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 + 4-s + 2·5-s − 8-s − 3·9-s − 2·10-s + 6·11-s + 4·13-s + 16-s + 6·17-s + 3·18-s − 8·19-s + 2·20-s − 6·22-s − 4·23-s − 25-s − 4·26-s − 2·29-s − 8·31-s − 32-s − 6·34-s − 3·36-s + 10·37-s + 8·38-s − 2·40-s − 2·41-s − 8·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 9-s − 0.632·10-s + 1.80·11-s + 1.10·13-s + 1/4·16-s + 1.45·17-s + 0.707·18-s − 1.83·19-s + 0.447·20-s − 1.27·22-s − 0.834·23-s − 1/5·25-s − 0.784·26-s − 0.371·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s − 1/2·36-s + 1.64·37-s + 1.29·38-s − 0.316·40-s − 0.312·41-s − 1.21·43-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  =  $(2,\ 142,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9425722449$
$L(\frac12)$  $\approx$  $0.9425722449$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−19.54874877222536, −18.58752254524498, −17.68577392873818, −16.85357041911525, −16.50837790807043, −14.76028470552878, −14.36791855197331, −13.13307129139540, −11.87036840604067, −11.06145732082172, −9.875254168076625, −9.023272714070286, −8.153392726763726, −6.433848720554304, −5.871421460660023, −3.676560468497131, −1.734941398968249, 1.734941398968249, 3.676560468497131, 5.871421460660023, 6.433848720554304, 8.153392726763726, 9.023272714070286, 9.875254168076625, 11.06145732082172, 11.87036840604067, 13.13307129139540, 14.36791855197331, 14.76028470552878, 16.50837790807043, 16.85357041911525, 17.68577392873818, 18.58752254524498, 19.54874877222536

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