Properties

Label 12-120e6-1.1-c1e6-0-1
Degree $12$
Conductor $2.986\times 10^{12}$
Sign $1$
Analytic cond. $0.774016$
Root an. cond. $0.978879$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4-s + 4·7-s + 2·8-s − 3·9-s + 8·14-s + 7·16-s + 12·17-s − 6·18-s − 8·23-s − 3·25-s + 4·28-s − 12·31-s + 10·32-s + 24·34-s − 3·36-s − 20·41-s − 16·46-s + 8·47-s + 2·49-s − 6·50-s + 8·56-s − 24·62-s − 12·63-s + 13·64-s + 12·68-s − 8·71-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 1.51·7-s + 0.707·8-s − 9-s + 2.13·14-s + 7/4·16-s + 2.91·17-s − 1.41·18-s − 1.66·23-s − 3/5·25-s + 0.755·28-s − 2.15·31-s + 1.76·32-s + 4.11·34-s − 1/2·36-s − 3.12·41-s − 2.35·46-s + 1.16·47-s + 2/7·49-s − 0.848·50-s + 1.06·56-s − 3.04·62-s − 1.51·63-s + 13/8·64-s + 1.45·68-s − 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{6} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(0.774016\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{6} \cdot 5^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.600084647\)
\(L(\frac12)\) \(\approx\) \(2.600084647\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T + 3 T^{2} - 3 p T^{3} + 3 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
3 \( ( 1 + T^{2} )^{3} \)
5 \( ( 1 + T^{2} )^{3} \)
good7 \( ( 1 - 2 T + 5 T^{2} - 12 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 87 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 - 22 T^{2} + 407 T^{4} - 7284 T^{6} + 407 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 - 6 T + 35 T^{2} - 172 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 2647 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 + 4 T + 57 T^{2} + 168 T^{3} + 57 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \)
31 \( ( 1 + 6 T + 77 T^{2} + 308 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 86 T^{2} + 6055 T^{4} - 248372 T^{6} + 6055 p^{2} T^{8} - 86 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 + 10 T + 87 T^{2} + 588 T^{3} + 87 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 114 T^{2} + 8087 T^{4} - 416540 T^{6} + 8087 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \)
47 \( ( 1 - 4 T + 49 T^{2} + 120 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{3} \)
59 \( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 33911 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 10759 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \)
67 \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{3} \)
71 \( ( 1 + 4 T + 101 T^{2} + 632 T^{3} + 101 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 6 T + p T^{2} )^{6} \)
79 \( ( 1 - 18 T + 317 T^{2} - 2908 T^{3} + 317 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 274 T^{2} + 37415 T^{4} - 3513756 T^{6} + 37415 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 + 14 T + 263 T^{2} + 2308 T^{3} + 263 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( ( 1 - 18 T + 287 T^{2} - 3164 T^{3} + 287 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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

−7.49987568892804079382098194100, −7.42258518526054279407657178323, −7.21449140090446490598010874412, −7.17919514397083014867525424051, −6.77109415876659883508988808593, −6.33451587212867048109917900785, −6.19335955431057575741683527318, −5.87784728064998666034187869190, −5.79464395975776817796016086050, −5.59585796279181435083924183329, −5.50882312973417180952929416950, −5.05224771178053274801949782614, −4.90516946629261015030257499076, −4.86310660401006800695125588711, −4.70243546252100533273111736366, −4.06699788751641656598834218495, −3.93951373413611225449352306003, −3.84356564904395697189356459591, −3.38564353872613129111733025248, −3.30506032899287401228828109181, −3.05304436726511319393488459990, −2.37764694920213258637818594049, −2.07685081005802612600182856735, −1.55441951206953152948658851411, −1.39947053199735115189171366965, 1.39947053199735115189171366965, 1.55441951206953152948658851411, 2.07685081005802612600182856735, 2.37764694920213258637818594049, 3.05304436726511319393488459990, 3.30506032899287401228828109181, 3.38564353872613129111733025248, 3.84356564904395697189356459591, 3.93951373413611225449352306003, 4.06699788751641656598834218495, 4.70243546252100533273111736366, 4.86310660401006800695125588711, 4.90516946629261015030257499076, 5.05224771178053274801949782614, 5.50882312973417180952929416950, 5.59585796279181435083924183329, 5.79464395975776817796016086050, 5.87784728064998666034187869190, 6.19335955431057575741683527318, 6.33451587212867048109917900785, 6.77109415876659883508988808593, 7.17919514397083014867525424051, 7.21449140090446490598010874412, 7.42258518526054279407657178323, 7.49987568892804079382098194100

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

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