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Properties

Label 12-2151e6-1.1-c0e6-0-0
Degree $12$
Conductor $9.905\times 10^{19}$
Sign $1$
Analytic cond. $1.53033$
Root an. cond. $1.03609$
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  − 3·3-s − 2·8-s + 3·9-s + 6·24-s + 2·27-s − 3·49-s + 64-s + 12·71-s − 6·72-s − 9·81-s − 6·101-s − 6·109-s + 3·113-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 9·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 3·3-s − 2·8-s + 3·9-s + 6·24-s + 2·27-s − 3·49-s + 64-s + 12·71-s − 6·72-s − 9·81-s − 6·101-s − 6·109-s + 3·113-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 9·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3·169-s + 173-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 239^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 239^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 239^{6}\)
Sign: $1$
Analytic conductor: \(1.53033\)
Root analytic conductor: \(1.03609\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 239^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1581568497\)
\(L(\frac12)\) \(\approx\) \(0.1581568497\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T + T^{2} )^{3} \)
239 \( ( 1 + T + T^{2} )^{3} \)
good2 \( ( 1 + T^{3} + T^{6} )^{2} \)
5 \( ( 1 + T^{3} + T^{6} )^{2} \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 + T^{3} + T^{6} )^{2} \)
13 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
17 \( ( 1 + T^{3} + T^{6} )^{2} \)
19 \( ( 1 - T )^{6}( 1 + T )^{6} \)
23 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
29 \( ( 1 + T^{3} + T^{6} )^{2} \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
43 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
47 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
61 \( ( 1 + T^{3} + T^{6} )^{2} \)
67 \( ( 1 + T^{3} + T^{6} )^{2} \)
71 \( ( 1 - T )^{12} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
83 \( ( 1 + T^{3} + T^{6} )^{2} \)
89 \( ( 1 - T )^{6}( 1 + T )^{6} \)
97 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
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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

−5.10563675585137393518917705686, −4.90239767203338450683981028336, −4.89191492470362284593141015459, −4.61893615851963534583311775643, −4.44879429440895800855772022491, −4.22976821137005796990788698326, −4.03140446766330096256140806525, −3.78626034075234929663284957098, −3.75577205103329397409624001980, −3.64457234661211147124053425892, −3.47770832589686805945734261048, −3.20314045939619209632830680398, −3.16309681529490038100986469478, −2.86253975413918395600563704371, −2.57364029131024282744214093932, −2.51953775668099392409197070350, −2.50437819724785868257449669641, −2.38668220169123174547162429190, −1.90619037866651601221210694555, −1.74885609067521600487305241119, −1.37011958547190538091790616067, −1.12689212121691924572331689639, −1.08051390806981075639607526879, −0.46456603179146260202630408286, −0.43921422556257897203297883536, 0.43921422556257897203297883536, 0.46456603179146260202630408286, 1.08051390806981075639607526879, 1.12689212121691924572331689639, 1.37011958547190538091790616067, 1.74885609067521600487305241119, 1.90619037866651601221210694555, 2.38668220169123174547162429190, 2.50437819724785868257449669641, 2.51953775668099392409197070350, 2.57364029131024282744214093932, 2.86253975413918395600563704371, 3.16309681529490038100986469478, 3.20314045939619209632830680398, 3.47770832589686805945734261048, 3.64457234661211147124053425892, 3.75577205103329397409624001980, 3.78626034075234929663284957098, 4.03140446766330096256140806525, 4.22976821137005796990788698326, 4.44879429440895800855772022491, 4.61893615851963534583311775643, 4.89191492470362284593141015459, 4.90239767203338450683981028336, 5.10563675585137393518917705686

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

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