# 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

# Learn more about

## 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.