Properties

Degree 2
Conductor $ 2 \cdot 71 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s + 4-s + 2·5-s − 3·6-s − 3·7-s − 8-s + 6·9-s − 2·10-s − 6·11-s + 3·12-s − 5·13-s + 3·14-s + 6·15-s + 16-s + 6·17-s − 6·18-s + 19-s + 2·20-s − 9·21-s + 6·22-s + 5·23-s − 3·24-s − 25-s + 5·26-s + 9·27-s − 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.894·5-s − 1.22·6-s − 1.13·7-s − 0.353·8-s + 2·9-s − 0.632·10-s − 1.80·11-s + 0.866·12-s − 1.38·13-s + 0.801·14-s + 1.54·15-s + 1/4·16-s + 1.45·17-s − 1.41·18-s + 0.229·19-s + 0.447·20-s − 1.96·21-s + 1.27·22-s + 1.04·23-s − 0.612·24-s − 1/5·25-s + 0.980·26-s + 1.73·27-s − 0.566·28-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  =  0
Selberg data  =  $(2,\ 142,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.300855191$
$L(\frac12)$  $\approx$  $1.300855191$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.39099929121939, −18.90634068873345, −18.07035564701808, −16.81019477362238, −15.90480700637484, −15.01224622282767, −14.15569246003640, −13.10988000360896, −12.60368438872146, −10.40842249661620, −9.755859107065146, −9.235323923464830, −7.863427877458433, −7.249325683219902, −5.471952658230112, −3.149967523192389, −2.363548517305414, 2.363548517305414, 3.149967523192389, 5.471952658230112, 7.249325683219902, 7.863427877458433, 9.235323923464830, 9.755859107065146, 10.40842249661620, 12.60368438872146, 13.10988000360896, 14.15569246003640, 15.01224622282767, 15.90480700637484, 16.81019477362238, 18.07035564701808, 18.90634068873345, 19.39099929121939

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