Properties

Label 12-1452e6-1.1-c3e6-0-3
Degree $12$
Conductor $9.371\times 10^{18}$
Sign $1$
Analytic cond. $3.95363\times 10^{11}$
Root an. cond. $9.25585$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 18·3-s + 13·5-s − 23·7-s + 189·9-s − 66·13-s + 234·15-s − 44·17-s − 270·19-s − 414·21-s − 124·23-s − 225·25-s + 1.51e3·27-s − 141·29-s − 253·31-s − 299·35-s + 288·37-s − 1.18e3·39-s − 428·41-s − 1.00e3·43-s + 2.45e3·45-s − 674·47-s − 674·49-s − 792·51-s − 773·53-s − 4.86e3·57-s − 17·59-s − 1.01e3·61-s + ⋯
L(s)  = 1  + 3.46·3-s + 1.16·5-s − 1.24·7-s + 7·9-s − 1.40·13-s + 4.02·15-s − 0.627·17-s − 3.26·19-s − 4.30·21-s − 1.12·23-s − 9/5·25-s + 10.7·27-s − 0.902·29-s − 1.46·31-s − 1.44·35-s + 1.27·37-s − 4.87·39-s − 1.63·41-s − 3.56·43-s + 8.13·45-s − 2.09·47-s − 1.96·49-s − 2.17·51-s − 2.00·53-s − 11.2·57-s − 0.0375·59-s − 2.13·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 3^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(3.95363\times 10^{11}\)
Root analytic conductor: \(9.25585\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{12} \cdot 3^{6} \cdot 11^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - p T )^{6} \)
11 \( 1 \)
good5 \( 1 - 13 T + 394 T^{2} - 4762 T^{3} + 66771 T^{4} - 161794 p T^{5} + 1648371 p T^{6} - 161794 p^{4} T^{7} + 66771 p^{6} T^{8} - 4762 p^{9} T^{9} + 394 p^{12} T^{10} - 13 p^{15} T^{11} + p^{18} T^{12} \)
7 \( 1 + 23 T + 1203 T^{2} + 29455 T^{3} + 854150 T^{4} + 16134743 T^{5} + 381227879 T^{6} + 16134743 p^{3} T^{7} + 854150 p^{6} T^{8} + 29455 p^{9} T^{9} + 1203 p^{12} T^{10} + 23 p^{15} T^{11} + p^{18} T^{12} \)
13 \( 1 + 66 T + 9156 T^{2} + 397210 T^{3} + 37189212 T^{4} + 1304230290 T^{5} + 98945146966 T^{6} + 1304230290 p^{3} T^{7} + 37189212 p^{6} T^{8} + 397210 p^{9} T^{9} + 9156 p^{12} T^{10} + 66 p^{15} T^{11} + p^{18} T^{12} \)
17 \( 1 + 44 T + 8477 T^{2} + 923978 T^{3} + 73719314 T^{4} + 5322292822 T^{5} + 518892195868 T^{6} + 5322292822 p^{3} T^{7} + 73719314 p^{6} T^{8} + 923978 p^{9} T^{9} + 8477 p^{12} T^{10} + 44 p^{15} T^{11} + p^{18} T^{12} \)
19 \( 1 + 270 T + 54379 T^{2} + 7189200 T^{3} + 828805960 T^{4} + 76701402200 T^{5} + 6836893032940 T^{6} + 76701402200 p^{3} T^{7} + 828805960 p^{6} T^{8} + 7189200 p^{9} T^{9} + 54379 p^{12} T^{10} + 270 p^{15} T^{11} + p^{18} T^{12} \)
23 \( 1 + 124 T + 56246 T^{2} + 5841418 T^{3} + 1461985958 T^{4} + 124480161308 T^{5} + 22544006048986 T^{6} + 124480161308 p^{3} T^{7} + 1461985958 p^{6} T^{8} + 5841418 p^{9} T^{9} + 56246 p^{12} T^{10} + 124 p^{15} T^{11} + p^{18} T^{12} \)
29 \( 1 + 141 T + 81699 T^{2} + 6922305 T^{3} + 3207355035 T^{4} + 208251057846 T^{5} + 91506047625506 T^{6} + 208251057846 p^{3} T^{7} + 3207355035 p^{6} T^{8} + 6922305 p^{9} T^{9} + 81699 p^{12} T^{10} + 141 p^{15} T^{11} + p^{18} T^{12} \)
31 \( 1 + 253 T + 109342 T^{2} + 28776244 T^{3} + 6990733475 T^{4} + 1405494050192 T^{5} + 272928641739047 T^{6} + 1405494050192 p^{3} T^{7} + 6990733475 p^{6} T^{8} + 28776244 p^{9} T^{9} + 109342 p^{12} T^{10} + 253 p^{15} T^{11} + p^{18} T^{12} \)
37 \( 1 - 288 T + 113632 T^{2} - 16989204 T^{3} + 5587963372 T^{4} - 443922413544 T^{5} + 209875557448466 T^{6} - 443922413544 p^{3} T^{7} + 5587963372 p^{6} T^{8} - 16989204 p^{9} T^{9} + 113632 p^{12} T^{10} - 288 p^{15} T^{11} + p^{18} T^{12} \)
41 \( 1 + 428 T + 342502 T^{2} + 97721310 T^{3} + 46604467954 T^{4} + 10098941823160 T^{5} + 3831460290264446 T^{6} + 10098941823160 p^{3} T^{7} + 46604467954 p^{6} T^{8} + 97721310 p^{9} T^{9} + 342502 p^{12} T^{10} + 428 p^{15} T^{11} + p^{18} T^{12} \)
43 \( 1 + 1006 T + 839516 T^{2} + 451564982 T^{3} + 210850471928 T^{4} + 75467001003262 T^{5} + 23891921313875546 T^{6} + 75467001003262 p^{3} T^{7} + 210850471928 p^{6} T^{8} + 451564982 p^{9} T^{9} + 839516 p^{12} T^{10} + 1006 p^{15} T^{11} + p^{18} T^{12} \)
47 \( 1 + 674 T + 695747 T^{2} + 320835338 T^{3} + 186998437214 T^{4} + 63820662447172 T^{5} + 26109183953673868 T^{6} + 63820662447172 p^{3} T^{7} + 186998437214 p^{6} T^{8} + 320835338 p^{9} T^{9} + 695747 p^{12} T^{10} + 674 p^{15} T^{11} + p^{18} T^{12} \)
53 \( 1 + 773 T + 477138 T^{2} + 235519794 T^{3} + 110426076459 T^{4} + 47857805874514 T^{5} + 21153543553472927 T^{6} + 47857805874514 p^{3} T^{7} + 110426076459 p^{6} T^{8} + 235519794 p^{9} T^{9} + 477138 p^{12} T^{10} + 773 p^{15} T^{11} + p^{18} T^{12} \)
59 \( 1 + 17 T + 820830 T^{2} + 60199956 T^{3} + 324707774607 T^{4} + 31376437157332 T^{5} + 80783553548705063 T^{6} + 31376437157332 p^{3} T^{7} + 324707774607 p^{6} T^{8} + 60199956 p^{9} T^{9} + 820830 p^{12} T^{10} + 17 p^{15} T^{11} + p^{18} T^{12} \)
61 \( 1 + 1016 T + 1304445 T^{2} + 914111632 T^{3} + 686466635712 T^{4} + 361967629532286 T^{5} + 200619996554838492 T^{6} + 361967629532286 p^{3} T^{7} + 686466635712 p^{6} T^{8} + 914111632 p^{9} T^{9} + 1304445 p^{12} T^{10} + 1016 p^{15} T^{11} + p^{18} T^{12} \)
67 \( 1 - 1836 T + 2911487 T^{2} - 2993982412 T^{3} + 40089169512 p T^{4} - 1865911532872528 T^{5} + 1142997412289947448 T^{6} - 1865911532872528 p^{3} T^{7} + 40089169512 p^{7} T^{8} - 2993982412 p^{9} T^{9} + 2911487 p^{12} T^{10} - 1836 p^{15} T^{11} + p^{18} T^{12} \)
71 \( 1 + 208 T + 1057837 T^{2} + 313047330 T^{3} + 523863849334 T^{4} + 232419716793410 T^{5} + 196255427246242076 T^{6} + 232419716793410 p^{3} T^{7} + 523863849334 p^{6} T^{8} + 313047330 p^{9} T^{9} + 1057837 p^{12} T^{10} + 208 p^{15} T^{11} + p^{18} T^{12} \)
73 \( 1 + 1521 T + 1978017 T^{2} + 1865063465 T^{3} + 1635891720495 T^{4} + 1212008590314846 T^{5} + 819948560883492526 T^{6} + 1212008590314846 p^{3} T^{7} + 1635891720495 p^{6} T^{8} + 1865063465 p^{9} T^{9} + 1978017 p^{12} T^{10} + 1521 p^{15} T^{11} + p^{18} T^{12} \)
79 \( 1 + 1425 T + 2595565 T^{2} + 2476471173 T^{3} + 2664591621430 T^{4} + 1976242795490865 T^{5} + 1615943392022944529 T^{6} + 1976242795490865 p^{3} T^{7} + 2664591621430 p^{6} T^{8} + 2476471173 p^{9} T^{9} + 2595565 p^{12} T^{10} + 1425 p^{15} T^{11} + p^{18} T^{12} \)
83 \( 1 + 3065 T + 7293643 T^{2} + 11322539823 T^{3} + 14679052898512 T^{4} + 14593020561845737 T^{5} + 12405601977882032561 T^{6} + 14593020561845737 p^{3} T^{7} + 14679052898512 p^{6} T^{8} + 11322539823 p^{9} T^{9} + 7293643 p^{12} T^{10} + 3065 p^{15} T^{11} + p^{18} T^{12} \)
89 \( 1 - 1444 T + 3814498 T^{2} - 4476581350 T^{3} + 73476855126 p T^{4} - 5852332783988720 T^{5} + 6130218815755081374 T^{6} - 5852332783988720 p^{3} T^{7} + 73476855126 p^{7} T^{8} - 4476581350 p^{9} T^{9} + 3814498 p^{12} T^{10} - 1444 p^{15} T^{11} + p^{18} T^{12} \)
97 \( 1 + 3887 T + 10303514 T^{2} + 19219632286 T^{3} + 29318869183703 T^{4} + 36159101867906234 T^{5} + 37928473655870402519 T^{6} + 36159101867906234 p^{3} T^{7} + 29318869183703 p^{6} T^{8} + 19219632286 p^{9} T^{9} + 10303514 p^{12} T^{10} + 3887 p^{15} T^{11} + p^{18} T^{12} \)
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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

−5.19794103235540326677959283779, −4.64908394078915974950650330066, −4.55867661085310352955329580268, −4.49462929010717053625927894229, −4.30042536721481508847453519324, −4.25893390783284540291982143565, −4.20429019002484175278901395435, −3.82839954627897407391174216343, −3.68606743948245557998579395262, −3.44129221178913265019969049025, −3.41738221871753586258339994566, −3.41320644799502148767569716668, −3.22846694948864126415516078177, −2.72393088161907501226091609733, −2.65776229510916167669269408234, −2.63414632707389927597894897753, −2.49861834453438752216727843561, −2.28562296463619814834989540744, −2.24894758527071368031501966574, −1.74290141971405579050444446675, −1.68095426589459545084822919458, −1.66758701117129657366021418542, −1.41501527567056069675377497482, −1.40426013326333939147657147673, −1.32450312035896482256139278348, 0, 0, 0, 0, 0, 0, 1.32450312035896482256139278348, 1.40426013326333939147657147673, 1.41501527567056069675377497482, 1.66758701117129657366021418542, 1.68095426589459545084822919458, 1.74290141971405579050444446675, 2.24894758527071368031501966574, 2.28562296463619814834989540744, 2.49861834453438752216727843561, 2.63414632707389927597894897753, 2.65776229510916167669269408234, 2.72393088161907501226091609733, 3.22846694948864126415516078177, 3.41320644799502148767569716668, 3.41738221871753586258339994566, 3.44129221178913265019969049025, 3.68606743948245557998579395262, 3.82839954627897407391174216343, 4.20429019002484175278901395435, 4.25893390783284540291982143565, 4.30042536721481508847453519324, 4.49462929010717053625927894229, 4.55867661085310352955329580268, 4.64908394078915974950650330066, 5.19794103235540326677959283779

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

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