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-s + 4-s + 6-s − 7-s + 8-s − 2·9-s + 12-s − 13-s − 14-s + 16-s − 2·18-s − 19-s − 21-s + 3·23-s + 24-s − 5·25-s − 26-s − 5·27-s − 28-s + 5·31-s + 32-s − 2·36-s − 4·37-s − 38-s − 39-s − 42-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.471·18-s − 0.229·19-s − 0.218·21-s + 0.625·23-s + 0.204·24-s − 25-s − 0.196·26-s − 0.962·27-s − 0.188·28-s + 0.898·31-s + 0.176·32-s − 1/3·36-s − 0.657·37-s − 0.162·38-s − 0.160·39-s − 0.154·42-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  =  $(2,\ 142,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.733348049$
$L(\frac12)$  $\approx$  $1.733348049$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−19.58024925031172, −19.06998719106987, −17.61800162822078, −16.79445936319647, −15.68682088550552, −14.92665109869052, −14.02033810524809, −13.32248972492364, −12.24215793074211, −11.31500525007428, −10.08859194652769, −8.920152760385419, −7.795105813975250, −6.520136540110099, −5.307677783577659, −3.776530019848495, −2.518743738692699, 2.518743738692699, 3.776530019848495, 5.307677783577659, 6.520136540110099, 7.795105813975250, 8.920152760385419, 10.08859194652769, 11.31500525007428, 12.24215793074211, 13.32248972492364, 14.02033810524809, 14.92665109869052, 15.68682088550552, 16.79445936319647, 17.61800162822078, 19.06998719106987, 19.58024925031172

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