# Properties

 Degree 2 Conductor $3 \cdot 47$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

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## 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