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-s + 4-s − 2·5-s + 6-s − 7-s − 8-s − 2·9-s + 2·10-s − 2·11-s − 12-s − 3·13-s + 14-s + 2·15-s + 16-s − 6·17-s + 2·18-s + 5·19-s − 2·20-s + 21-s + 2·22-s − 23-s + 24-s − 25-s + 3·26-s + 5·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.632·10-s − 0.603·11-s − 0.288·12-s − 0.832·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.471·18-s + 1.14·19-s − 0.447·20-s + 0.218·21-s + 0.426·22-s − 0.208·23-s + 0.204·24-s − 1/5·25-s + 0.588·26-s + 0.962·27-s − 0.188·28-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 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−19.81277593637068, −19.22356244622491, −17.98903644255477, −17.52182898802246, −16.33271631513259, −15.83958368956920, −14.86105356380939, −13.52394926255452, −12.20412994837287, −11.57220845793572, −10.67984973493383, −9.544113424423370, −8.356442494540894, −7.391410364234232, −6.213317267162534, −4.761417136870512, −2.880766729048032, 0, 2.880766729048032, 4.761417136870512, 6.213317267162534, 7.391410364234232, 8.356442494540894, 9.544113424423370, 10.67984973493383, 11.57220845793572, 12.20412994837287, 13.52394926255452, 14.86105356380939, 15.83958368956920, 16.33271631513259, 17.52182898802246, 17.98903644255477, 19.22356244622491, 19.81277593637068

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