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Properties

Label 12-567e6-1.1-c1e6-0-5
Degree $12$
Conductor $3.323\times 10^{16}$
Sign $1$
Analytic cond. $8613.12$
Root an. cond. $2.12779$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 6·4-s + 5-s − 4·7-s − 6·8-s − 4·10-s + 7·11-s − 2·13-s + 16·14-s + 11·16-s − 5·19-s + 6·20-s − 28·22-s + 6·23-s + 9·25-s + 8·26-s − 24·28-s + 13·29-s − 16·31-s − 22·32-s − 4·35-s + 8·37-s + 20·38-s − 6·40-s + 2·41-s + 9·43-s + 42·44-s + ⋯
L(s)  = 1  − 2.82·2-s + 3·4-s + 0.447·5-s − 1.51·7-s − 2.12·8-s − 1.26·10-s + 2.11·11-s − 0.554·13-s + 4.27·14-s + 11/4·16-s − 1.14·19-s + 1.34·20-s − 5.96·22-s + 1.25·23-s + 9/5·25-s + 1.56·26-s − 4.53·28-s + 2.41·29-s − 2.87·31-s − 3.88·32-s − 0.676·35-s + 1.31·37-s + 3.24·38-s − 0.948·40-s + 0.312·41-s + 1.37·43-s + 6.33·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{24} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(8613.12\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{24} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6207489084\)
\(L(\frac12)\) \(\approx\) \(0.6207489084\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + 4 T + 2 p T^{2} + 55 T^{3} + 2 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( ( 1 + p T + 3 T^{2} + 5 T^{3} + 3 p T^{4} + p^{3} T^{5} + p^{3} T^{6} )^{2} \)
5 \( 1 - T - 8 T^{2} + 17 T^{3} + 23 T^{4} - 52 T^{5} - 11 T^{6} - 52 p T^{7} + 23 p^{2} T^{8} + 17 p^{3} T^{9} - 8 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 7 T + 4 T^{2} - T^{3} + 431 T^{4} - 982 T^{5} - 893 T^{6} - 982 p T^{7} + 431 p^{2} T^{8} - p^{3} T^{9} + 4 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 2 T - 16 T^{2} + 30 T^{3} + 10 p T^{4} - 602 T^{5} - 1457 T^{6} - 602 p T^{7} + 10 p^{3} T^{8} + 30 p^{3} T^{9} - 16 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 18 T^{2} + 18 T^{3} + 18 T^{4} - 162 T^{5} + 4399 T^{6} - 162 p T^{7} + 18 p^{2} T^{8} + 18 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 + 5 T - 28 T^{2} - 3 p T^{3} + 997 T^{4} + 268 T^{5} - 22757 T^{6} + 268 p T^{7} + 997 p^{2} T^{8} - 3 p^{4} T^{9} - 28 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T - 36 T^{2} + 102 T^{3} + 1926 T^{4} - 2526 T^{5} - 42653 T^{6} - 2526 p T^{7} + 1926 p^{2} T^{8} + 102 p^{3} T^{9} - 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 13 T + 52 T^{2} + 5 T^{3} + 13 p T^{4} - 11986 T^{5} + 91837 T^{6} - 11986 p T^{7} + 13 p^{3} T^{8} + 5 p^{3} T^{9} + 52 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 8 T + 94 T^{2} + 427 T^{3} + 94 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 8 T - 42 T^{2} + 150 T^{3} + 3322 T^{4} - 1094 T^{5} - 153041 T^{6} - 1094 p T^{7} + 3322 p^{2} T^{8} + 150 p^{3} T^{9} - 42 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 2 T - 14 T^{2} - 482 T^{3} + 32 T^{4} + 3604 T^{5} + 167563 T^{6} + 3604 p T^{7} + 32 p^{2} T^{8} - 482 p^{3} T^{9} - 14 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 9 T - 36 T^{2} + 293 T^{3} + 2601 T^{4} - 4824 T^{5} - 126453 T^{6} - 4824 p T^{7} + 2601 p^{2} T^{8} + 293 p^{3} T^{9} - 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 9 T + 99 T^{2} + 855 T^{3} + 99 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 - 24 T + 252 T^{2} - 2202 T^{3} + 20916 T^{4} - 146148 T^{5} + 883411 T^{6} - 146148 p T^{7} + 20916 p^{2} T^{8} - 2202 p^{3} T^{9} + 252 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( ( 1 - 15 T + 243 T^{2} - 1851 T^{3} + 243 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + T + 134 T^{2} + T^{3} + 134 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 - 14 T + 254 T^{2} - 1907 T^{3} + 254 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 - 3 T + 105 T^{2} - 669 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 7 T - 36 T^{2} + 513 T^{3} + 733 T^{4} - 46082 T^{5} - 8831 T^{6} - 46082 p T^{7} + 733 p^{2} T^{8} + 513 p^{3} T^{9} - 36 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 - 6 T + 168 T^{2} - 821 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 3 T - 60 T^{2} + 2247 T^{3} - 4647 T^{4} - 70968 T^{5} + 2191939 T^{6} - 70968 p T^{7} - 4647 p^{2} T^{8} + 2247 p^{3} T^{9} - 60 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 5 T - 149 T^{2} + 68 T^{3} + 12785 T^{4} - 45481 T^{5} - 1321850 T^{6} - 45481 p T^{7} + 12785 p^{2} T^{8} + 68 p^{3} T^{9} - 149 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 14 T - 111 T^{2} + 1086 T^{3} + 26782 T^{4} - 124358 T^{5} - 1914107 T^{6} - 124358 p T^{7} + 26782 p^{2} T^{8} + 1086 p^{3} T^{9} - 111 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.68243195044757005334614324587, −5.66153130214581415464833478639, −5.60625095215384137566938806175, −5.60272252631564518666869170724, −5.12243449147293868634570045290, −4.84030127523679185061860983234, −4.68411611565113668024993684365, −4.52079994950196417827139982353, −4.44019114723370315064225436348, −4.10162593327172015482885255041, −3.80223298498029479262976145162, −3.77784846303759789923358263789, −3.36379724793488595934464380471, −3.33612676183136691489729675805, −3.18909649567410138811963264065, −3.04513398179748031451229800023, −2.55505911822643246321723439498, −2.39551526956320008215740957342, −2.07568563717449980737874038988, −1.88862971718474534095367212400, −1.87290111519287062066616107922, −0.894053063294227549349329949713, −0.876359553905391236420701543855, −0.66737402094787153529529167900, −0.61705857262361039095979511366, 0.61705857262361039095979511366, 0.66737402094787153529529167900, 0.876359553905391236420701543855, 0.894053063294227549349329949713, 1.87290111519287062066616107922, 1.88862971718474534095367212400, 2.07568563717449980737874038988, 2.39551526956320008215740957342, 2.55505911822643246321723439498, 3.04513398179748031451229800023, 3.18909649567410138811963264065, 3.33612676183136691489729675805, 3.36379724793488595934464380471, 3.77784846303759789923358263789, 3.80223298498029479262976145162, 4.10162593327172015482885255041, 4.44019114723370315064225436348, 4.52079994950196417827139982353, 4.68411611565113668024993684365, 4.84030127523679185061860983234, 5.12243449147293868634570045290, 5.60272252631564518666869170724, 5.60625095215384137566938806175, 5.66153130214581415464833478639, 5.68243195044757005334614324587

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

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