Properties

Label 12-2888e6-1.1-c0e6-0-10
Degree $12$
Conductor $5.802\times 10^{20}$
Sign $1$
Analytic cond. $8.96449$
Root an. cond. $1.20054$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8-s + 3·11-s + 2·27-s − 3·49-s + 3·83-s + 3·88-s + 6·107-s + 6·113-s + 6·121-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 + 2·216-s + ⋯
L(s)  = 1  + 8-s + 3·11-s + 2·27-s − 3·49-s + 3·83-s + 3·88-s + 6·107-s + 6·113-s + 6·121-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 + 2·216-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(8.96449\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 19^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.684657238\)
\(L(\frac12)\) \(\approx\) \(3.684657238\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
19 \( 1 \)
good3 \( ( 1 - T^{3} + T^{6} )^{2} \)
5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
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 \( ( 1 - T^{3} + T^{6} )^{2} \)
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 \( ( 1 - T^{3} + T^{6} )^{2} \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 - T^{3} + T^{6} )^{2} \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
89 \( ( 1 - T^{3} + T^{6} )^{2} \)
97 \( ( 1 - T^{3} + T^{6} )^{2} \)
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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

−4.71573727638766214188246433440, −4.66916094282393279337459524968, −4.50151199209770002992900826768, −4.48165354001040694099559989639, −4.01037405205281024457426025419, −3.97161662395227114114006426602, −3.93622838396439939386872983214, −3.83063502472055764933929353401, −3.50341200695774901342190382091, −3.40664048697879073138435065179, −3.30487519179585972210789711705, −3.09160034479100497144954793305, −2.95897381007437452914083944871, −2.88975911218576581705740135183, −2.82641898652551433707878187510, −2.08805506983884222471264065174, −2.07093508508354573964051111952, −1.97008531857657750542271789287, −1.96394389029809764817339897498, −1.88754771767008998070881159250, −1.51130941652604428453138521343, −1.18755028154528213202417086321, −0.923645274946067624005142048956, −0.869734020773896771011876679620, −0.806607480943845926828533750502, 0.806607480943845926828533750502, 0.869734020773896771011876679620, 0.923645274946067624005142048956, 1.18755028154528213202417086321, 1.51130941652604428453138521343, 1.88754771767008998070881159250, 1.96394389029809764817339897498, 1.97008531857657750542271789287, 2.07093508508354573964051111952, 2.08805506983884222471264065174, 2.82641898652551433707878187510, 2.88975911218576581705740135183, 2.95897381007437452914083944871, 3.09160034479100497144954793305, 3.30487519179585972210789711705, 3.40664048697879073138435065179, 3.50341200695774901342190382091, 3.83063502472055764933929353401, 3.93622838396439939386872983214, 3.97161662395227114114006426602, 4.01037405205281024457426025419, 4.48165354001040694099559989639, 4.50151199209770002992900826768, 4.66916094282393279337459524968, 4.71573727638766214188246433440

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

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