Properties

Degree 12
Conductor $ 7^{6} \cdot 11^{6} \cdot 43^{6} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·7-s + 2·8-s − 6·11-s + 6·23-s + 6·43-s + 21·49-s − 12·56-s + 64-s + 36·77-s − 12·88-s + 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 36·161-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 12·184-s + 191-s + ⋯
L(s)  = 1  − 6·7-s + 2·8-s − 6·11-s + 6·23-s + 6·43-s + 21·49-s − 12·56-s + 64-s + 36·77-s − 12·88-s + 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 36·161-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 12·184-s + 191-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(7^{6} \cdot 11^{6} \cdot 43^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{3311} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(12,\ 7^{6} \cdot 11^{6} \cdot 43^{6} ,\ ( \ : [0]^{6} ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $0.5825329066$
$L(\frac12)$  $\approx$  $0.5825329066$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;11,\;43\}$, \(F_p\) is a polynomial of degree 12. If $p \in \{7,\;11,\;43\}$, then $F_p$ is a polynomial of degree at most 11.
$p$$F_p$
bad7 \( ( 1 + T )^{6} \)
11 \( ( 1 + T )^{6} \)
43 \( ( 1 - T )^{6} \)
good2 \( ( 1 - T^{3} + T^{6} )^{2} \)
3 \( 1 - T^{6} + T^{12} \)
5 \( 1 - T^{6} + T^{12} \)
13 \( ( 1 - T^{2} + T^{4} )^{3} \)
17 \( 1 - T^{6} + T^{12} \)
19 \( ( 1 - T )^{6}( 1 + T )^{6} \)
23 \( ( 1 - T + T^{2} )^{6} \)
29 \( ( 1 + T^{3} + T^{6} )^{2} \)
31 \( ( 1 - T )^{6}( 1 + T )^{6} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( 1 - T^{6} + T^{12} \)
47 \( ( 1 - T )^{6}( 1 + T )^{6} \)
53 \( ( 1 + T^{3} + T^{6} )^{2} \)
59 \( ( 1 - T )^{6}( 1 + T )^{6} \)
61 \( ( 1 - T )^{6}( 1 + T )^{6} \)
67 \( ( 1 + T^{3} + T^{6} )^{2} \)
71 \( ( 1 - T )^{6}( 1 + T )^{6} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 - T )^{6}( 1 + T )^{6} \)
83 \( 1 - T^{6} + T^{12} \)
89 \( ( 1 + T^{2} )^{6} \)
97 \( ( 1 - T )^{6}( 1 + T )^{6} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.62845025381119585935137093097, −4.56807208083852890955816004050, −4.30592251167204352831845041925, −4.24918905050462287344085091642, −4.22750218161341761354355288077, −3.89051929061376608274061306591, −3.73364855203391378193108226767, −3.36696554176594540161337435032, −3.36114866313686682902960155952, −3.12834159359762304708597431312, −3.11268737081187728429117616230, −3.07792495138543732068890134561, −2.96619327250389796491487991684, −2.77607906931930552135157304236, −2.63768066237565615846388638570, −2.47600462797563496725773151650, −2.32042870900223378047411027926, −2.28007286410743235673800277247, −2.16954680516121578815263408390, −1.76626982298360311091601119003, −1.15314755041358650459711425266, −0.981465568049694033529470700852, −0.864784803723543020446377103744, −0.58239319241444963534525937350, −0.45422024122663926346917650619, 0.45422024122663926346917650619, 0.58239319241444963534525937350, 0.864784803723543020446377103744, 0.981465568049694033529470700852, 1.15314755041358650459711425266, 1.76626982298360311091601119003, 2.16954680516121578815263408390, 2.28007286410743235673800277247, 2.32042870900223378047411027926, 2.47600462797563496725773151650, 2.63768066237565615846388638570, 2.77607906931930552135157304236, 2.96619327250389796491487991684, 3.07792495138543732068890134561, 3.11268737081187728429117616230, 3.12834159359762304708597431312, 3.36114866313686682902960155952, 3.36696554176594540161337435032, 3.73364855203391378193108226767, 3.89051929061376608274061306591, 4.22750218161341761354355288077, 4.24918905050462287344085091642, 4.30592251167204352831845041925, 4.56807208083852890955816004050, 4.62845025381119585935137093097

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

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