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Properties

Label 12-3381e6-1.1-c1e6-0-3
Degree $12$
Conductor $1.494\times 10^{21}$
Sign $1$
Analytic cond. $3.87198\times 10^{8}$
Root an. cond. $5.19590$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 6·3-s + 2·4-s − 2·5-s − 18·6-s + 2·8-s + 21·9-s + 6·10-s − 3·11-s + 12·12-s − 12·15-s − 4·16-s − 8·17-s − 63·18-s − 19·19-s − 4·20-s + 9·22-s − 6·23-s + 12·24-s − 13·25-s + 56·27-s − 12·29-s + 36·30-s + 2·31-s − 18·33-s + 24·34-s + 42·36-s + ⋯
L(s)  = 1  − 2.12·2-s + 3.46·3-s + 4-s − 0.894·5-s − 7.34·6-s + 0.707·8-s + 7·9-s + 1.89·10-s − 0.904·11-s + 3.46·12-s − 3.09·15-s − 16-s − 1.94·17-s − 14.8·18-s − 4.35·19-s − 0.894·20-s + 1.91·22-s − 1.25·23-s + 2.44·24-s − 2.59·25-s + 10.7·27-s − 2.22·29-s + 6.57·30-s + 0.359·31-s − 3.13·33-s + 4.11·34-s + 7·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 7^{12} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(3.87198\times 10^{8}\)
Root analytic conductor: \(5.19590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{6} \cdot 7^{12} \cdot 23^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T )^{6} \)
7 \( 1 \)
23 \( ( 1 + T )^{6} \)
good2 \( 1 + 3 T + 7 T^{2} + 13 T^{3} + 23 T^{4} + 41 T^{5} + 15 p^{2} T^{6} + 41 p T^{7} + 23 p^{2} T^{8} + 13 p^{3} T^{9} + 7 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 2 T + 17 T^{2} + 34 T^{3} + 156 T^{4} + 292 T^{5} + 944 T^{6} + 292 p T^{7} + 156 p^{2} T^{8} + 34 p^{3} T^{9} + 17 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 3 T + 28 T^{2} + 43 T^{3} + 324 T^{4} + 431 T^{5} + 3610 T^{6} + 431 p T^{7} + 324 p^{2} T^{8} + 43 p^{3} T^{9} + 28 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 37 T^{2} + 36 T^{3} + 750 T^{4} + 924 T^{5} + 11452 T^{6} + 924 p T^{7} + 750 p^{2} T^{8} + 36 p^{3} T^{9} + 37 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 8 T + 72 T^{2} + 354 T^{3} + 2100 T^{4} + 8360 T^{5} + 41090 T^{6} + 8360 p T^{7} + 2100 p^{2} T^{8} + 354 p^{3} T^{9} + 72 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + p T + 184 T^{2} + 1263 T^{3} + 7512 T^{4} + 40731 T^{5} + 193142 T^{6} + 40731 p T^{7} + 7512 p^{2} T^{8} + 1263 p^{3} T^{9} + 184 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
29 \( 1 + 12 T + 108 T^{2} + 514 T^{3} + 1608 T^{4} - 2940 T^{5} - 34870 T^{6} - 2940 p T^{7} + 1608 p^{2} T^{8} + 514 p^{3} T^{9} + 108 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + 4446 T^{4} + 3142 T^{5} + 138366 T^{6} + 3142 p T^{7} + 4446 p^{2} T^{8} - 2 p^{4} T^{9} + 102 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 10 T + 198 T^{2} + 1566 T^{3} + 17098 T^{4} + 105634 T^{5} + 824306 T^{6} + 105634 p T^{7} + 17098 p^{2} T^{8} + 1566 p^{3} T^{9} + 198 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 3 T + 84 T^{2} + 197 T^{3} + 5372 T^{4} + 9739 T^{5} + 214354 T^{6} + 9739 p T^{7} + 5372 p^{2} T^{8} + 197 p^{3} T^{9} + 84 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 2 T + 133 T^{2} + 412 T^{3} + 8850 T^{4} + 37052 T^{5} + 428672 T^{6} + 37052 p T^{7} + 8850 p^{2} T^{8} + 412 p^{3} T^{9} + 133 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 7 T + 114 T^{2} + 933 T^{3} + 9391 T^{4} + 61470 T^{5} + 587004 T^{6} + 61470 p T^{7} + 9391 p^{2} T^{8} + 933 p^{3} T^{9} + 114 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 5 T + 233 T^{2} + 892 T^{3} + 24754 T^{4} + 74098 T^{5} + 1613358 T^{6} + 74098 p T^{7} + 24754 p^{2} T^{8} + 892 p^{3} T^{9} + 233 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 21 T + 383 T^{2} + 4874 T^{3} + 55520 T^{4} + 512716 T^{5} + 4340148 T^{6} + 512716 p T^{7} + 55520 p^{2} T^{8} + 4874 p^{3} T^{9} + 383 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 29 T + 519 T^{2} + 6864 T^{3} + 76610 T^{4} + 725530 T^{5} + 6068108 T^{6} + 725530 p T^{7} + 76610 p^{2} T^{8} + 6864 p^{3} T^{9} + 519 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 16 T + 331 T^{2} - 3372 T^{3} + 42142 T^{4} - 325644 T^{5} + 3289140 T^{6} - 325644 p T^{7} + 42142 p^{2} T^{8} - 3372 p^{3} T^{9} + 331 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 6 T + 81 T^{2} + 954 T^{3} + 16088 T^{4} + 109494 T^{5} + 789148 T^{6} + 109494 p T^{7} + 16088 p^{2} T^{8} + 954 p^{3} T^{9} + 81 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 8 T + 338 T^{2} - 2864 T^{3} + 51790 T^{4} - 413816 T^{5} + 4736906 T^{6} - 413816 p T^{7} + 51790 p^{2} T^{8} - 2864 p^{3} T^{9} + 338 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 12 T + 286 T^{2} + 2932 T^{3} + 46918 T^{4} + 386436 T^{5} + 4532214 T^{6} + 386436 p T^{7} + 46918 p^{2} T^{8} + 2932 p^{3} T^{9} + 286 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 24 T + 464 T^{2} + 5982 T^{3} + 73184 T^{4} + 760172 T^{5} + 7511590 T^{6} + 760172 p T^{7} + 73184 p^{2} T^{8} + 5982 p^{3} T^{9} + 464 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 18 T + 585 T^{2} + 7670 T^{3} + 135752 T^{4} + 1335970 T^{5} + 16308088 T^{6} + 1335970 p T^{7} + 135752 p^{2} T^{8} + 7670 p^{3} T^{9} + 585 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 22 T + 676 T^{2} + 9762 T^{3} + 170704 T^{4} + 1808334 T^{5} + 22292226 T^{6} + 1808334 p T^{7} + 170704 p^{2} T^{8} + 9762 p^{3} T^{9} + 676 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.60387599038693879389197001477, −4.43201808743421905150059339239, −4.30360964867235372109403523331, −4.29786875351333055888089363136, −4.25035573330722416586333746104, −4.21493543646419435044029706030, −4.14610531584599256449781275423, −3.74587017071873507121096677577, −3.61533533737376503704328919413, −3.60014920199487992867846640822, −3.42487891359107071306545194698, −3.34313880353716223785951020248, −3.05905584124788354508410806851, −2.82022903561912023989601551605, −2.72698661407350540706066793777, −2.40901266980875218710276725360, −2.39587780714431376081937638854, −2.29747751688301724102113944949, −2.20229774883464492330590479409, −1.93835737393683193798709089209, −1.78480316280788411719374453686, −1.58881995484391372019891699466, −1.50106218460774980427961585974, −1.33781953647340100063610603109, −1.18892304476342608801604582573, 0, 0, 0, 0, 0, 0, 1.18892304476342608801604582573, 1.33781953647340100063610603109, 1.50106218460774980427961585974, 1.58881995484391372019891699466, 1.78480316280788411719374453686, 1.93835737393683193798709089209, 2.20229774883464492330590479409, 2.29747751688301724102113944949, 2.39587780714431376081937638854, 2.40901266980875218710276725360, 2.72698661407350540706066793777, 2.82022903561912023989601551605, 3.05905584124788354508410806851, 3.34313880353716223785951020248, 3.42487891359107071306545194698, 3.60014920199487992867846640822, 3.61533533737376503704328919413, 3.74587017071873507121096677577, 4.14610531584599256449781275423, 4.21493543646419435044029706030, 4.25035573330722416586333746104, 4.29786875351333055888089363136, 4.30360964867235372109403523331, 4.43201808743421905150059339239, 4.60387599038693879389197001477

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

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