Properties

Label 12-14e6-1.1-c9e6-0-1
Degree $12$
Conductor $7529536$
Sign $1$
Analytic cond. $140537.$
Root an. cond. $2.68523$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 48·2-s + 71·3-s + 768·4-s − 1.08e3·5-s + 3.40e3·6-s − 6.79e3·7-s − 8.19e3·8-s + 2.74e4·9-s − 5.20e4·10-s + 2.55e3·11-s + 5.45e4·12-s + 3.61e4·13-s − 3.26e5·14-s − 7.70e4·15-s − 5.89e5·16-s + 2.07e4·17-s + 1.31e6·18-s + 1.22e6·19-s − 8.33e5·20-s − 4.82e5·21-s + 1.22e5·22-s − 1.96e6·23-s − 5.81e5·24-s + 3.08e6·25-s + 1.73e6·26-s + 4.69e6·27-s − 5.21e6·28-s + ⋯
L(s)  = 1  + 2.12·2-s + 0.506·3-s + 3/2·4-s − 0.776·5-s + 1.07·6-s − 1.06·7-s − 0.707·8-s + 1.39·9-s − 1.64·10-s + 0.0526·11-s + 0.759·12-s + 0.350·13-s − 2.26·14-s − 0.392·15-s − 9/4·16-s + 0.0602·17-s + 2.96·18-s + 2.14·19-s − 1.16·20-s − 0.541·21-s + 0.111·22-s − 1.46·23-s − 0.357·24-s + 1.58·25-s + 0.744·26-s + 1.69·27-s − 1.60·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7529536 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7529536 ^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7529536\)    =    \(2^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(140537.\)
Root analytic conductor: \(2.68523\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7529536,\ (\ :[9/2]^{6}),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(10.46880252\)
\(L(\frac12)\) \(\approx\) \(10.46880252\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p^{4} T + p^{8} T^{2} )^{3} \)
7 \( 1 + 6796 T - 635515 p^{2} T^{2} - 25759976 p^{5} T^{3} - 635515 p^{11} T^{4} + 6796 p^{18} T^{5} + p^{27} T^{6} \)
good3 \( 1 - 71 T - 22451 T^{2} - 381370 p T^{3} + 2296387 p^{4} T^{4} + 1415873303 p^{3} T^{5} - 17472309986 p^{4} T^{6} + 1415873303 p^{12} T^{7} + 2296387 p^{22} T^{8} - 381370 p^{28} T^{9} - 22451 p^{36} T^{10} - 71 p^{45} T^{11} + p^{54} T^{12} \)
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 \( ( 1 + 2262918094 T + 3679792263586873919 T^{2} + \)\(35\!\cdots\!36\)\( T^{3} + 3679792263586873919 p^{9} T^{4} + 2262918094 p^{18} T^{5} + p^{27} T^{6} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.384576392835783561870823139684, −9.279807580912588274149717239787, −8.851368649284368415064024725890, −8.381416905889511643979105141567, −8.219204014963705242011530924024, −7.967126662607652403091882262658, −7.24320180417348290639237061150, −6.95698128348836074955227327666, −6.93702263886968952486089283688, −6.89142093130583675919529851951, −6.14573766690199134306524848058, −5.49882248627150216124032924102, −5.36474338217687532026267019500, −5.35058363748220981566030719703, −4.94805609011934341831533276933, −4.00689270866333608640289995398, −3.88438116571029492503093097543, −3.78386950716898048662296284090, −3.74479079233610291051878332201, −3.06484796766458959250157072646, −2.46515160371259556897164843742, −2.25701673191906726057769793808, −1.32387113186369133833605957055, −0.811276141623970159815430949866, −0.41326273800254849928972115574, 0.41326273800254849928972115574, 0.811276141623970159815430949866, 1.32387113186369133833605957055, 2.25701673191906726057769793808, 2.46515160371259556897164843742, 3.06484796766458959250157072646, 3.74479079233610291051878332201, 3.78386950716898048662296284090, 3.88438116571029492503093097543, 4.00689270866333608640289995398, 4.94805609011934341831533276933, 5.35058363748220981566030719703, 5.36474338217687532026267019500, 5.49882248627150216124032924102, 6.14573766690199134306524848058, 6.89142093130583675919529851951, 6.93702263886968952486089283688, 6.95698128348836074955227327666, 7.24320180417348290639237061150, 7.967126662607652403091882262658, 8.219204014963705242011530924024, 8.381416905889511643979105141567, 8.851368649284368415064024725890, 9.279807580912588274149717239787, 9.384576392835783561870823139684

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.