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Properties

Label 12-3e36-1.1-c1e6-0-10
Degree $12$
Conductor $1.501\times 10^{17}$
Sign $1$
Analytic cond. $38907.0$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 6·5-s + 9·7-s − 9·8-s + 18·10-s + 3·11-s + 9·13-s + 27·14-s − 9·16-s + 9·17-s + 3·19-s + 9·22-s + 15·23-s + 27·25-s + 27·26-s − 15·29-s + 9·31-s + 3·32-s + 27·34-s + 54·35-s + 3·37-s + 9·38-s − 54·40-s − 3·41-s + 9·43-s + 45·46-s − 15·47-s + ⋯
L(s)  = 1  + 2.12·2-s + 2.68·5-s + 3.40·7-s − 3.18·8-s + 5.69·10-s + 0.904·11-s + 2.49·13-s + 7.21·14-s − 9/4·16-s + 2.18·17-s + 0.688·19-s + 1.91·22-s + 3.12·23-s + 27/5·25-s + 5.29·26-s − 2.78·29-s + 1.61·31-s + 0.530·32-s + 4.63·34-s + 9.12·35-s + 0.493·37-s + 1.45·38-s − 8.53·40-s − 0.468·41-s + 1.37·43-s + 6.63·46-s − 2.18·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{36}\)
Sign: $1$
Analytic conductor: \(38907.0\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{36} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(52.75729872\)
\(L(\frac12)\) \(\approx\) \(52.75729872\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 3 T + 9 T^{2} - 9 p T^{3} + 9 p^{2} T^{4} - 57 T^{5} + 91 T^{6} - 57 p T^{7} + 9 p^{4} T^{8} - 9 p^{4} T^{9} + 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 6 T + 9 T^{2} + 27 T^{3} - 99 T^{4} - 33 T^{5} + 514 T^{6} - 33 p T^{7} - 99 p^{2} T^{8} + 27 p^{3} T^{9} + 9 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 9 T + 36 T^{2} - 10 p T^{3} + 513 T^{5} - 1923 T^{6} + 513 p T^{7} - 10 p^{4} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T + 36 T^{2} - 126 T^{3} + 684 T^{4} - 2451 T^{5} + 8677 T^{6} - 2451 p T^{7} + 684 p^{2} T^{8} - 126 p^{3} T^{9} + 36 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 9 T + 36 T^{2} - 4 p T^{3} - 162 T^{4} + 1917 T^{5} - 8889 T^{6} + 1917 p T^{7} - 162 p^{2} T^{8} - 4 p^{4} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{3} \)
19 \( 1 - 3 T - 24 T^{2} + 131 T^{3} + 117 T^{4} - 1116 T^{5} + 3003 T^{6} - 1116 p T^{7} + 117 p^{2} T^{8} + 131 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 15 T + 90 T^{2} - 162 T^{3} - 1017 T^{4} + 10101 T^{5} - 54485 T^{6} + 10101 p T^{7} - 1017 p^{2} T^{8} - 162 p^{3} T^{9} + 90 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 15 T + 144 T^{2} + 1188 T^{3} + 8163 T^{4} + 49317 T^{5} + 277705 T^{6} + 49317 p T^{7} + 8163 p^{2} T^{8} + 1188 p^{3} T^{9} + 144 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 9 T + 36 T^{2} + 164 T^{3} - 27 p T^{4} - 1377 T^{5} + 53805 T^{6} - 1377 p T^{7} - 27 p^{3} T^{8} + 164 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T - 78 T^{2} + 5 p T^{3} + 3681 T^{4} - 4194 T^{5} - 141339 T^{6} - 4194 p T^{7} + 3681 p^{2} T^{8} + 5 p^{4} T^{9} - 78 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 3 T + 36 T^{2} - 72 T^{3} + 18 p T^{4} + 1119 T^{5} + 93799 T^{6} + 1119 p T^{7} + 18 p^{3} T^{8} - 72 p^{3} T^{9} + 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 9 T + 36 T^{2} + 308 T^{3} - 1485 T^{4} + 2403 T^{5} + 82749 T^{6} + 2403 p T^{7} - 1485 p^{2} T^{8} + 308 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 15 T + 90 T^{2} + 126 T^{3} + 765 T^{4} + 40155 T^{5} + 461953 T^{6} + 40155 p T^{7} + 765 p^{2} T^{8} + 126 p^{3} T^{9} + 90 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 18 T + 240 T^{2} + 1989 T^{3} + 240 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 6 T + 36 T^{2} - 999 T^{3} + 4329 T^{4} - 34053 T^{5} + 664957 T^{6} - 34053 p T^{7} + 4329 p^{2} T^{8} - 999 p^{3} T^{9} + 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 18 T + 153 T^{2} - 745 T^{3} - 3915 T^{4} + 107703 T^{5} - 1034862 T^{6} + 107703 p T^{7} - 3915 p^{2} T^{8} - 745 p^{3} T^{9} + 153 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 9 T + 126 T^{2} + 758 T^{3} + 8937 T^{4} + 25569 T^{5} + 327585 T^{6} + 25569 p T^{7} + 8937 p^{2} T^{8} + 758 p^{3} T^{9} + 126 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 9 T + 30 T^{2} - 99 T^{3} - 3531 T^{4} - 5580 T^{5} + 200671 T^{6} - 5580 p T^{7} - 3531 p^{2} T^{8} - 99 p^{3} T^{9} + 30 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 6 T - 114 T^{2} - 58 T^{3} + 8676 T^{4} - 27576 T^{5} - 826869 T^{6} - 27576 p T^{7} + 8676 p^{2} T^{8} - 58 p^{3} T^{9} - 114 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 9 T + 144 T^{2} - 1096 T^{3} + 19413 T^{4} - 150957 T^{5} + 1716951 T^{6} - 150957 p T^{7} + 19413 p^{2} T^{8} - 1096 p^{3} T^{9} + 144 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 21 T + 198 T^{2} - 1386 T^{3} + 1953 T^{4} + 125691 T^{5} - 1739735 T^{6} + 125691 p T^{7} + 1953 p^{2} T^{8} - 1386 p^{3} T^{9} + 198 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 78 T^{2} - 1998 T^{3} - 858 T^{4} + 77922 T^{5} + 1866463 T^{6} + 77922 p T^{7} - 858 p^{2} T^{8} - 1998 p^{3} T^{9} - 78 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 - 36 T + 576 T^{2} - 4498 T^{3} - 4860 T^{4} + 578988 T^{5} - 7833165 T^{6} + 578988 p T^{7} - 4860 p^{2} T^{8} - 4498 p^{3} T^{9} + 576 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.54030650392550113103358919537, −5.38271481210261889698086677718, −5.07172410766409037885572274729, −4.97162961189259660385637688532, −4.85536853846760588303286308595, −4.83795533021789518208528383653, −4.82526419880881362444051952003, −4.34748643567407616672423956551, −4.29422544905455761499982613800, −4.14684791338315242107953857003, −3.82905360015540609018045200664, −3.68136969772610726992969723728, −3.63018272960433066693562651135, −3.27037687333684196147575573079, −3.14360952021820085942727265727, −2.89814604064344455828142901284, −2.65832921717516611532379786473, −2.60277388037505361012481497282, −2.07359958268742720250439342234, −1.68475625065057087344874778634, −1.59003019231981783883332441344, −1.43703667937789102383478970181, −1.32792992481616282169851449775, −1.00415111810951139295098173432, −0.911335230770448958832362849511, 0.911335230770448958832362849511, 1.00415111810951139295098173432, 1.32792992481616282169851449775, 1.43703667937789102383478970181, 1.59003019231981783883332441344, 1.68475625065057087344874778634, 2.07359958268742720250439342234, 2.60277388037505361012481497282, 2.65832921717516611532379786473, 2.89814604064344455828142901284, 3.14360952021820085942727265727, 3.27037687333684196147575573079, 3.63018272960433066693562651135, 3.68136969772610726992969723728, 3.82905360015540609018045200664, 4.14684791338315242107953857003, 4.29422544905455761499982613800, 4.34748643567407616672423956551, 4.82526419880881362444051952003, 4.83795533021789518208528383653, 4.85536853846760588303286308595, 4.97162961189259660385637688532, 5.07172410766409037885572274729, 5.38271481210261889698086677718, 5.54030650392550113103358919537

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

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