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Properties

Label 12-292e6-1.1-c0e6-0-1
Degree $12$
Conductor $6.199\times 10^{14}$
Sign $1$
Analytic cond. $9.57722\times 10^{-6}$
Root an. cond. $0.381742$
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 − 3·9-s + 6·13-s − 3·29-s − 3·37-s − 3·41-s − 3·49-s − 3·61-s + 3·72-s + 3·81-s − 3·89-s − 3·101-s − 6·104-s + 6·113-s − 18·117-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 21·169-s + 173-s + ⋯
L(s)  = 1  − 8-s − 3·9-s + 6·13-s − 3·29-s − 3·37-s − 3·41-s − 3·49-s − 3·61-s + 3·72-s + 3·81-s − 3·89-s − 3·101-s − 6·104-s + 6·113-s − 18·117-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 21·169-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 73^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 73^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 73^{6}\)
Sign: $1$
Analytic conductor: \(9.57722\times 10^{-6}\)
Root analytic conductor: \(0.381742\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 73^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1868066906\)
\(L(\frac12)\) \(\approx\) \(0.1868066906\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{3} + T^{6} \)
73 \( 1 + T^{3} + T^{6} \)
good3 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
5 \( ( 1 + T^{3} + T^{6} )^{2} \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
13 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 + T^{3} + T^{6} )^{2} \)
19 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
37 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
41 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 + T^{3} + T^{6} )^{2} \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T )^{6}( 1 + T )^{6} \)
89 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
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

−6.64478925230890205008118691217, −6.40406670359715935015369111981, −6.30111562365042925765581848294, −6.10349964521253102376734966071, −6.09680923786599468800932208883, −5.64038278551307982457041403717, −5.62330124221652176216274859222, −5.62270995835903669182754211095, −5.41120397224169023538613279302, −5.22833033403699823181039499449, −4.99936035479020552619275380397, −4.43021530078673615580397567257, −4.33281311370947678918162704253, −4.15651762876425665885448597071, −3.76365495877705418529816032127, −3.61454872618260626343308962082, −3.39187649758692499370594552564, −3.26917632658866722205608767076, −3.09254502616858086623365264659, −3.05697379638769290720833041982, −2.91592575733250746916440530451, −1.82041604095478822135301560981, −1.79794215476465530771151072883, −1.78681493551263075298504868838, −1.37615098528970397593758210245, 1.37615098528970397593758210245, 1.78681493551263075298504868838, 1.79794215476465530771151072883, 1.82041604095478822135301560981, 2.91592575733250746916440530451, 3.05697379638769290720833041982, 3.09254502616858086623365264659, 3.26917632658866722205608767076, 3.39187649758692499370594552564, 3.61454872618260626343308962082, 3.76365495877705418529816032127, 4.15651762876425665885448597071, 4.33281311370947678918162704253, 4.43021530078673615580397567257, 4.99936035479020552619275380397, 5.22833033403699823181039499449, 5.41120397224169023538613279302, 5.62270995835903669182754211095, 5.62330124221652176216274859222, 5.64038278551307982457041403717, 6.09680923786599468800932208883, 6.10349964521253102376734966071, 6.30111562365042925765581848294, 6.40406670359715935015369111981, 6.64478925230890205008118691217

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

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