Properties

Degree 12
Conductor $ 3^{6} \cdot 7^{6} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 9·3-s − 11·5-s − 9·6-s − 13·7-s + 21·8-s + 27·9-s + 11·10-s − 35·11-s + 124·13-s + 13·14-s − 99·15-s − 31·16-s − 48·17-s − 27·18-s + 202·19-s − 117·21-s + 35·22-s − 216·23-s + 189·24-s + 183·25-s − 124·26-s − 54·27-s + 106·29-s + 99·30-s + 95·31-s − 64·32-s + ⋯
L(s)  = 1  − 0.353·2-s + 1.73·3-s − 0.983·5-s − 0.612·6-s − 0.701·7-s + 0.928·8-s + 9-s + 0.347·10-s − 0.959·11-s + 2.64·13-s + 0.248·14-s − 1.70·15-s − 0.484·16-s − 0.684·17-s − 0.353·18-s + 2.43·19-s − 1.21·21-s + 0.339·22-s − 1.95·23-s + 1.60·24-s + 1.46·25-s − 0.935·26-s − 0.384·27-s + 0.678·29-s + 0.602·30-s + 0.550·31-s − 0.353·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85766121 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85766121 ^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(85766121\)    =    \(3^{6} \cdot 7^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{21} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(12,\ 85766121,\ (\ :[3/2]^{6}),\ 1)$
$L(2)$  $\approx$  $1.74113$
$L(\frac12)$  $\approx$  $1.74113$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7\}$,\(F_p(T)\) is a polynomial of degree 12. If $p \in \{3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad3 \( ( 1 - p T + p^{2} T^{2} )^{3} \)
7 \( 1 + 13 T + 236 T^{2} + 1735 p T^{3} + 236 p^{3} T^{4} + 13 p^{6} T^{5} + p^{9} T^{6} \)
good2 \( 1 + T + T^{2} - 5 p^{2} T^{3} - 5 p T^{4} + p^{6} T^{5} + 265 p^{2} T^{6} + p^{9} T^{7} - 5 p^{7} T^{8} - 5 p^{11} T^{9} + p^{12} T^{10} + p^{15} T^{11} + p^{18} T^{12} \)
5 \( 1 + 11 T - 62 T^{2} - 203 p T^{3} - 1208 p T^{4} - 54313 T^{5} + 121696 T^{6} - 54313 p^{3} T^{7} - 1208 p^{7} T^{8} - 203 p^{10} T^{9} - 62 p^{12} T^{10} + 11 p^{15} T^{11} + p^{18} T^{12} \)
11 \( 1 + 35 T - 1400 T^{2} - 113593 T^{3} - 198940 T^{4} + 87110135 T^{5} + 3928586038 T^{6} + 87110135 p^{3} T^{7} - 198940 p^{6} T^{8} - 113593 p^{9} T^{9} - 1400 p^{12} T^{10} + 35 p^{15} T^{11} + p^{18} T^{12} \)
13 \( ( 1 - 62 T + 7016 T^{2} - 253976 T^{3} + 7016 p^{3} T^{4} - 62 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
17 \( 1 + 48 T - 10035 T^{2} - 125232 T^{3} + 74409318 T^{4} - 234420432 T^{5} - 437742983351 T^{6} - 234420432 p^{3} T^{7} + 74409318 p^{6} T^{8} - 125232 p^{9} T^{9} - 10035 p^{12} T^{10} + 48 p^{15} T^{11} + p^{18} T^{12} \)
19 \( 1 - 202 T + 7946 T^{2} - 627636 T^{3} + 247297462 T^{4} - 17185599794 T^{5} + 349471935958 T^{6} - 17185599794 p^{3} T^{7} + 247297462 p^{6} T^{8} - 627636 p^{9} T^{9} + 7946 p^{12} T^{10} - 202 p^{15} T^{11} + p^{18} T^{12} \)
23 \( 1 + 216 T + 10827 T^{2} + 387864 T^{3} + 53856198 T^{4} - 24653558952 T^{5} - 5413409425505 T^{6} - 24653558952 p^{3} T^{7} + 53856198 p^{6} T^{8} + 387864 p^{9} T^{9} + 10827 p^{12} T^{10} + 216 p^{15} T^{11} + p^{18} T^{12} \)
29 \( ( 1 - 53 T + 52695 T^{2} - 3410210 T^{3} + 52695 p^{3} T^{4} - 53 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
31 \( 1 - 95 T - 70347 T^{2} + 3756594 T^{3} + 3398738767 T^{4} - 83374434539 T^{5} - 110906046363338 T^{6} - 83374434539 p^{3} T^{7} + 3398738767 p^{6} T^{8} + 3756594 p^{9} T^{9} - 70347 p^{12} T^{10} - 95 p^{15} T^{11} + p^{18} T^{12} \)
37 \( 1 + 262 T - 97404 T^{2} - 9678072 T^{3} + 12194182072 T^{4} + 680381910454 T^{5} - 605701122868778 T^{6} + 680381910454 p^{3} T^{7} + 12194182072 p^{6} T^{8} - 9678072 p^{9} T^{9} - 97404 p^{12} T^{10} + 262 p^{15} T^{11} + p^{18} T^{12} \)
41 \( ( 1 - 244 T + 187983 T^{2} - 33933832 T^{3} + 187983 p^{3} T^{4} - 244 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
43 \( ( 1 - 360 T + 166158 T^{2} - 38975294 T^{3} + 166158 p^{3} T^{4} - 360 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
47 \( 1 - 210 T - 20853 T^{2} + 83809446 T^{3} - 12756928590 T^{4} - 2596137940074 T^{5} + 3698984470026571 T^{6} - 2596137940074 p^{3} T^{7} - 12756928590 p^{6} T^{8} + 83809446 p^{9} T^{9} - 20853 p^{12} T^{10} - 210 p^{15} T^{11} + p^{18} T^{12} \)
53 \( 1 + 393 T - 211446 T^{2} - 23899125 T^{3} + 46453564620 T^{4} - 3425920762143 T^{5} - 9724787230272680 T^{6} - 3425920762143 p^{3} T^{7} + 46453564620 p^{6} T^{8} - 23899125 p^{9} T^{9} - 211446 p^{12} T^{10} + 393 p^{15} T^{11} + p^{18} T^{12} \)
59 \( 1 + 1143 T + 557208 T^{2} + 118327563 T^{3} - 14314666608 T^{4} - 458696646099 p T^{5} - 16891447327378130 T^{6} - 458696646099 p^{4} T^{7} - 14314666608 p^{6} T^{8} + 118327563 p^{9} T^{9} + 557208 p^{12} T^{10} + 1143 p^{15} T^{11} + p^{18} T^{12} \)
61 \( 1 - 70 T - 335143 T^{2} - 129510330 T^{3} + 42145697866 T^{4} + 25171752927730 T^{5} - 316289217432887 T^{6} + 25171752927730 p^{3} T^{7} + 42145697866 p^{6} T^{8} - 129510330 p^{9} T^{9} - 335143 p^{12} T^{10} - 70 p^{15} T^{11} + p^{18} T^{12} \)
67 \( 1 - 628 T - 202942 T^{2} + 436381932 T^{3} - 77667044702 T^{4} - 1097444953952 p T^{5} + 16943668917790 p^{2} T^{6} - 1097444953952 p^{4} T^{7} - 77667044702 p^{6} T^{8} + 436381932 p^{9} T^{9} - 202942 p^{12} T^{10} - 628 p^{15} T^{11} + p^{18} T^{12} \)
71 \( ( 1 - 318 T + 742929 T^{2} - 256167372 T^{3} + 742929 p^{3} T^{4} - 318 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
73 \( 1 + 988 T - 186552 T^{2} - 102237300 T^{3} + 281568890272 T^{4} - 16988127696596 T^{5} - 164639785652996186 T^{6} - 16988127696596 p^{3} T^{7} + 281568890272 p^{6} T^{8} - 102237300 p^{9} T^{9} - 186552 p^{12} T^{10} + 988 p^{15} T^{11} + p^{18} T^{12} \)
79 \( 1 + 861 T - 479895 T^{2} - 258646666 T^{3} + 325257480351 T^{4} - 27564282842211 T^{5} - 246706047980056146 T^{6} - 27564282842211 p^{3} T^{7} + 325257480351 p^{6} T^{8} - 258646666 p^{9} T^{9} - 479895 p^{12} T^{10} + 861 p^{15} T^{11} + p^{18} T^{12} \)
83 \( ( 1 - 519 T + 1583745 T^{2} - 545598870 T^{3} + 1583745 p^{3} T^{4} - 519 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
89 \( 1 + 1766 T + 725929 T^{2} - 728159446 T^{3} - 335534377858 T^{4} + 846551335831238 T^{5} + 1249625385561159997 T^{6} + 846551335831238 p^{3} T^{7} - 335534377858 p^{6} T^{8} - 728159446 p^{9} T^{9} + 725929 p^{12} T^{10} + 1766 p^{15} T^{11} + p^{18} T^{12} \)
97 \( ( 1 - 19 T + 2168419 T^{2} + 10094878 T^{3} + 2168419 p^{3} T^{4} - 19 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
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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

−10.48528527387765232793952545781, −9.974220816746377401584436604345, −9.670961843150247179522507042355, −9.418284138730015966557182287993, −9.332721178444415412381209580040, −8.868632507400261035981468343767, −8.722991757986355289959678706162, −8.445349222325137440194941910230, −8.119892828999079687461125020766, −7.952594403274095311263756886792, −7.79543120621363543950968364786, −7.32358574739059563002668662537, −7.18989616090266510259301698319, −6.93249795294349278054850478660, −6.04758516490262056537973288899, −6.03416664316828508044427140253, −5.88817185904019278055773599757, −5.16970437169280649720345940778, −4.49089376306427944573774469808, −4.35379303846738909285594228489, −3.77821328450074966821513454856, −3.38228381260687071477557465693, −3.05831546450298624563908847700, −2.48372776722578918437887004989, −1.28503207381873994848961598039, 1.28503207381873994848961598039, 2.48372776722578918437887004989, 3.05831546450298624563908847700, 3.38228381260687071477557465693, 3.77821328450074966821513454856, 4.35379303846738909285594228489, 4.49089376306427944573774469808, 5.16970437169280649720345940778, 5.88817185904019278055773599757, 6.03416664316828508044427140253, 6.04758516490262056537973288899, 6.93249795294349278054850478660, 7.18989616090266510259301698319, 7.32358574739059563002668662537, 7.79543120621363543950968364786, 7.952594403274095311263756886792, 8.119892828999079687461125020766, 8.445349222325137440194941910230, 8.722991757986355289959678706162, 8.868632507400261035981468343767, 9.332721178444415412381209580040, 9.418284138730015966557182287993, 9.670961843150247179522507042355, 9.974220816746377401584436604345, 10.48528527387765232793952545781

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

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