↑

Properties

Label 12-2888e6-1.1-c0e6-0-1
Degree $12$
Conductor $5.802\times 10^{20}$
Sign $1$
Analytic cond. $8.96449$
Root an. cond. $1.20054$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 8-s + 3·9-s + 3·24-s − 27-s − 6·41-s − 3·49-s − 6·59-s + 3·67-s + 3·72-s − 3·73-s − 6·81-s + 3·97-s − 3·107-s − 18·123-s + 127-s + 131-s + 137-s + 139-s − 9·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 18·177-s + ⋯
L(s)  = 1  + 3·3-s + 8-s + 3·9-s + 3·24-s − 27-s − 6·41-s − 3·49-s − 6·59-s + 3·67-s + 3·72-s − 3·73-s − 6·81-s + 3·97-s − 3·107-s − 18·123-s + 127-s + 131-s + 137-s + 139-s − 9·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 18·177-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(8.96449\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 19^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.340032309\)
\(L(\frac12)\) \(\approx\) \(1.340032309\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
19 \( 1 \)
good3 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 + T^{3} + T^{6} )^{2} \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \)
43 \( ( 1 + T^{3} + T^{6} )^{2} \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 + T^{3} + T^{6} )^{2} \)
89 \( ( 1 - T^{3} + T^{6} )^{2} \)
97 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.55722562736011883808942773113, −4.46389024739387989570389346269, −4.36280300032940901537312320996, −4.33540087818991102298854061720, −4.18113332209920085117323736000, −3.94332968671675169952241773074, −3.89263359034411314783900896050, −3.46813126143027143253076556522, −3.29945266027797467412227052557, −3.28862571742974189249721026601, −3.25559517346108979056305973019, −3.08001907118846920551201387259, −3.01477067710761075055830759071, −2.95986115928077477386995662327, −2.83941151805954343301759843271, −2.44304109337064591319266065701, −2.12936087948683535619722696861, −1.93464264334566206416743839366, −1.89164294543425103488461011377, −1.85088629190448296435541598871, −1.77459382425977294508705141417, −1.50046052541945540019839182565, −1.25934674060609043023673179723, −0.970248826454434833075147303003, −0.23125042384434741832610571197, 0.23125042384434741832610571197, 0.970248826454434833075147303003, 1.25934674060609043023673179723, 1.50046052541945540019839182565, 1.77459382425977294508705141417, 1.85088629190448296435541598871, 1.89164294543425103488461011377, 1.93464264334566206416743839366, 2.12936087948683535619722696861, 2.44304109337064591319266065701, 2.83941151805954343301759843271, 2.95986115928077477386995662327, 3.01477067710761075055830759071, 3.08001907118846920551201387259, 3.25559517346108979056305973019, 3.28862571742974189249721026601, 3.29945266027797467412227052557, 3.46813126143027143253076556522, 3.89263359034411314783900896050, 3.94332968671675169952241773074, 4.18113332209920085117323736000, 4.33540087818991102298854061720, 4.36280300032940901537312320996, 4.46389024739387989570389346269, 4.55722562736011883808942773113

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

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