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Properties

Label 12-2736e6-1.1-c0e6-0-1
Degree $12$
Conductor $4.195\times 10^{20}$
Sign $1$
Analytic cond. $6.48095$
Root an. cond. $1.16852$
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  + 3·13-s − 3·19-s + 3·43-s − 6·61-s + 3·67-s − 3·73-s + 6·79-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 3·13-s − 3·19-s + 3·43-s − 6·61-s + 3·67-s − 3·73-s + 6·79-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(6.48095\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{12} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.346040585\)
\(L(\frac12)\) \(\approx\) \(1.346040585\)
\(L(1)\) 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 \( 1 \)
19 \( ( 1 + T + T^{2} )^{3} \)
good5 \( 1 - T^{6} + T^{12} \)
7 \( ( 1 + T^{3} + T^{6} )^{2} \)
11 \( ( 1 - T^{2} + T^{4} )^{3} \)
13 \( ( 1 - T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
17 \( 1 - T^{6} + T^{12} \)
23 \( 1 - T^{6} + T^{12} \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
37 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
47 \( 1 - T^{6} + T^{12} \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \)
67 \( ( 1 - T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
79 \( ( 1 - T )^{6}( 1 - T^{3} + T^{6} ) \)
83 \( ( 1 - T^{2} + T^{4} )^{3} \)
89 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
97 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
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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

−4.75435580176048061653701780178, −4.58293949417983782347736193770, −4.57819724793505743885516438634, −4.24668603952314352125614261726, −4.19131193662584885211295872997, −4.16317568230237485694964151403, −3.83338636893518699278408941860, −3.64453450186779995115191786747, −3.60817876515959616251396104805, −3.57434527778619797696092959446, −3.30504962496349480154310217232, −3.20508390441640231911417072977, −3.08927906755982887035253645996, −2.65059519333702155460061645144, −2.58612481982095659005971066799, −2.42584975755551579093993924770, −2.29972583153627645105063306829, −2.03357108038411536476503018909, −1.92465077296939244059607431080, −1.84994584453421408842044431216, −1.30633350690278709621312099625, −1.27074827134321347631009450844, −1.17747433708757185479128214790, −0.927118342063271800821952834942, −0.39379786947555079377368709751, 0.39379786947555079377368709751, 0.927118342063271800821952834942, 1.17747433708757185479128214790, 1.27074827134321347631009450844, 1.30633350690278709621312099625, 1.84994584453421408842044431216, 1.92465077296939244059607431080, 2.03357108038411536476503018909, 2.29972583153627645105063306829, 2.42584975755551579093993924770, 2.58612481982095659005971066799, 2.65059519333702155460061645144, 3.08927906755982887035253645996, 3.20508390441640231911417072977, 3.30504962496349480154310217232, 3.57434527778619797696092959446, 3.60817876515959616251396104805, 3.64453450186779995115191786747, 3.83338636893518699278408941860, 4.16317568230237485694964151403, 4.19131193662584885211295872997, 4.24668603952314352125614261726, 4.57819724793505743885516438634, 4.58293949417983782347736193770, 4.75435580176048061653701780178

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

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