↑

Properties

Label 12-9680e6-1.1-c1e6-0-6
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 $6$

Origins

Origins of factors

Downloads

Learn more

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: \(6\)
Selberg data: \((12,\ 2^{24} \cdot 5^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( 1 \)
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 \( 1 - 20 T + 252 T^{2} - 1744 T^{3} + 6800 T^{4} + 4116 T^{5} - 327526 T^{6} + 4116 p T^{7} + 6800 p^{2} T^{8} - 1744 p^{3} T^{9} + 252 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
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

−4.12791480301829488089968477563, −3.98543970992048378310616274773, −3.91099934232330730894930405259, −3.90184846867746142194137751302, −3.72248788517454990329908771380, −3.57425443371537060788018701913, −3.50460351102748501008402021925, −3.35967957535718434805338237627, −3.18901805274289107406295828260, −3.15778673568530512557954271399, −3.09255822209714699303524283435, −3.09038462668101423216846823686, −2.85132765061511690938851817358, −2.45844540440870857260481476757, −2.44600672700995550358820198639, −2.33105241941224269420676036752, −2.27194331202934570702947831511, −1.99110709242913170760573882646, −1.74264767301961022639920967991, −1.54331932399030158757641909001, −1.40625297428764331289782453449, −1.06726842875133247139894967281, −1.00202886326342819353812226526, −0.875643041188690133806141013087, −0.849978971022228472089532559919, 0, 0, 0, 0, 0, 0, 0.849978971022228472089532559919, 0.875643041188690133806141013087, 1.00202886326342819353812226526, 1.06726842875133247139894967281, 1.40625297428764331289782453449, 1.54331932399030158757641909001, 1.74264767301961022639920967991, 1.99110709242913170760573882646, 2.27194331202934570702947831511, 2.33105241941224269420676036752, 2.44600672700995550358820198639, 2.45844540440870857260481476757, 2.85132765061511690938851817358, 3.09038462668101423216846823686, 3.09255822209714699303524283435, 3.15778673568530512557954271399, 3.18901805274289107406295828260, 3.35967957535718434805338237627, 3.50460351102748501008402021925, 3.57425443371537060788018701913, 3.72248788517454990329908771380, 3.90184846867746142194137751302, 3.91099934232330730894930405259, 3.98543970992048378310616274773, 4.12791480301829488089968477563

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

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