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Properties

Label 12-1368e6-1.1-c0e6-0-8
Degree $12$
Conductor $6.554\times 10^{18}$
Sign $1$
Analytic cond. $0.101264$
Root an. cond. $0.826269$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8-s + 3·41-s − 3·49-s + 3·59-s − 3·67-s + 6·73-s − 3·97-s − 3·107-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 8-s + 3·41-s − 3·49-s + 3·59-s − 3·67-s + 6·73-s − 3·97-s − 3·107-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(0.101264\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{12} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.206099994\)
\(L(\frac12)\) \(\approx\) \(1.206099994\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
3 \( 1 \)
19 \( 1 + T^{3} + T^{6} \)
good5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T^{3} + T^{6} )^{2} \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
43 \( ( 1 + T^{3} + T^{6} )^{2} \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T^{3} + T^{6} )^{2} \)
89 \( ( 1 - T^{3} + T^{6} )^{2} \)
97 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
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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

−5.34917796674674607365311047076, −5.10322673611250113940444738004, −4.87527718899260544314940461264, −4.82474859965014723984427307759, −4.60591103935087721741175043268, −4.60120072680533270478644518944, −4.32776739984497326455100462331, −4.06014770763908497093717848947, −4.02334268265414628921121604209, −3.75770325074225807551660191206, −3.70676610531026564349526626369, −3.59426921854885338489579199497, −3.51170385586158802213469212714, −3.00082475588478984637833464627, −2.86846714021961662671051209902, −2.71659461309361275623900224654, −2.52076218624501440137204669672, −2.39820065182450849368983229293, −2.38080897339282886204784830074, −1.89424456873575858825120713011, −1.58900805298940183490429102612, −1.47620533384632479922595983506, −1.44625138924848356055359883944, −0.936260874972603277636415509724, −0.69525114263984237989861565491, 0.69525114263984237989861565491, 0.936260874972603277636415509724, 1.44625138924848356055359883944, 1.47620533384632479922595983506, 1.58900805298940183490429102612, 1.89424456873575858825120713011, 2.38080897339282886204784830074, 2.39820065182450849368983229293, 2.52076218624501440137204669672, 2.71659461309361275623900224654, 2.86846714021961662671051209902, 3.00082475588478984637833464627, 3.51170385586158802213469212714, 3.59426921854885338489579199497, 3.70676610531026564349526626369, 3.75770325074225807551660191206, 4.02334268265414628921121604209, 4.06014770763908497093717848947, 4.32776739984497326455100462331, 4.60120072680533270478644518944, 4.60591103935087721741175043268, 4.82474859965014723984427307759, 4.87527718899260544314940461264, 5.10322673611250113940444738004, 5.34917796674674607365311047076

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

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