Properties

Degree 2
Conductor $ 3 \cdot 47 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 5-s − 3·7-s + 9-s − 3·11-s + 2·12-s − 4·13-s + 15-s + 4·16-s + 8·17-s − 6·19-s + 2·20-s + 3·21-s + 3·23-s − 4·25-s − 27-s + 6·28-s − 29-s + 4·31-s + 3·33-s + 3·35-s − 2·36-s + 37-s + 4·39-s − 10·41-s − 8·43-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.577·12-s − 1.10·13-s + 0.258·15-s + 16-s + 1.94·17-s − 1.37·19-s + 0.447·20-s + 0.654·21-s + 0.625·23-s − 4/5·25-s − 0.192·27-s + 1.13·28-s − 0.185·29-s + 0.718·31-s + 0.522·33-s + 0.507·35-s − 1/3·36-s + 0.164·37-s + 0.640·39-s − 1.56·41-s − 1.21·43-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  =  1
Selberg data  =  $(2,\ 141,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−19.65638073882631, −18.98206520119263, −18.44566268172067, −17.08150269707128, −16.75743588819683, −15.51599358927305, −14.64890592912454, −13.35526816566080, −12.66579327489625, −11.90753988390856, −10.19500279746488, −9.896550436222926, −8.405267645813231, −7.307368496935090, −5.834860529322476, −4.753498304195796, −3.318478101145632, 0, 3.318478101145632, 4.753498304195796, 5.834860529322476, 7.307368496935090, 8.405267645813231, 9.896550436222926, 10.19500279746488, 11.90753988390856, 12.66579327489625, 13.35526816566080, 14.64890592912454, 15.51599358927305, 16.75743588819683, 17.08150269707128, 18.44566268172067, 18.98206520119263, 19.65638073882631

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