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Properties

Label 12-9680e6-1.1-c1e6-0-0
Degree $12$
Conductor $8.227\times 10^{23}$
Sign $1$
Analytic cond. $2.13262\times 10^{11}$
Root an. cond. $8.79176$
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  − 2·3-s − 6·5-s + 6·7-s − 2·9-s + 6·13-s + 12·15-s − 11·17-s − 11·19-s − 12·21-s − 18·23-s + 21·25-s + 9·27-s + 6·29-s − 31-s − 36·35-s + 4·37-s − 12·39-s + 4·41-s + 3·43-s + 12·45-s − 14·47-s + 49-s + 22·51-s + 14·53-s + 22·57-s − 2·59-s + 4·61-s + ⋯
L(s)  = 1  − 1.15·3-s − 2.68·5-s + 2.26·7-s − 2/3·9-s + 1.66·13-s + 3.09·15-s − 2.66·17-s − 2.52·19-s − 2.61·21-s − 3.75·23-s + 21/5·25-s + 1.73·27-s + 1.11·29-s − 0.179·31-s − 6.08·35-s + 0.657·37-s − 1.92·39-s + 0.624·41-s + 0.457·43-s + 1.78·45-s − 2.04·47-s + 1/7·49-s + 3.08·51-s + 1.92·53-s + 2.91·57-s − 0.260·59-s + 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 5^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(2.13262\times 10^{11}\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 5^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4475983806\)
\(L(\frac12)\) \(\approx\) \(0.4475983806\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 + T )^{6} \)
11 \( 1 \)
good3 \( 1 + 2 T + 2 p T^{2} + 7 T^{3} + 4 p T^{4} + 23 T^{5} + 38 T^{6} + 23 p T^{7} + 4 p^{3} T^{8} + 7 p^{3} T^{9} + 2 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 6 T + 5 p T^{2} - 149 T^{3} + 554 T^{4} - 1723 T^{5} + 5023 T^{6} - 1723 p T^{7} + 554 p^{2} T^{8} - 149 p^{3} T^{9} + 5 p^{5} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 6 T + 53 T^{2} - 249 T^{3} + 1412 T^{4} - 5379 T^{5} + 23041 T^{6} - 5379 p T^{7} + 1412 p^{2} T^{8} - 249 p^{3} T^{9} + 53 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 11 T + 108 T^{2} + 719 T^{3} + 4498 T^{4} + 21852 T^{5} + 99850 T^{6} + 21852 p T^{7} + 4498 p^{2} T^{8} + 719 p^{3} T^{9} + 108 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 11 T + 129 T^{2} + 918 T^{3} + 6319 T^{4} + 32631 T^{5} + 160757 T^{6} + 32631 p T^{7} + 6319 p^{2} T^{8} + 918 p^{3} T^{9} + 129 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 18 T + 188 T^{2} + 1501 T^{3} + 10104 T^{4} + 58774 T^{5} + 299719 T^{6} + 58774 p T^{7} + 10104 p^{2} T^{8} + 1501 p^{3} T^{9} + 188 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 6 T + 138 T^{2} - 851 T^{3} + 298 p T^{4} - 48657 T^{5} + 317862 T^{6} - 48657 p T^{7} + 298 p^{3} T^{8} - 851 p^{3} T^{9} + 138 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + T + 111 T^{2} + 8 p T^{3} + 6474 T^{4} + 14069 T^{5} + 250608 T^{6} + 14069 p T^{7} + 6474 p^{2} T^{8} + 8 p^{4} T^{9} + 111 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 4 T + 132 T^{2} - 403 T^{3} + 9146 T^{4} - 21636 T^{5} + 401873 T^{6} - 21636 p T^{7} + 9146 p^{2} T^{8} - 403 p^{3} T^{9} + 132 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 4 T + 70 T^{2} - 229 T^{3} + 4484 T^{4} - 16062 T^{5} + 232749 T^{6} - 16062 p T^{7} + 4484 p^{2} T^{8} - 229 p^{3} T^{9} + 70 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T + 95 T^{2} + 260 T^{3} + 4732 T^{4} + 17257 T^{5} + 289940 T^{6} + 17257 p T^{7} + 4732 p^{2} T^{8} + 260 p^{3} T^{9} + 95 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 14 T + 277 T^{2} + 2787 T^{3} + 31854 T^{4} + 239597 T^{5} + 1979131 T^{6} + 239597 p T^{7} + 31854 p^{2} T^{8} + 2787 p^{3} T^{9} + 277 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 14 T + 375 T^{2} - 3761 T^{3} + 54050 T^{4} - 400979 T^{5} + 3920079 T^{6} - 400979 p T^{7} + 54050 p^{2} T^{8} - 3761 p^{3} T^{9} + 375 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 2 T + 135 T^{2} + 333 T^{3} + 5894 T^{4} + 21621 T^{5} + 166061 T^{6} + 21621 p T^{7} + 5894 p^{2} T^{8} + 333 p^{3} T^{9} + 135 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 4 T + 129 T^{2} + 310 T^{3} + 7030 T^{4} + 57366 T^{5} + 421860 T^{6} + 57366 p T^{7} + 7030 p^{2} T^{8} + 310 p^{3} T^{9} + 129 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 11 T + 336 T^{2} + 2995 T^{3} + 51392 T^{4} + 365850 T^{5} + 4451546 T^{6} + 365850 p T^{7} + 51392 p^{2} T^{8} + 2995 p^{3} T^{9} + 336 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 7 T + 268 T^{2} + 2027 T^{3} + 38448 T^{4} + 253714 T^{5} + 3408986 T^{6} + 253714 p T^{7} + 38448 p^{2} T^{8} + 2027 p^{3} T^{9} + 268 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 9 T + 358 T^{2} - 2261 T^{3} + 53912 T^{4} - 254976 T^{5} + 4836866 T^{6} - 254976 p T^{7} + 53912 p^{2} T^{8} - 2261 p^{3} T^{9} + 358 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 36 T + 818 T^{2} - 13731 T^{3} + 186992 T^{4} - 2113647 T^{5} + 20335402 T^{6} - 2113647 p T^{7} + 186992 p^{2} T^{8} - 13731 p^{3} T^{9} + 818 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 45 T + 1177 T^{2} + 21836 T^{3} + 316222 T^{4} + 3737621 T^{5} + 37080044 T^{6} + 3737621 p T^{7} + 316222 p^{2} T^{8} + 21836 p^{3} T^{9} + 1177 p^{4} T^{10} + 45 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - T + 115 T^{2} - 899 T^{3} + 7226 T^{4} - 16021 T^{5} + 1054107 T^{6} - 16021 p T^{7} + 7226 p^{2} T^{8} - 899 p^{3} T^{9} + 115 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 20 T + 252 T^{2} - 1744 T^{3} + 6800 T^{4} + 4116 T^{5} - 327526 T^{6} + 4116 p T^{7} + 6800 p^{2} T^{8} - 1744 p^{3} T^{9} + 252 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
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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

−3.88324515612983174016677720289, −3.87281414114363094492645057075, −3.53157972520906985903761836119, −3.50443398430394104743470959979, −3.48514768993126474665846745103, −3.44622894217803074835314973632, −3.34469115981564830805821302263, −2.76108442213346926883932747766, −2.72176603642704429402501772591, −2.68354075089112795594847405675, −2.57291358681910736976615378772, −2.38105779858667443147727319724, −2.36211406185625796640986798483, −1.92683926558309815776409552994, −1.87400418147478726143578182248, −1.81978182023038862506030483755, −1.58595751706749187304547954519, −1.53765150161831239943367704010, −1.51525155345407261238792498016, −0.990929318509037426035827373345, −0.71970284091677976590493871173, −0.66820516707056679494995137469, −0.51949558430695909496294226570, −0.27007992197075977548903331705, −0.13348651705392643560160971405, 0.13348651705392643560160971405, 0.27007992197075977548903331705, 0.51949558430695909496294226570, 0.66820516707056679494995137469, 0.71970284091677976590493871173, 0.990929318509037426035827373345, 1.51525155345407261238792498016, 1.53765150161831239943367704010, 1.58595751706749187304547954519, 1.81978182023038862506030483755, 1.87400418147478726143578182248, 1.92683926558309815776409552994, 2.36211406185625796640986798483, 2.38105779858667443147727319724, 2.57291358681910736976615378772, 2.68354075089112795594847405675, 2.72176603642704429402501772591, 2.76108442213346926883932747766, 3.34469115981564830805821302263, 3.44622894217803074835314973632, 3.48514768993126474665846745103, 3.50443398430394104743470959979, 3.53157972520906985903761836119, 3.87281414114363094492645057075, 3.88324515612983174016677720289

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

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