↑

Properties

Label 12-91e12-1.1-c1e6-0-5
Degree $12$
Conductor $3.225\times 10^{23}$
Sign $1$
Analytic cond. $8.35909\times 10^{10}$
Root an. cond. $8.13167$
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  − 2·2-s − 3-s − 2·4-s − 5-s + 2·6-s + 7·8-s − 10·9-s + 2·10-s − 4·11-s + 2·12-s + 15-s − 4·16-s − 5·17-s + 20·18-s + 19-s + 2·20-s + 8·22-s + 23-s − 7·24-s − 18·25-s + 10·27-s − 3·29-s − 2·30-s − 16·31-s − 4·32-s + 4·33-s + 10·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s − 4-s − 0.447·5-s + 0.816·6-s + 2.47·8-s − 3.33·9-s + 0.632·10-s − 1.20·11-s + 0.577·12-s + 0.258·15-s − 16-s − 1.21·17-s + 4.71·18-s + 0.229·19-s + 0.447·20-s + 1.70·22-s + 0.208·23-s − 1.42·24-s − 3.59·25-s + 1.92·27-s − 0.557·29-s − 0.365·30-s − 2.87·31-s − 0.707·32-s + 0.696·33-s + 1.71·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(8.35909\times 10^{10}\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + p T + 3 p T^{2} + 9 T^{3} + 5 p^{2} T^{4} + 7 p^{2} T^{5} + 51 T^{6} + 7 p^{3} T^{7} + 5 p^{4} T^{8} + 9 p^{3} T^{9} + 3 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
3 \( 1 + T + 11 T^{2} + 11 T^{3} + 19 p T^{4} + 55 T^{5} + 197 T^{6} + 55 p T^{7} + 19 p^{3} T^{8} + 11 p^{3} T^{9} + 11 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + T + 19 T^{2} + 7 T^{3} + 161 T^{4} - 3 T^{5} + 913 T^{6} - 3 p T^{7} + 161 p^{2} T^{8} + 7 p^{3} T^{9} + 19 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 4 T + 45 T^{2} + 144 T^{3} + 972 T^{4} + 2539 T^{5} + 13237 T^{6} + 2539 p T^{7} + 972 p^{2} T^{8} + 144 p^{3} T^{9} + 45 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 5 T + 90 T^{2} + 411 T^{3} + 3539 T^{4} + 13744 T^{5} + 78123 T^{6} + 13744 p T^{7} + 3539 p^{2} T^{8} + 411 p^{3} T^{9} + 90 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - T + 50 T^{2} + 16 T^{3} + 1180 T^{4} + 1175 T^{5} + 23331 T^{6} + 1175 p T^{7} + 1180 p^{2} T^{8} + 16 p^{3} T^{9} + 50 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - T + 32 T^{2} - 52 T^{3} + 1214 T^{4} - 1383 T^{5} + 21935 T^{6} - 1383 p T^{7} + 1214 p^{2} T^{8} - 52 p^{3} T^{9} + 32 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T + 96 T^{2} + 191 T^{3} + 4061 T^{4} + 5126 T^{5} + 122643 T^{6} + 5126 p T^{7} + 4061 p^{2} T^{8} + 191 p^{3} T^{9} + 96 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 16 T + 236 T^{2} + 2185 T^{3} + 18573 T^{4} + 122325 T^{5} + 755039 T^{6} + 122325 p T^{7} + 18573 p^{2} T^{8} + 2185 p^{3} T^{9} + 236 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 13 T + 184 T^{2} - 1054 T^{3} + 7158 T^{4} - 10573 T^{5} + 113729 T^{6} - 10573 p T^{7} + 7158 p^{2} T^{8} - 1054 p^{3} T^{9} + 184 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 8 T + 225 T^{2} - 1362 T^{3} + 21488 T^{4} - 101725 T^{5} + 1145451 T^{6} - 101725 p T^{7} + 21488 p^{2} T^{8} - 1362 p^{3} T^{9} + 225 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 11 T + 259 T^{2} - 2099 T^{3} + 27622 T^{4} - 170696 T^{5} + 1576761 T^{6} - 170696 p T^{7} + 27622 p^{2} T^{8} - 2099 p^{3} T^{9} + 259 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - T + 105 T^{2} - 147 T^{3} + 5543 T^{4} - 3359 T^{5} + 246951 T^{6} - 3359 p T^{7} + 5543 p^{2} T^{8} - 147 p^{3} T^{9} + 105 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 2 T + 218 T^{2} - 344 T^{3} + 22040 T^{4} - 26940 T^{5} + 1409201 T^{6} - 26940 p T^{7} + 22040 p^{2} T^{8} - 344 p^{3} T^{9} + 218 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 13 T + 5 p T^{2} + 2839 T^{3} + 38957 T^{4} + 294699 T^{5} + 2963017 T^{6} + 294699 p T^{7} + 38957 p^{2} T^{8} + 2839 p^{3} T^{9} + 5 p^{5} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 5 T + 165 T^{2} - 599 T^{3} + 15743 T^{4} - 53393 T^{5} + 1179159 T^{6} - 53393 p T^{7} + 15743 p^{2} T^{8} - 599 p^{3} T^{9} + 165 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 11 T + 296 T^{2} - 2796 T^{3} + 41197 T^{4} - 326168 T^{5} + 3447813 T^{6} - 326168 p T^{7} + 41197 p^{2} T^{8} - 2796 p^{3} T^{9} + 296 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 6 T + 285 T^{2} + 14 p T^{3} + 35468 T^{4} + 74185 T^{5} + 2901951 T^{6} + 74185 p T^{7} + 35468 p^{2} T^{8} + 14 p^{4} T^{9} + 285 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 30 T + 676 T^{2} - 10699 T^{3} + 141027 T^{4} - 1519265 T^{5} + 14149139 T^{6} - 1519265 p T^{7} + 141027 p^{2} T^{8} - 10699 p^{3} T^{9} + 676 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 7 T + 326 T^{2} + 2455 T^{3} + 54096 T^{4} + 344443 T^{5} + 5474643 T^{6} + 344443 p T^{7} + 54096 p^{2} T^{8} + 2455 p^{3} T^{9} + 326 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 27 T + 656 T^{2} + 10802 T^{3} + 153994 T^{4} + 1760871 T^{5} + 17670883 T^{6} + 1760871 p T^{7} + 153994 p^{2} T^{8} + 10802 p^{3} T^{9} + 656 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 4 T + 167 T^{2} + 1648 T^{3} + 21035 T^{4} + 202110 T^{5} + 2204075 T^{6} + 202110 p T^{7} + 21035 p^{2} T^{8} + 1648 p^{3} T^{9} + 167 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 35 T + 947 T^{2} - 18161 T^{3} + 281670 T^{4} - 3629766 T^{5} + 38644781 T^{6} - 3629766 p T^{7} + 281670 p^{2} T^{8} - 18161 p^{3} T^{9} + 947 p^{4} T^{10} - 35 p^{5} T^{11} + p^{6} T^{12} \)
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

−4.38323668572006794303991424944, −4.09111035815516949106470639132, −4.05086090158787778710198752280, −3.98842479807383135502089488988, −3.87290722887860888616943126735, −3.81603974126337240789914592031, −3.69032498453840001442144172539, −3.43021937395472587128169804628, −3.17822103064131551297991985295, −3.16991285188783987996213202201, −3.01020709077548520364937826400, −2.72631438152404130390482317136, −2.72344644393127322432433967727, −2.54632148219581551337442497540, −2.53879277040556516444517951049, −2.24381646596488040773145883951, −2.08923350315352947219284672496, −2.01560096299128982294216681482, −1.88591816080869985898938771015, −1.83304595444883949530609874929, −1.35366139191036065750431918708, −1.20106405442967128958482364420, −0.858547216551596530603189231690, −0.830673765686925700893816157909, −0.794769609623520459294333432021, 0, 0, 0, 0, 0, 0, 0.794769609623520459294333432021, 0.830673765686925700893816157909, 0.858547216551596530603189231690, 1.20106405442967128958482364420, 1.35366139191036065750431918708, 1.83304595444883949530609874929, 1.88591816080869985898938771015, 2.01560096299128982294216681482, 2.08923350315352947219284672496, 2.24381646596488040773145883951, 2.53879277040556516444517951049, 2.54632148219581551337442497540, 2.72344644393127322432433967727, 2.72631438152404130390482317136, 3.01020709077548520364937826400, 3.16991285188783987996213202201, 3.17822103064131551297991985295, 3.43021937395472587128169804628, 3.69032498453840001442144172539, 3.81603974126337240789914592031, 3.87290722887860888616943126735, 3.98842479807383135502089488988, 4.05086090158787778710198752280, 4.09111035815516949106470639132, 4.38323668572006794303991424944

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

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