Properties

 Degree 2 Conductor $2 \cdot 5 \cdot 11 \cdot 43$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + 2-s − 3.45·3-s + 4-s − 5-s − 3.45·6-s − 3.54·7-s + 8-s + 8.96·9-s − 10-s + 11-s − 3.45·12-s + 5.54·13-s − 3.54·14-s + 3.45·15-s + 16-s − 6.38·17-s + 8.96·18-s + 7.62·19-s − 20-s + 12.2·21-s + 22-s − 4.31·23-s − 3.45·24-s + 25-s + 5.54·26-s − 20.6·27-s − 3.54·28-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1.99·3-s + 0.5·4-s − 0.447·5-s − 1.41·6-s − 1.33·7-s + 0.353·8-s + 2.98·9-s − 0.316·10-s + 0.301·11-s − 0.998·12-s + 1.53·13-s − 0.947·14-s + 0.893·15-s + 0.250·16-s − 1.54·17-s + 2.11·18-s + 1.74·19-s − 0.223·20-s + 2.67·21-s + 0.213·22-s − 0.899·23-s − 0.706·24-s + 0.200·25-s + 1.08·26-s − 3.96·27-s − 0.669·28-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$4730$$    =    $$2 \cdot 5 \cdot 11 \cdot 43$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{4730} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 4730,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $1.013402037$ $L(\frac12)$ $\approx$ $1.013402037$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5,\;11,\;43\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;5,\;11,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - T$$
5 $$1 + T$$
11 $$1 - T$$
43 $$1 - T$$
good3 $$1 + 3.45T + 3T^{2}$$
7 $$1 + 3.54T + 7T^{2}$$
13 $$1 - 5.54T + 13T^{2}$$
17 $$1 + 6.38T + 17T^{2}$$
19 $$1 - 7.62T + 19T^{2}$$
23 $$1 + 4.31T + 23T^{2}$$
29 $$1 - 2.64T + 29T^{2}$$
31 $$1 + 1.22T + 31T^{2}$$
37 $$1 + 2.16T + 37T^{2}$$
41 $$1 + 3.56T + 41T^{2}$$
47 $$1 + 11.4T + 47T^{2}$$
53 $$1 - 2.96T + 53T^{2}$$
59 $$1 + 10.9T + 59T^{2}$$
61 $$1 + 0.437T + 61T^{2}$$
67 $$1 - 6.14T + 67T^{2}$$
71 $$1 + 5.21T + 71T^{2}$$
73 $$1 - 12.2T + 73T^{2}$$
79 $$1 - 11.4T + 79T^{2}$$
83 $$1 - 10.2T + 83T^{2}$$
89 $$1 + 13.7T + 89T^{2}$$
97 $$1 - 10.0T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}