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Properties

Label 12-1944e6-1.1-c0e6-0-0
Degree $12$
Conductor $5.397\times 10^{19}$
Sign $1$
Analytic cond. $0.833912$
Root an. cond. $0.984978$
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  + 8-s − 6·11-s − 6·41-s − 3·43-s + 3·59-s − 3·67-s − 6·88-s − 3·89-s + 6·97-s + 21·121-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 + ⋯
L(s)  = 1  + 8-s − 6·11-s − 6·41-s − 3·43-s + 3·59-s − 3·67-s − 6·88-s − 3·89-s + 6·97-s + 21·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{30}\)
Sign: $1$
Analytic conductor: \(0.833912\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{30} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3408308474\)
\(L(\frac12)\) \(\approx\) \(0.3408308474\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
3 \( 1 \)
good5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
11 \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T^{3} + T^{6} )^{2} \)
19 \( ( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
37 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
41 \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \)
43 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
73 \( ( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T^{3} + T^{6} )^{2} \)
89 \( ( 1 + T )^{6}( 1 - T + T^{2} )^{3} \)
97 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
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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

−5.13619935354431997665822605743, −4.73019895524261238653757005801, −4.72843888119058415264620621686, −4.71409941221615438235013347329, −4.55544866675015027259945620086, −4.51243200769593784976302239152, −4.28810742579248703809278881907, −3.60063365439144641765397653015, −3.55751829014377090533045860120, −3.53934852535729187330428183508, −3.50714251151926768098134352075, −3.47863453984749721220659713187, −3.06610146756619313699446218112, −3.00509583650460613396024623902, −2.56481035797116139691459758888, −2.52106935241553668683576681407, −2.47538976066937424056853430939, −2.40381144938195629884787005015, −2.15705915063208319161240804429, −1.66005251536660642911708995972, −1.63677577307350172972356075262, −1.58901617012545022043932721233, −1.39076086364621085132858389440, −0.46075950541962928712663926373, −0.41315284090464139611703878691, 0.41315284090464139611703878691, 0.46075950541962928712663926373, 1.39076086364621085132858389440, 1.58901617012545022043932721233, 1.63677577307350172972356075262, 1.66005251536660642911708995972, 2.15705915063208319161240804429, 2.40381144938195629884787005015, 2.47538976066937424056853430939, 2.52106935241553668683576681407, 2.56481035797116139691459758888, 3.00509583650460613396024623902, 3.06610146756619313699446218112, 3.47863453984749721220659713187, 3.50714251151926768098134352075, 3.53934852535729187330428183508, 3.55751829014377090533045860120, 3.60063365439144641765397653015, 4.28810742579248703809278881907, 4.51243200769593784976302239152, 4.55544866675015027259945620086, 4.71409941221615438235013347329, 4.72843888119058415264620621686, 4.73019895524261238653757005801, 5.13619935354431997665822605743

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

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