Properties

Label 12-864e6-1.1-c0e6-0-0
Degree $12$
Conductor $4.160\times 10^{17}$
Sign $1$
Analytic cond. $0.00642725$
Root an. cond. $0.656652$
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  + 3·11-s + 27-s − 3·41-s + 3·43-s − 6·59-s + 3·67-s + 3·89-s − 3·97-s + 3·121-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 + ⋯
L(s)  = 1  + 3·11-s + 27-s − 3·41-s + 3·43-s − 6·59-s + 3·67-s + 3·89-s − 3·97-s + 3·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{30} \cdot 3^{18}\)
Sign: $1$
Analytic conductor: \(0.00642725\)
Root analytic conductor: \(0.656652\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{864} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{30} \cdot 3^{18} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9324120623\)
\(L(\frac12)\) \(\approx\) \(0.9324120623\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−5.82120811786269020488655950824, −5.27353005254842872102181965405, −5.25725246553453368970832890388, −5.12943223799191162585901759655, −4.96465952706016470716110972001, −4.83887828113249312189562565079, −4.77382009248142497053692253420, −4.29690578283899426179963838290, −4.25728877682293028888567227115, −4.14925594983012218675113337399, −3.97024591520817943054651417412, −3.81160705595061180820198190542, −3.48038191394335792638921780288, −3.47266272589055496995336154381, −3.41635591593603092218720761330, −2.95271955397086025546705325804, −2.80530936422477318736258737986, −2.58004789456972379620689128136, −2.54837736362177468293033411053, −2.01109970149798648382494256556, −1.89445748414108061807889473431, −1.60065576960816549234678278242, −1.30184874323710865942771920751, −1.27439171901337151242175599893, −0.923867651047348099398639848761, 0.923867651047348099398639848761, 1.27439171901337151242175599893, 1.30184874323710865942771920751, 1.60065576960816549234678278242, 1.89445748414108061807889473431, 2.01109970149798648382494256556, 2.54837736362177468293033411053, 2.58004789456972379620689128136, 2.80530936422477318736258737986, 2.95271955397086025546705325804, 3.41635591593603092218720761330, 3.47266272589055496995336154381, 3.48038191394335792638921780288, 3.81160705595061180820198190542, 3.97024591520817943054651417412, 4.14925594983012218675113337399, 4.25728877682293028888567227115, 4.29690578283899426179963838290, 4.77382009248142497053692253420, 4.83887828113249312189562565079, 4.96465952706016470716110972001, 5.12943223799191162585901759655, 5.25725246553453368970832890388, 5.27353005254842872102181965405, 5.82120811786269020488655950824

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

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