Properties

Degree 2
Conductor $ 2 \cdot 71 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 4-s − 4·5-s − 3·6-s − 3·7-s + 8-s + 6·9-s − 4·10-s − 3·12-s + 13-s − 3·14-s + 12·15-s + 16-s + 6·18-s − 5·19-s − 4·20-s + 9·21-s − 7·23-s − 3·24-s + 11·25-s + 26-s − 9·27-s − 3·28-s − 8·29-s + 12·30-s + 7·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.78·5-s − 1.22·6-s − 1.13·7-s + 0.353·8-s + 2·9-s − 1.26·10-s − 0.866·12-s + 0.277·13-s − 0.801·14-s + 3.09·15-s + 1/4·16-s + 1.41·18-s − 1.14·19-s − 0.894·20-s + 1.96·21-s − 1.45·23-s − 0.612·24-s + 11/5·25-s + 0.196·26-s − 1.73·27-s − 0.566·28-s − 1.48·29-s + 2.19·30-s + 1.25·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(142\)    =    \(2 \cdot 71\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{142} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 142,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;71\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;71\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
71 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.82155763409596, −19.18178768603098, −18.27508303353933, −16.93634020792167, −16.21178065567619, −15.81056737523845, −14.89029176685616, −13.07323328130018, −12.47781640487055, −11.66428490243199, −11.09563244780362, −10.00722131740449, −8.015127034089896, −6.796658899889500, −6.069565651126277, −4.593405476508064, −3.696521183586152, 0, 3.696521183586152, 4.593405476508064, 6.069565651126277, 6.796658899889500, 8.015127034089896, 10.00722131740449, 11.09563244780362, 11.66428490243199, 12.47781640487055, 13.07323328130018, 14.89029176685616, 15.81056737523845, 16.21178065567619, 16.93634020792167, 18.27508303353933, 19.18178768603098, 19.82155763409596

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