Properties

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

Related objects

Dirichlet series

 L(s)  = 1 + 0.618·2-s − 1.61·3-s − 0.618·4-s − 1.00·6-s + 0.618·7-s − 8-s + 1.61·9-s + 0.999·12-s + 0.381·14-s − 1.61·17-s + 1.00·18-s − 1.00·21-s + 1.61·24-s + 25-s − 27-s − 0.381·28-s + 0.999·32-s − 1.00·34-s − 0.999·36-s − 1.61·37-s − 0.618·42-s + 47-s − 0.618·49-s + 0.618·50-s + 2.61·51-s + 0.618·53-s − 0.618·54-s + ⋯
 L(s)  = 1 + 0.618·2-s − 1.61·3-s − 0.618·4-s − 1.00·6-s + 0.618·7-s − 8-s + 1.61·9-s + 0.999·12-s + 0.381·14-s − 1.61·17-s + 1.00·18-s − 1.00·21-s + 1.61·24-s + 25-s − 27-s − 0.381·28-s + 0.999·32-s − 1.00·34-s − 0.999·36-s − 1.61·37-s − 0.618·42-s + 47-s − 0.618·49-s + 0.618·50-s + 2.61·51-s + 0.618·53-s − 0.618·54-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$47$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : $\chi_{47} (46, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 47,\ (\ :0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $0.3597283851$ $L(\frac12)$ $\approx$ $0.3597283851$ $L(1)$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 47$, $$F_p$$ is a polynomial of degree 2. If $p = 47$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad47 $$1 - T$$
good2 $$1 - 0.618T + T^{2}$$
3 $$1 + 1.61T + T^{2}$$
5 $$1 - T^{2}$$
7 $$1 - 0.618T + T^{2}$$
11 $$1 - T^{2}$$
13 $$1 - T^{2}$$
17 $$1 + 1.61T + T^{2}$$
19 $$1 - T^{2}$$
23 $$1 - T^{2}$$
29 $$1 - T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + 1.61T + T^{2}$$
41 $$1 - T^{2}$$
43 $$1 - T^{2}$$
53 $$1 - 0.618T + T^{2}$$
59 $$1 - 0.618T + T^{2}$$
61 $$1 - 0.618T + T^{2}$$
67 $$1 - T^{2}$$
71 $$1 + 1.61T + T^{2}$$
73 $$1 - T^{2}$$
79 $$1 + 1.61T + T^{2}$$
83 $$1 - 2T + T^{2}$$
89 $$1 - 0.618T + T^{2}$$
97 $$1 - 0.618T + T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}