Properties

Degree 2
Conductor $ 3 \cdot 47 $
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 + 2·5-s − 6-s + 3·8-s + 9-s − 2·10-s + 4·11-s − 12-s − 2·13-s + 2·15-s − 16-s + 2·17-s − 18-s − 2·20-s − 4·22-s + 3·24-s − 25-s + 2·26-s + 27-s − 6·29-s − 2·30-s − 4·31-s − 5·32-s + 4·33-s − 2·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.447·20-s − 0.852·22-s + 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 1.11·29-s − 0.365·30-s − 0.718·31-s − 0.883·32-s + 0.696·33-s − 0.342·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(141\)    =    \(3 \cdot 47\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{141} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 141,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9624779691$
$L(\frac12)$  $\approx$  $0.9624779691$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;47\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;47\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
47 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−19.35631543239757, −18.77740204924927, −17.65319384360445, −17.20695735393980, −16.26716073005224, −14.75891576058830, −14.13631296987067, −13.33645536469428, −12.25269126968811, −10.75662405333874, −9.575443434008071, −9.310688144069273, −8.104359289936234, −6.919959083244517, −5.362704434971122, −3.829866538039977, −1.746301502258159, 1.746301502258159, 3.829866538039977, 5.362704434971122, 6.919959083244517, 8.104359289936234, 9.310688144069273, 9.575443434008071, 10.75662405333874, 12.25269126968811, 13.33645536469428, 14.13631296987067, 14.75891576058830, 16.26716073005224, 17.20695735393980, 17.65319384360445, 18.77740204924927, 19.35631543239757

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