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Properties

Label 12-3e36-1.1-c1e6-0-4
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  + 6·2-s + 18·4-s + 3·5-s − 9·7-s + 36·8-s + 18·10-s + 15·11-s − 9·13-s − 54·14-s + 54·16-s − 9·17-s + 3·19-s + 54·20-s + 90·22-s + 12·23-s − 9·25-s − 54·26-s − 162·28-s − 3·29-s − 9·31-s + 69·32-s − 54·34-s − 27·35-s + 3·37-s + 18·38-s + 108·40-s + 21·41-s + ⋯
L(s)  = 1  + 4.24·2-s + 9·4-s + 1.34·5-s − 3.40·7-s + 12.7·8-s + 5.69·10-s + 4.52·11-s − 2.49·13-s − 14.4·14-s + 27/2·16-s − 2.18·17-s + 0.688·19-s + 12.0·20-s + 19.1·22-s + 2.50·23-s − 9/5·25-s − 10.5·26-s − 30.6·28-s − 0.557·29-s − 1.61·31-s + 12.1·32-s − 9.26·34-s − 4.56·35-s + 0.493·37-s + 2.91·38-s + 17.0·40-s + 3.27·41-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

\(L(1)\) \(\approx\) \(36.50361927\)
\(L(\frac12)\) \(\approx\) \(36.50361927\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 p T + 9 p T^{2} - 9 p^{2} T^{3} + 27 p T^{4} - 69 T^{5} + 91 T^{6} - 69 p T^{7} + 27 p^{3} T^{8} - 9 p^{5} T^{9} + 9 p^{5} T^{10} - 3 p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 3 T + 18 T^{2} - 54 T^{3} + 189 T^{4} - 453 T^{5} + 1189 T^{6} - 453 p T^{7} + 189 p^{2} T^{8} - 54 p^{3} T^{9} + 18 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 9 T + 45 T^{2} + 173 T^{3} + 594 T^{4} + 1782 T^{5} + 4881 T^{6} + 1782 p T^{7} + 594 p^{2} T^{8} + 173 p^{3} T^{9} + 45 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 15 T + 9 p T^{2} - 333 T^{3} + 162 T^{4} + 4098 T^{5} - 21023 T^{6} + 4098 p T^{7} + 162 p^{2} T^{8} - 333 p^{3} T^{9} + 9 p^{5} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 9 T + 45 T^{2} + 191 T^{3} + 810 T^{4} + 2916 T^{5} + 10065 T^{6} + 2916 p T^{7} + 810 p^{2} T^{8} + 191 p^{3} T^{9} + 45 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 12 T + 99 T^{2} - 675 T^{3} + 4455 T^{4} - 24051 T^{5} + 121258 T^{6} - 24051 p T^{7} + 4455 p^{2} T^{8} - 675 p^{3} T^{9} + 99 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T - 36 T^{2} - 270 T^{3} - 945 T^{4} + 5637 T^{5} + 70993 T^{6} + 5637 p T^{7} - 945 p^{2} T^{8} - 270 p^{3} T^{9} - 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 9 T + 72 T^{2} + 650 T^{3} + 4293 T^{4} + 24921 T^{5} + 159267 T^{6} + 24921 p T^{7} + 4293 p^{2} T^{8} + 650 p^{3} T^{9} + 72 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 21 T + 207 T^{2} - 1089 T^{3} + 378 T^{4} + 52980 T^{5} - 510623 T^{6} + 52980 p T^{7} + 378 p^{2} T^{8} - 1089 p^{3} T^{9} + 207 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 9 T + 72 T^{2} + 794 T^{3} + 5697 T^{4} + 34965 T^{5} + 270831 T^{6} + 34965 p T^{7} + 5697 p^{2} T^{8} + 794 p^{3} T^{9} + 72 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 15 T + 180 T^{2} - 1476 T^{3} + 10665 T^{4} - 64275 T^{5} + 398503 T^{6} - 64275 p T^{7} + 10665 p^{2} T^{8} - 1476 p^{3} T^{9} + 180 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 3 T - 36 T^{2} + 972 T^{3} - 3402 T^{4} - 23781 T^{5} + 658585 T^{6} - 23781 p T^{7} - 3402 p^{2} T^{8} + 972 p^{3} T^{9} - 36 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 27 T + 270 T^{2} + 794 T^{3} - 6183 T^{4} - 65691 T^{5} - 407355 T^{6} - 65691 p T^{7} - 6183 p^{2} T^{8} + 794 p^{3} T^{9} + 270 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 27 T + 324 T^{2} - 2320 T^{3} + 4941 T^{4} + 121797 T^{5} - 1636989 T^{6} + 121797 p T^{7} + 4941 p^{2} T^{8} - 2320 p^{3} T^{9} + 324 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 - 18 T + 153 T^{2} - 853 T^{3} - 5697 T^{4} + 179145 T^{5} - 1911282 T^{6} + 179145 p T^{7} - 5697 p^{2} T^{8} - 853 p^{3} T^{9} + 153 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 15 T + 72 T^{2} + 360 T^{3} - 3159 T^{4} - 68631 T^{5} + 989803 T^{6} - 68631 p T^{7} - 3159 p^{2} T^{8} + 360 p^{3} T^{9} + 72 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
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 \( 1 + 36 T + 396 T^{2} - 610 T^{3} - 33480 T^{4} + 134136 T^{5} + 5366595 T^{6} + 134136 p T^{7} - 33480 p^{2} T^{8} - 610 p^{3} T^{9} + 396 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.52390280477022704590930241657, −5.49862282659755609221003639342, −5.33269060486007741017773960552, −5.12878695926888308230188449306, −4.65406348395341026597175109249, −4.53048941685112571740676081969, −4.50979878941300395611775285849, −4.24592421787371157229347102037, −4.20491600599441402437801394430, −4.06624149375713664370003393497, −3.87058737135610416430504888321, −3.67488832941401382268702382640, −3.58674815090456858342734524728, −3.34418341427621822412155128061, −3.21265623618778195207849547519, −3.16180772946964283637857601367, −2.47714323123982049486035058775, −2.44603266873717591675447308919, −2.43363313207530016194271628879, −2.40932787013470605733790149564, −2.05901429862715177125290657500, −1.63788609229976415449466335929, −1.13271226667858826554727503923, −1.04486583328236232143977769941, −0.41452916364938497017210930200, 0.41452916364938497017210930200, 1.04486583328236232143977769941, 1.13271226667858826554727503923, 1.63788609229976415449466335929, 2.05901429862715177125290657500, 2.40932787013470605733790149564, 2.43363313207530016194271628879, 2.44603266873717591675447308919, 2.47714323123982049486035058775, 3.16180772946964283637857601367, 3.21265623618778195207849547519, 3.34418341427621822412155128061, 3.58674815090456858342734524728, 3.67488832941401382268702382640, 3.87058737135610416430504888321, 4.06624149375713664370003393497, 4.20491600599441402437801394430, 4.24592421787371157229347102037, 4.50979878941300395611775285849, 4.53048941685112571740676081969, 4.65406348395341026597175109249, 5.12878695926888308230188449306, 5.33269060486007741017773960552, 5.49862282659755609221003639342, 5.52390280477022704590930241657

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

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