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  + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s − 3·7-s + 9-s − 2·10-s + 11-s + 2·12-s − 2·13-s − 6·14-s − 15-s − 4·16-s + 2·17-s + 2·18-s + 6·19-s − 2·20-s − 3·21-s + 2·22-s + 3·23-s − 4·25-s − 4·26-s + 27-s − 6·28-s + 3·29-s − 2·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 1.13·7-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s − 0.554·13-s − 1.60·14-s − 0.258·15-s − 16-s + 0.485·17-s + 0.471·18-s + 1.37·19-s − 0.447·20-s − 0.654·21-s + 0.426·22-s + 0.625·23-s − 4/5·25-s − 0.784·26-s + 0.192·27-s − 1.13·28-s + 0.557·29-s − 0.365·30-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  =  $(2,\ 141,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.107535578$
$L(\frac12)$  $\approx$  $2.107535578$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−19.67971175162188, −19.09523995231317, −17.83426205003420, −16.38121520610161, −15.71155894301007, −14.88265784993754, −13.98456106931414, −13.27824273167304, −12.33792187853332, −11.68018802766311, −10.03012465082033, −9.085897389420590, −7.525398570351447, −6.464878707154972, −5.148855718214433, −3.772578814910722, −2.936570543267435, 2.936570543267435, 3.772578814910722, 5.148855718214433, 6.464878707154972, 7.525398570351447, 9.085897389420590, 10.03012465082033, 11.68018802766311, 12.33792187853332, 13.27824273167304, 13.98456106931414, 14.88265784993754, 15.71155894301007, 16.38121520610161, 17.83426205003420, 19.09523995231317, 19.67971175162188

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