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Properties

Label 12-4026e6-1.1-c1e6-0-1
Degree $12$
Conductor $4.258\times 10^{21}$
Sign $1$
Analytic cond. $1.10383\times 10^{9}$
Root an. cond. $5.66990$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s − 6·3-s + 21·4-s − 6·5-s − 36·6-s + 7-s + 56·8-s + 21·9-s − 36·10-s − 6·11-s − 126·12-s + 2·13-s + 6·14-s + 36·15-s + 126·16-s − 13·17-s + 126·18-s + 19-s − 126·20-s − 6·21-s − 36·22-s − 11·23-s − 336·24-s + 7·25-s + 12·26-s − 56·27-s + 21·28-s + ⋯
L(s)  = 1  + 4.24·2-s − 3.46·3-s + 21/2·4-s − 2.68·5-s − 14.6·6-s + 0.377·7-s + 19.7·8-s + 7·9-s − 11.3·10-s − 1.80·11-s − 36.3·12-s + 0.554·13-s + 1.60·14-s + 9.29·15-s + 63/2·16-s − 3.15·17-s + 29.6·18-s + 0.229·19-s − 28.1·20-s − 1.30·21-s − 7.67·22-s − 2.29·23-s − 68.5·24-s + 7/5·25-s + 2.35·26-s − 10.7·27-s + 3.96·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 11^{6} \cdot 61^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 11^{6} \cdot 61^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 11^{6} \cdot 61^{6}\)
Sign: $1$
Analytic conductor: \(1.10383\times 10^{9}\)
Root analytic conductor: \(5.66990\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 11^{6} \cdot 61^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad2 \( ( 1 - T )^{6} \)
3 \( ( 1 + T )^{6} \)
11 \( ( 1 + T )^{6} \)
61 \( ( 1 - T )^{6} \)
good5 \( 1 + 6 T + 29 T^{2} + 117 T^{3} + 372 T^{4} + 1023 T^{5} + 2516 T^{6} + 1023 p T^{7} + 372 p^{2} T^{8} + 117 p^{3} T^{9} + 29 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - T + 24 T^{2} - 5 p T^{3} + 307 T^{4} - 415 T^{5} + 2648 T^{6} - 415 p T^{7} + 307 p^{2} T^{8} - 5 p^{4} T^{9} + 24 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 2 T + 60 T^{2} - 108 T^{3} + 1668 T^{4} - 2487 T^{5} + 27484 T^{6} - 2487 p T^{7} + 1668 p^{2} T^{8} - 108 p^{3} T^{9} + 60 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 13 T + 101 T^{2} + 479 T^{3} + 1466 T^{4} + 1913 T^{5} + 885 T^{6} + 1913 p T^{7} + 1466 p^{2} T^{8} + 479 p^{3} T^{9} + 101 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - T + 20 T^{2} - 113 T^{3} + 113 T^{4} + 189 T^{5} + 7116 T^{6} + 189 p T^{7} + 113 p^{2} T^{8} - 113 p^{3} T^{9} + 20 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 11 T + 113 T^{2} + 593 T^{3} + 3232 T^{4} + 9762 T^{5} + 53552 T^{6} + 9762 p T^{7} + 3232 p^{2} T^{8} + 593 p^{3} T^{9} + 113 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 14 T + 169 T^{2} + 1529 T^{3} + 11755 T^{4} + 77809 T^{5} + 446222 T^{6} + 77809 p T^{7} + 11755 p^{2} T^{8} + 1529 p^{3} T^{9} + 169 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 5 T + 94 T^{2} + 368 T^{3} + 4111 T^{4} + 12623 T^{5} + 133948 T^{6} + 12623 p T^{7} + 4111 p^{2} T^{8} + 368 p^{3} T^{9} + 94 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 6 T + 157 T^{2} + 492 T^{3} + 9721 T^{4} + 15403 T^{5} + 396453 T^{6} + 15403 p T^{7} + 9721 p^{2} T^{8} + 492 p^{3} T^{9} + 157 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 25 T + 371 T^{2} + 3752 T^{3} + 30437 T^{4} + 211679 T^{5} + 1388622 T^{6} + 211679 p T^{7} + 30437 p^{2} T^{8} + 3752 p^{3} T^{9} + 371 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 19 T + 289 T^{2} - 2719 T^{3} + 23018 T^{4} - 149938 T^{5} + 1049156 T^{6} - 149938 p T^{7} + 23018 p^{2} T^{8} - 2719 p^{3} T^{9} + 289 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 10 T + 260 T^{2} + 2040 T^{3} + 29132 T^{4} + 178673 T^{5} + 1795972 T^{6} + 178673 p T^{7} + 29132 p^{2} T^{8} + 2040 p^{3} T^{9} + 260 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 17 T + 370 T^{2} + 77 p T^{3} + 50917 T^{4} + 409681 T^{5} + 3620938 T^{6} + 409681 p T^{7} + 50917 p^{2} T^{8} + 77 p^{4} T^{9} + 370 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 14 T + 254 T^{2} + 2290 T^{3} + 25172 T^{4} + 172633 T^{5} + 1628952 T^{6} + 172633 p T^{7} + 25172 p^{2} T^{8} + 2290 p^{3} T^{9} + 254 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 12 T + 260 T^{2} - 2692 T^{3} + 36308 T^{4} - 306087 T^{5} + 2989418 T^{6} - 306087 p T^{7} + 36308 p^{2} T^{8} - 2692 p^{3} T^{9} + 260 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 6 T + 279 T^{2} - 1265 T^{3} + 36464 T^{4} - 129133 T^{5} + 3056036 T^{6} - 129133 p T^{7} + 36464 p^{2} T^{8} - 1265 p^{3} T^{9} + 279 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 29 T + 649 T^{2} + 9813 T^{3} + 128754 T^{4} + 1353010 T^{5} + 12719488 T^{6} + 1353010 p T^{7} + 128754 p^{2} T^{8} + 9813 p^{3} T^{9} + 649 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 24 T + 318 T^{2} + 2904 T^{3} + 29300 T^{4} + 312897 T^{5} + 3138590 T^{6} + 312897 p T^{7} + 29300 p^{2} T^{8} + 2904 p^{3} T^{9} + 318 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 9 T - 3 T^{2} - 625 T^{3} + 1196 T^{4} + 19774 T^{5} + 127172 T^{6} + 19774 p T^{7} + 1196 p^{2} T^{8} - 625 p^{3} T^{9} - 3 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 4 T + 171 T^{2} + 1179 T^{3} + 28542 T^{4} + 151791 T^{5} + 2823324 T^{6} + 151791 p T^{7} + 28542 p^{2} T^{8} + 1179 p^{3} T^{9} + 171 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 5 T + 334 T^{2} + 1183 T^{3} + 56850 T^{4} + 146998 T^{5} + 6383687 T^{6} + 146998 p T^{7} + 56850 p^{2} T^{8} + 1183 p^{3} T^{9} + 334 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.81625550579031294061474149282, −4.53848426953079850473838517314, −4.44071384247411066180378883859, −4.40120866847078279051432903483, −4.32548047180078634043398804189, −4.17700260136035382122556047690, −4.12397396829483378373278269079, −3.76524095095624779966625101928, −3.62409989859867215843129123480, −3.60148431227892455869481849570, −3.58816325402585388787910826041, −3.58372307040851870079948599879, −3.46989814236019027658431186077, −2.92001937987053975942732179071, −2.74034741034405961075940659534, −2.60813869936107821381264463025, −2.54865183163965335659207529303, −2.38926441536564147898421262551, −2.26033909355797893785173711367, −1.73411199290295858794871102990, −1.69316590072136543261404620486, −1.58346327313316676378374994541, −1.53643018894913713552099416987, −1.33769512714066077661193302865, −1.23726867704080035218000680397, 0, 0, 0, 0, 0, 0, 1.23726867704080035218000680397, 1.33769512714066077661193302865, 1.53643018894913713552099416987, 1.58346327313316676378374994541, 1.69316590072136543261404620486, 1.73411199290295858794871102990, 2.26033909355797893785173711367, 2.38926441536564147898421262551, 2.54865183163965335659207529303, 2.60813869936107821381264463025, 2.74034741034405961075940659534, 2.92001937987053975942732179071, 3.46989814236019027658431186077, 3.58372307040851870079948599879, 3.58816325402585388787910826041, 3.60148431227892455869481849570, 3.62409989859867215843129123480, 3.76524095095624779966625101928, 4.12397396829483378373278269079, 4.17700260136035382122556047690, 4.32548047180078634043398804189, 4.40120866847078279051432903483, 4.44071384247411066180378883859, 4.53848426953079850473838517314, 4.81625550579031294061474149282

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

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