Properties

Degree 12
Conductor $ 43^{6} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 7·3-s + 5·4-s + 43·5-s + 42·6-s + 8·7-s − 40·8-s − 16·9-s + 258·10-s − 28·11-s + 35·12-s + 56·13-s + 48·14-s + 301·15-s − 141·16-s + 19·17-s − 96·18-s − 75·19-s + 215·20-s + 56·21-s − 168·22-s + 131·23-s − 280·24-s + 602·25-s + 336·26-s − 210·27-s + 40·28-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.34·3-s + 5/8·4-s + 3.84·5-s + 2.85·6-s + 0.431·7-s − 1.76·8-s − 0.592·9-s + 8.15·10-s − 0.767·11-s + 0.841·12-s + 1.19·13-s + 0.916·14-s + 5.18·15-s − 2.20·16-s + 0.271·17-s − 1.25·18-s − 0.905·19-s + 2.40·20-s + 0.581·21-s − 1.62·22-s + 1.18·23-s − 2.38·24-s + 4.81·25-s + 2.53·26-s − 1.49·27-s + 0.269·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(43^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{43} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((12,\ 43^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\)
\(L(2)\)  \(\approx\)  \(19.0140\)
\(L(\frac12)\)  \(\approx\)  \(19.0140\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 12. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad43 \( ( 1 + p T )^{6} \)
good2 \( 1 - 3 p T + 31 T^{2} - 29 p^{2} T^{3} + 221 p T^{4} - 23 p^{6} T^{5} + 1167 p^{2} T^{6} - 23 p^{9} T^{7} + 221 p^{7} T^{8} - 29 p^{11} T^{9} + 31 p^{12} T^{10} - 3 p^{16} T^{11} + p^{18} T^{12} \)
3 \( 1 - 7 T + 65 T^{2} - 119 p T^{3} + 2599 T^{4} - 15158 T^{5} + 96098 T^{6} - 15158 p^{3} T^{7} + 2599 p^{6} T^{8} - 119 p^{10} T^{9} + 65 p^{12} T^{10} - 7 p^{15} T^{11} + p^{18} T^{12} \)
5 \( 1 - 43 T + 1247 T^{2} - 26367 T^{3} + 452519 T^{4} - 6421366 T^{5} + 77975134 T^{6} - 6421366 p^{3} T^{7} + 452519 p^{6} T^{8} - 26367 p^{9} T^{9} + 1247 p^{12} T^{10} - 43 p^{15} T^{11} + p^{18} T^{12} \)
7 \( 1 - 8 T + 886 T^{2} - 4080 T^{3} + 63165 p T^{4} - 3221432 T^{5} + 186242508 T^{6} - 3221432 p^{3} T^{7} + 63165 p^{7} T^{8} - 4080 p^{9} T^{9} + 886 p^{12} T^{10} - 8 p^{15} T^{11} + p^{18} T^{12} \)
11 \( 1 + 28 T + 3144 T^{2} + 41080 T^{3} + 3977536 T^{4} + 285452 p T^{5} + 4111332998 T^{6} + 285452 p^{4} T^{7} + 3977536 p^{6} T^{8} + 41080 p^{9} T^{9} + 3144 p^{12} T^{10} + 28 p^{15} T^{11} + p^{18} T^{12} \)
13 \( 1 - 56 T + 8776 T^{2} - 530884 T^{3} + 37473880 T^{4} - 2150765192 T^{5} + 100690118558 T^{6} - 2150765192 p^{3} T^{7} + 37473880 p^{6} T^{8} - 530884 p^{9} T^{9} + 8776 p^{12} T^{10} - 56 p^{15} T^{11} + p^{18} T^{12} \)
17 \( 1 - 19 T + 23143 T^{2} - 543393 T^{3} + 241535186 T^{4} - 5650313095 T^{5} + 1493450611759 T^{6} - 5650313095 p^{3} T^{7} + 241535186 p^{6} T^{8} - 543393 p^{9} T^{9} + 23143 p^{12} T^{10} - 19 p^{15} T^{11} + p^{18} T^{12} \)
19 \( 1 + 75 T + 38259 T^{2} + 2365781 T^{3} + 629552155 T^{4} + 31151517862 T^{5} + 5681041321490 T^{6} + 31151517862 p^{3} T^{7} + 629552155 p^{6} T^{8} + 2365781 p^{9} T^{9} + 38259 p^{12} T^{10} + 75 p^{15} T^{11} + p^{18} T^{12} \)
23 \( 1 - 131 T + 52195 T^{2} - 3677795 T^{3} + 974730114 T^{4} - 33210519163 T^{5} + 11858751245947 T^{6} - 33210519163 p^{3} T^{7} + 974730114 p^{6} T^{8} - 3677795 p^{9} T^{9} + 52195 p^{12} T^{10} - 131 p^{15} T^{11} + p^{18} T^{12} \)
29 \( 1 - 515 T + 204583 T^{2} - 58068807 T^{3} + 13900558631 T^{4} - 2733986934494 T^{5} + 464005745217070 T^{6} - 2733986934494 p^{3} T^{7} + 13900558631 p^{6} T^{8} - 58068807 p^{9} T^{9} + 204583 p^{12} T^{10} - 515 p^{15} T^{11} + p^{18} T^{12} \)
31 \( 1 - 237 T + 125373 T^{2} - 21016589 T^{3} + 7321362670 T^{4} - 1031063540341 T^{5} + 275109610824401 T^{6} - 1031063540341 p^{3} T^{7} + 7321362670 p^{6} T^{8} - 21016589 p^{9} T^{9} + 125373 p^{12} T^{10} - 237 p^{15} T^{11} + p^{18} T^{12} \)
37 \( 1 - 269 T + 126311 T^{2} - 30748693 T^{3} + 8177560635 T^{4} - 1302357216602 T^{5} + 425119347961066 T^{6} - 1302357216602 p^{3} T^{7} + 8177560635 p^{6} T^{8} - 30748693 p^{9} T^{9} + 126311 p^{12} T^{10} - 269 p^{15} T^{11} + p^{18} T^{12} \)
41 \( 1 - 471 T + 349763 T^{2} - 112600045 T^{3} + 50459129866 T^{4} - 12922113800443 T^{5} + 4372223136871043 T^{6} - 12922113800443 p^{3} T^{7} + 50459129866 p^{6} T^{8} - 112600045 p^{9} T^{9} + 349763 p^{12} T^{10} - 471 p^{15} T^{11} + p^{18} T^{12} \)
47 \( 1 - 415 T + 421631 T^{2} - 116866317 T^{3} + 77411983523 T^{4} - 16052264514750 T^{5} + 9154052369892234 T^{6} - 16052264514750 p^{3} T^{7} + 77411983523 p^{6} T^{8} - 116866317 p^{9} T^{9} + 421631 p^{12} T^{10} - 415 p^{15} T^{11} + p^{18} T^{12} \)
53 \( 1 - 450 T + 321704 T^{2} - 149982378 T^{3} + 77929548632 T^{4} - 28654270442506 T^{5} + 12746558079363422 T^{6} - 28654270442506 p^{3} T^{7} + 77929548632 p^{6} T^{8} - 149982378 p^{9} T^{9} + 321704 p^{12} T^{10} - 450 p^{15} T^{11} + p^{18} T^{12} \)
59 \( 1 - 356 T + 1153950 T^{2} - 347607228 T^{3} + 570632518215 T^{4} - 139273869185096 T^{5} + 154351054402988548 T^{6} - 139273869185096 p^{3} T^{7} + 570632518215 p^{6} T^{8} - 347607228 p^{9} T^{9} + 1153950 p^{12} T^{10} - 356 p^{15} T^{11} + p^{18} T^{12} \)
61 \( 1 + 1328 T + 1795994 T^{2} + 1454429624 T^{3} + 1136828745699 T^{4} + 649067768079368 T^{5} + 354455789917301172 T^{6} + 649067768079368 p^{3} T^{7} + 1136828745699 p^{6} T^{8} + 1454429624 p^{9} T^{9} + 1795994 p^{12} T^{10} + 1328 p^{15} T^{11} + p^{18} T^{12} \)
67 \( 1 + 632 T + 927628 T^{2} + 253029932 T^{3} + 179674044568 T^{4} - 62778009874096 T^{5} - 5212390617577006 T^{6} - 62778009874096 p^{3} T^{7} + 179674044568 p^{6} T^{8} + 253029932 p^{9} T^{9} + 927628 p^{12} T^{10} + 632 p^{15} T^{11} + p^{18} T^{12} \)
71 \( 1 + 144 T + 863230 T^{2} + 72744496 T^{3} + 475313592223 T^{4} + 62333254020128 T^{5} + 209643801201434276 T^{6} + 62333254020128 p^{3} T^{7} + 475313592223 p^{6} T^{8} + 72744496 p^{9} T^{9} + 863230 p^{12} T^{10} + 144 p^{15} T^{11} + p^{18} T^{12} \)
73 \( 1 - 864 T + 1080658 T^{2} - 439706488 T^{3} + 458263245867 T^{4} - 209583356925592 T^{5} + 222637509631027524 T^{6} - 209583356925592 p^{3} T^{7} + 458263245867 p^{6} T^{8} - 439706488 p^{9} T^{9} + 1080658 p^{12} T^{10} - 864 p^{15} T^{11} + p^{18} T^{12} \)
79 \( 1 + 1613 T + 2731081 T^{2} + 3130876751 T^{3} + 3268965374139 T^{4} + 2799662350208018 T^{5} + 2111366737833161318 T^{6} + 2799662350208018 p^{3} T^{7} + 3268965374139 p^{6} T^{8} + 3130876751 p^{9} T^{9} + 2731081 p^{12} T^{10} + 1613 p^{15} T^{11} + p^{18} T^{12} \)
83 \( 1 + 682 T + 1120004 T^{2} + 1783287782 T^{3} + 1291656570448 T^{4} + 1149121488958882 T^{5} + 1227368803599345682 T^{6} + 1149121488958882 p^{3} T^{7} + 1291656570448 p^{6} T^{8} + 1783287782 p^{9} T^{9} + 1120004 p^{12} T^{10} + 682 p^{15} T^{11} + p^{18} T^{12} \)
89 \( 1 - 3378 T + 7851850 T^{2} - 13055260850 T^{3} + 17370332054203 T^{4} - 19017874978895668 T^{5} + 17369747195795800052 T^{6} - 19017874978895668 p^{3} T^{7} + 17370332054203 p^{6} T^{8} - 13055260850 p^{9} T^{9} + 7851850 p^{12} T^{10} - 3378 p^{15} T^{11} + p^{18} T^{12} \)
97 \( 1 + 55 T + 2496871 T^{2} - 940317011 T^{3} + 3317883854770 T^{4} - 1706175158728421 T^{5} + 3579842987764450575 T^{6} - 1706175158728421 p^{3} T^{7} + 3317883854770 p^{6} T^{8} - 940317011 p^{9} T^{9} + 2496871 p^{12} T^{10} + 55 p^{15} T^{11} + p^{18} T^{12} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.957652713935566756127072289002, −8.446643352522337062228285424485, −8.253588136707017446862521317260, −8.150115608227947422150358821166, −8.037654713362921648572062740172, −7.65373109568997716005663893938, −7.04045787079637627980159620522, −6.73845211370475354777199837234, −6.37730194293215393695599291147, −6.28400983251440728734163494312, −5.95291669955338180588365150927, −5.83976303890908103743692306354, −5.76253988828029033383401738990, −5.15330130451801237043567001255, −5.02358574249926313807328819262, −4.92099093914120585181442148797, −4.38867928280824437368629379094, −4.18402275716880518852285964863, −3.99448284606684095514476385754, −3.01978979060949517898100838695, −2.90431880886791947813790758202, −2.77663447143581097021597185479, −2.28025810147586165925541206680, −1.83297946493770618690766357526, −1.12975603138289563510018708281, 1.12975603138289563510018708281, 1.83297946493770618690766357526, 2.28025810147586165925541206680, 2.77663447143581097021597185479, 2.90431880886791947813790758202, 3.01978979060949517898100838695, 3.99448284606684095514476385754, 4.18402275716880518852285964863, 4.38867928280824437368629379094, 4.92099093914120585181442148797, 5.02358574249926313807328819262, 5.15330130451801237043567001255, 5.76253988828029033383401738990, 5.83976303890908103743692306354, 5.95291669955338180588365150927, 6.28400983251440728734163494312, 6.37730194293215393695599291147, 6.73845211370475354777199837234, 7.04045787079637627980159620522, 7.65373109568997716005663893938, 8.037654713362921648572062740172, 8.150115608227947422150358821166, 8.253588136707017446862521317260, 8.446643352522337062228285424485, 8.957652713935566756127072289002

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

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