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Properties

Label 12-3744e6-1.1-c0e6-0-1
Degree $12$
Conductor $2.754\times 10^{21}$
Sign $1$
Analytic cond. $42.5557$
Root an. cond. $1.36693$
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  + 3·13-s + 27-s + 3·31-s − 12·107-s + 3·113-s − 3·121-s + 2·125-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 + 227-s + ⋯
L(s)  = 1  + 3·13-s + 27-s + 3·31-s − 12·107-s + 3·113-s − 3·121-s + 2·125-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 + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{30} \cdot 3^{12} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(42.5557\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{30} \cdot 3^{12} \cdot 13^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.948798205\)
\(L(\frac12)\) \(\approx\) \(2.948798205\)
\(L(1)\) 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 \( 1 - T^{3} + T^{6} \)
13 \( ( 1 - T + T^{2} )^{3} \)
good5 \( ( 1 - T^{3} + T^{6} )^{2} \)
7 \( ( 1 + T^{3} + T^{6} )^{2} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
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 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
31 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
37 \( ( 1 - T^{3} + T^{6} )^{2} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
43 \( ( 1 - T^{3} + T^{6} )^{2} \)
47 \( ( 1 + T^{3} + T^{6} )^{2} \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
61 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
67 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
71 \( ( 1 + T^{3} + T^{6} )^{2} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
83 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
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

−4.49719197996130716771057535405, −4.44570526440947321995955933817, −4.30822187932156393532369697701, −4.15326817501144025280541996273, −3.94573534024520707209492164041, −3.91190634924372386119770913778, −3.74931965196939253762283800839, −3.63189400667665096835793635687, −3.41313418946060608837177018751, −3.33257256759645255391418777525, −3.17176191733616277393082727845, −2.93563920650582969014263299834, −2.71691718403907785979194125584, −2.65603964865468624861587147953, −2.51569730872888658402410492592, −2.43566852611020266306635428573, −2.42675058333601504373646258366, −1.75821262403368109136986224545, −1.65722199959778919819522399452, −1.58888006257969945872339103854, −1.36867557474791387746263336947, −1.28977832532480544903708330089, −1.03930106763587803135405709687, −0.808699302684641340685947552365, −0.51500662070888455246833975896, 0.51500662070888455246833975896, 0.808699302684641340685947552365, 1.03930106763587803135405709687, 1.28977832532480544903708330089, 1.36867557474791387746263336947, 1.58888006257969945872339103854, 1.65722199959778919819522399452, 1.75821262403368109136986224545, 2.42675058333601504373646258366, 2.43566852611020266306635428573, 2.51569730872888658402410492592, 2.65603964865468624861587147953, 2.71691718403907785979194125584, 2.93563920650582969014263299834, 3.17176191733616277393082727845, 3.33257256759645255391418777525, 3.41313418946060608837177018751, 3.63189400667665096835793635687, 3.74931965196939253762283800839, 3.91190634924372386119770913778, 3.94573534024520707209492164041, 4.15326817501144025280541996273, 4.30822187932156393532369697701, 4.44570526440947321995955933817, 4.49719197996130716771057535405

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

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