$32\!\cdots\!94$$ T^{6} - $$55\!\cdots\!99$$ p^{9} T^{7} + $$44\!\cdots\!79$$ p^{18} T^{8} - $$25\!\cdots\!86$$ p^{27} T^{9} - 150981519635583 p^{36} T^{10} + 48391269 p^{45} T^{11} + p^{54} T^{12} $
	53	$ 1 + 102186411 T - 2234880947557977 T^{2} - $$73\!\cdots\!52$$ T^{3} + $$92\!\cdots\!85$$ p T^{4} + $$12\!\cdots\!01$$ T^{5} - $$10\!\cdots\!02$$ T^{6} + $$12\!\cdots\!01$$ p^{9} T^{7} + $$92\!\cdots\!85$$ p^{19} T^{8} - $$73\!\cdots\!52$$ p^{27} T^{9} - 2234880947557977 p^{36} T^{10} + 102186411 p^{45} T^{11} + p^{54} T^{12} $
	59	$ 1 + 144220135 T - 11778297934463291 T^{2} - $$48\!\cdots\!50$$ T^{3} + $$40\!\cdots\!07$$ T^{4} + $$13\!\cdots\!35$$ T^{5} - $$27\!\cdots\!46$$ T^{6} + $$13\!\cdots\!35$$ p^{9} T^{7} + $$40\!\cdots\!07$$ p^{18} T^{8} - $$48\!\cdots\!50$$ p^{27} T^{9} - 11778297934463291 p^{36} T^{10} + 144220135 p^{45} T^{11} + p^{54} T^{12} $
	61	$ 1 - 280936871 T + 36546579532150335 T^{2} - $$22\!\cdots\!92$$ T^{3} - $$49\!\cdots\!67$$ T^{4} + $$36\!\cdots\!79$$ T^{5} - $$54\!\cdots\!66$$ T^{6} + $$36\!\cdots\!79$$ p^{9} T^{7} - $$49\!\cdots\!67$$ p^{18} T^{8} - $$22\!\cdots\!92$$ p^{27} T^{9} + 36546579532150335 p^{36} T^{10} - 280936871 p^{45} T^{11} + p^{54} T^{12} $
	67	$ 1 - 170710399 T - 4886073723330499 T^{2} + $$19\!\cdots\!22$$ T^{3} - $$43\!\cdots\!89$$ T^{4} + $$57\!\cdots\!09$$ T^{5} + $$23\!\cdots\!58$$ T^{6} + $$57\!\cdots\!09$$ p^{9} T^{7} - $$43\!\cdots\!89$$ p^{18} T^{8} + $$19\!\cdots\!22$$ p^{27} T^{9} - 4886073723330499 p^{36} T^{10} - 170710399 p^{45} T^{11} + p^{54} T^{12} $
	71	$ ( 1 + 469758688 T + 193321396251048789 T^{2} + $$44\!\cdots\!96$$ T^{3} + 193321396251048789 p^{9} T^{4} + 469758688 p^{18} T^{5} + p^{27} T^{6} )^{2} $
	73	$ 1 - 613838539 T + 123154674922492043 T^{2} - $$21\!\cdots\!72$$ T^{3} + $$80\!\cdots\!05$$ T^{4} - $$87\!\cdots\!29$$ T^{5} - $$15\!\cdots\!22$$ T^{6} - $$87\!\cdots\!29$$ p^{9} T^{7} + $$80\!\cdots\!05$$ p^{18} T^{8} - $$21\!\cdots\!72$$ p^{27} T^{9} + 123154674922492043 p^{36} T^{10} - 613838539 p^{45} T^{11} + p^{54} T^{12} $
	79	$ 1 - 197445809 T - 73823225419941879 T^{2} + $$16\!\cdots\!74$$ T^{3} - $$46\!\cdots\!53$$ T^{4} + $$31\!\cdots\!51$$ T^{5} + $$17\!\cdots\!26$$ T^{6} + $$31\!\cdots\!51$$ p^{9} T^{7} - $$46\!\cdots\!53$$ p^{18} T^{8} + $$16\!\cdots\!74$$ p^{27} T^{9} - 73823225419941879 p^{36} T^{10} - 197445809 p^{45} T^{11} + p^{54} T^{12} $
	83	$ ( 1 - 1074181436 T + 687268151069374857 T^{2} - $$31\!\cdots\!16$$ T^{3} + 687268151069374857 p^{9} T^{4} - 1074181436 p^{18} T^{5} + p^{27} T^{6} )^{2} $
	89	$ 1 + 805730427 T + 56355264573351195 T^{2} - $$43\!\cdots\!04$$ T^{3} - $$22\!\cdots\!47$$ T^{4} + $$35\!\cdots\!37$$ T^{5} + $$92\!\cdots\!26$$ T^{6} + $$35\!\cdots\!37$$ p^{9} T^{7} - $$22\!\cdots\!47$$ p^{18} T^{8} - $$43\!\cdots\!04$$ p^{27} T^{9} + 56355264573351195 p^{36} T^{10} + 805730427 p^{45} T^{11} + p^{54} T^{12} $
	97	$ ( 1 + 2262918094 T + 3679792263586873919 T^{2} + $$35\!\cdots\!36$$ T^{3} + 3679792263586873919 p^{9} T^{4} + 2262918094 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

−5.73338239090019793568707079706, −5.59870659118635399897433366381, −5.08818702526891777905448633257, −4.95561279675679261228459250392, −4.92202032246724826372680983039, −4.80654435155791862312698448151, −4.53720740828823928128683765122, −4.04448231382805036442114520929, −3.74091209834951488230750612234, −3.62692728255455887577748008638, −3.41107685557963836871453943047, −3.24269450629463210522821012921, −3.00673022655664051079499690641, −2.74177656289950821513241116519, −2.54814142923988436315365942212, −2.24654971724369714024042335345, −1.76690780674848964033978427034, −1.73156990059614754955231815976, −1.62916531378648847123345425858, −1.14764767466989110505691576374, −0.975087492190289130404358726401, −0.818037147728003755817435817184, −0.77543062979521664492894639107, −0.33122733547940426601478132591, −0.02943523185028159663653747819, 0.02943523185028159663653747819, 0.33122733547940426601478132591, 0.77543062979521664492894639107, 0.818037147728003755817435817184, 0.975087492190289130404358726401, 1.14764767466989110505691576374, 1.62916531378648847123345425858, 1.73156990059614754955231815976, 1.76690780674848964033978427034, 2.24654971724369714024042335345, 2.54814142923988436315365942212, 2.74177656289950821513241116519, 3.00673022655664051079499690641, 3.24269450629463210522821012921, 3.41107685557963836871453943047, 3.62692728255455887577748008638, 3.74091209834951488230750612234, 4.04448231382805036442114520929, 4.53720740828823928128683765122, 4.80654435155791862312698448151, 4.92202032246724826372680983039, 4.95561279675679261228459250392, 5.08818702526891777905448633257, 5.59870659118635399897433366381, 5.73338239090019793568707079706

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(0.03413597216\)
\(L(\frac12)\)	\(\approx\)	\(0.03413597216\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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