↑

Properties

Label 12-273e6-1.1-c1e6-0-3
Degree $12$
Conductor $4.140\times 10^{14}$
Sign $1$
Analytic cond. $107.309$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 3·4-s − 3·5-s + 2·8-s + 3·9-s + 3·11-s + 9·12-s + 6·13-s − 9·15-s + 6·16-s − 9·17-s − 3·19-s − 9·20-s + 6·24-s + 15·25-s − 2·27-s + 12·29-s − 12·31-s + 9·32-s + 9·33-s + 9·36-s + 3·37-s + 18·39-s − 6·40-s − 12·43-s + 9·44-s − 9·45-s + ⋯
L(s)  = 1  + 1.73·3-s + 3/2·4-s − 1.34·5-s + 0.707·8-s + 9-s + 0.904·11-s + 2.59·12-s + 1.66·13-s − 2.32·15-s + 3/2·16-s − 2.18·17-s − 0.688·19-s − 2.01·20-s + 1.22·24-s + 3·25-s − 0.384·27-s + 2.22·29-s − 2.15·31-s + 1.59·32-s + 1.56·33-s + 3/2·36-s + 0.493·37-s + 2.88·39-s − 0.948·40-s − 1.82·43-s + 1.35·44-s − 1.34·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 7^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(107.309\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 7^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.168875744\)
\(L(\frac12)\) \(\approx\) \(6.168875744\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T + T^{2} )^{3} \)
7 \( 1 - 17 T^{3} + p^{3} T^{6} \)
13 \( ( 1 - T )^{6} \)
good2 \( 1 - 3 T^{2} - p T^{3} + 3 T^{4} + 3 T^{5} - T^{6} + 3 p T^{7} + 3 p^{2} T^{8} - p^{4} T^{9} - 3 p^{4} T^{10} + p^{6} T^{12} \)
5 \( 1 + 3 T - 6 T^{2} - 13 T^{3} + 63 T^{4} + 12 p T^{5} - 259 T^{6} + 12 p^{2} T^{7} + 63 p^{2} T^{8} - 13 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T - 24 T^{2} + 31 T^{3} + 531 T^{4} - 30 p T^{5} - 6181 T^{6} - 30 p^{2} T^{7} + 531 p^{2} T^{8} + 31 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 9 T + 6 T^{2} + 29 T^{3} + 1443 T^{4} + 228 p T^{5} - 287 p T^{6} + 228 p^{2} T^{7} + 1443 p^{2} T^{8} + 29 p^{3} T^{9} + 6 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 39 T^{2} - 78 T^{3} + 1059 T^{4} + 939 T^{5} - 20986 T^{6} + 939 p T^{7} + 1059 p^{2} T^{8} - 78 p^{3} T^{9} - 39 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T^{2} + 18 T^{3} - 102 T^{4} - 54 T^{5} + 23587 T^{6} - 54 p T^{7} - 102 p^{2} T^{8} + 18 p^{3} T^{9} - 6 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 6 T + 36 T^{2} - 59 T^{3} + 36 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 12 T + 12 T^{2} + 58 T^{3} + 4176 T^{4} + 14040 T^{5} - 36777 T^{6} + 14040 p T^{7} + 4176 p^{2} T^{8} + 58 p^{3} T^{9} + 12 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T - 102 T^{2} + 109 T^{3} + 7551 T^{4} - 108 p T^{5} - 318051 T^{6} - 108 p^{2} T^{7} + 7551 p^{2} T^{8} + 109 p^{3} T^{9} - 102 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 96 T^{2} - 27 T^{3} + 96 p T^{4} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 6 T + 120 T^{2} + 465 T^{3} + 120 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 6 T - 69 T^{2} - 450 T^{3} + 2850 T^{4} + 10734 T^{5} - 98453 T^{6} + 10734 p T^{7} + 2850 p^{2} T^{8} - 450 p^{3} T^{9} - 69 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 3 T - 150 T^{2} + 153 T^{3} + 15909 T^{4} - 150 p T^{5} - 968195 T^{6} - 150 p^{2} T^{7} + 15909 p^{2} T^{8} + 153 p^{3} T^{9} - 150 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 3 T + 48 T^{2} - 153 T^{3} + 4227 T^{4} - 29514 T^{5} + 307915 T^{6} - 29514 p T^{7} + 4227 p^{2} T^{8} - 153 p^{3} T^{9} + 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 15 T + 114 T^{2} + 19 T^{3} - 6795 T^{4} - 55962 T^{5} - 457515 T^{6} - 55962 p T^{7} - 6795 p^{2} T^{8} + 19 p^{3} T^{9} + 114 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 9 T - 72 T^{2} + 889 T^{3} + 2961 T^{4} - 29232 T^{5} - 46797 T^{6} - 29232 p T^{7} + 2961 p^{2} T^{8} + 889 p^{3} T^{9} - 72 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 21 T + 267 T^{2} - 2385 T^{3} + 267 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 90 T^{2} - 142 T^{3} + 1530 T^{4} + 6390 T^{5} + 191775 T^{6} + 6390 p T^{7} + 1530 p^{2} T^{8} - 142 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 - 24 T + 240 T^{2} - 1522 T^{3} + 7848 T^{4} + 25272 T^{5} - 828081 T^{6} + 25272 p T^{7} + 7848 p^{2} T^{8} - 1522 p^{3} T^{9} + 240 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 12 T + 294 T^{2} + 2045 T^{3} + 294 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 15 T - 78 T^{2} - 617 T^{3} + 26127 T^{4} + 1032 p T^{5} - 21143 p T^{6} + 1032 p^{2} T^{7} + 26127 p^{2} T^{8} - 617 p^{3} T^{9} - 78 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 21 T + 375 T^{2} - 3805 T^{3} + 375 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.70764942363058714527513982113, −6.57353749757084942768590017156, −6.23162276516275759833462516153, −6.22492053912946405691276892648, −5.88513707041164825195723053751, −5.62092558154097685886872637320, −5.23409632077389567229641349997, −5.20479787124233952593703390590, −4.90238070902066351556779660012, −4.66976418244345027695388234427, −4.50563588954114942163101744582, −4.19465953983801381491283976564, −4.15094583458727162217313061058, −3.73439487558590327048987781051, −3.66877417844796264299611007698, −3.37395727014626571350323287754, −3.36326774071587102728682926231, −2.93351789920650251845436177056, −2.71962472121342457220243379205, −2.59059504175149591801815944807, −2.02929172386769140725004579519, −1.99493139200842315950946286956, −1.76776999518867029654714118961, −1.17507455901818428429550140513, −0.819986593895102864130326887990, 0.819986593895102864130326887990, 1.17507455901818428429550140513, 1.76776999518867029654714118961, 1.99493139200842315950946286956, 2.02929172386769140725004579519, 2.59059504175149591801815944807, 2.71962472121342457220243379205, 2.93351789920650251845436177056, 3.36326774071587102728682926231, 3.37395727014626571350323287754, 3.66877417844796264299611007698, 3.73439487558590327048987781051, 4.15094583458727162217313061058, 4.19465953983801381491283976564, 4.50563588954114942163101744582, 4.66976418244345027695388234427, 4.90238070902066351556779660012, 5.20479787124233952593703390590, 5.23409632077389567229641349997, 5.62092558154097685886872637320, 5.88513707041164825195723053751, 6.22492053912946405691276892648, 6.23162276516275759833462516153, 6.57353749757084942768590017156, 6.70764942363058714527513982113

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

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