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 + 6-s + 4·7-s + 3·8-s + 9-s + 12-s + 6·13-s − 4·14-s − 16-s − 6·17-s − 18-s + 2·19-s − 4·21-s + 4·23-s − 3·24-s − 5·25-s − 6·26-s − 27-s − 4·28-s + 8·29-s + 6·31-s − 5·32-s + 6·34-s − 36-s − 6·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.288·12-s + 1.66·13-s − 1.06·14-s − 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.872·21-s + 0.834·23-s − 0.612·24-s − 25-s − 1.17·26-s − 0.192·27-s − 0.755·28-s + 1.48·29-s + 1.07·31-s − 0.883·32-s + 1.02·34-s − 1/6·36-s − 0.986·37-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  =  $(2,\ 141,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6902788027$
$L(\frac12)$  $\approx$  $0.6902788027$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−19.30918498335306, −18.35515841045995, −17.71869815357687, −17.38591503020114, −16.10891340610229, −15.25488526965811, −13.84239392845284, −13.39265884605229, −11.75312513610494, −11.05397435276072, −10.19203032281564, −8.698507358810087, −8.261405216068068, −6.792054951134437, −5.229813966660965, −4.239130972511684, −1.388121523866083, 1.388121523866083, 4.239130972511684, 5.229813966660965, 6.792054951134437, 8.261405216068068, 8.698507358810087, 10.19203032281564, 11.05397435276072, 11.75312513610494, 13.39265884605229, 13.84239392845284, 15.25488526965811, 16.10891340610229, 17.38591503020114, 17.71869815357687, 18.35515841045995, 19.30918498335306

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