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Properties

Label 12-3312e6-1.1-c0e6-0-0
Degree $12$
Conductor $1.320\times 10^{21}$
Sign $1$
Analytic cond. $20.3931$
Root an. cond. $1.28565$
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  + 3·23-s − 3·25-s + 27-s − 3·49-s − 3·59-s + 3·101-s − 3·121-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 + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 3·23-s − 3·25-s + 27-s − 3·49-s − 3·59-s + 3·101-s − 3·121-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 + 223-s + 227-s + 229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(20.3931\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{12} \cdot 23^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.421402275\)
\(L(\frac12)\) \(\approx\) \(1.421402275\)
\(L(1)\) 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 \( 1 - T^{3} + T^{6} \)
23 \( ( 1 - T + T^{2} )^{3} \)
good5 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 + T^{3} + T^{6} )^{2} \)
17 \( ( 1 - T )^{6}( 1 + T )^{6} \)
19 \( ( 1 - T )^{6}( 1 + T )^{6} \)
29 \( ( 1 + T^{3} + T^{6} )^{2} \)
31 \( ( 1 - T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 + T^{3} + T^{6} )^{2} \)
43 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
47 \( ( 1 - T^{3} + T^{6} )^{2} \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 + T )^{6}( 1 - T + T^{2} )^{3} \)
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 \( ( 1 + T^{3} + T^{6} )^{2} \)
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

−4.76141666439504725237647968640, −4.70169740180242502520211415921, −4.58631222271144710655455521927, −4.07840462782250195333438567221, −3.91708101081398597790005577963, −3.89637088662067014838544300489, −3.82967319322874241211766562271, −3.73244920145056174546961957911, −3.45312113410931377753015273528, −3.44376244381988360942480114083, −3.15132058807160964189710449331, −3.00024021823520945383790710754, −2.79123495075364308532954693092, −2.71328754233653665205258922930, −2.62427628479996310591380233318, −2.56058687177002190591109350044, −2.11564951866663963784259987469, −2.01077290908997860599356854043, −1.83570701715485786169430284921, −1.46512663811570902440759519743, −1.45426631760614867826947508116, −1.38091654251952449792719728941, −1.19595653589390158223348912010, −0.57620412015851202555279693057, −0.46698019097156645973380625799, 0.46698019097156645973380625799, 0.57620412015851202555279693057, 1.19595653589390158223348912010, 1.38091654251952449792719728941, 1.45426631760614867826947508116, 1.46512663811570902440759519743, 1.83570701715485786169430284921, 2.01077290908997860599356854043, 2.11564951866663963784259987469, 2.56058687177002190591109350044, 2.62427628479996310591380233318, 2.71328754233653665205258922930, 2.79123495075364308532954693092, 3.00024021823520945383790710754, 3.15132058807160964189710449331, 3.44376244381988360942480114083, 3.45312113410931377753015273528, 3.73244920145056174546961957911, 3.82967319322874241211766562271, 3.89637088662067014838544300489, 3.91708101081398597790005577963, 4.07840462782250195333438567221, 4.58631222271144710655455521927, 4.70169740180242502520211415921, 4.76141666439504725237647968640

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

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