# Properties

 Degree 2 Conductor 71 Sign $1$ Motivic weight 0 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.80·2-s − 0.445·3-s + 2.24·4-s + 1.24·5-s + 0.801·6-s − 2.24·8-s − 0.801·9-s − 2.24·10-s − 12-s − 0.554·15-s + 1.80·16-s + 1.44·18-s − 1.80·19-s + 2.80·20-s + 1.00·24-s + 0.554·25-s + 0.801·27-s − 0.445·29-s + 0.999·30-s − 1.00·32-s − 1.80·36-s − 1.80·37-s + 3.24·38-s − 2.80·40-s + 1.24·43-s − 45-s − 0.801·48-s + ⋯
 L(s)  = 1 − 1.80·2-s − 0.445·3-s + 2.24·4-s + 1.24·5-s + 0.801·6-s − 2.24·8-s − 0.801·9-s − 2.24·10-s − 12-s − 0.554·15-s + 1.80·16-s + 1.44·18-s − 1.80·19-s + 2.80·20-s + 1.00·24-s + 0.554·25-s + 0.801·27-s − 0.445·29-s + 0.999·30-s − 1.00·32-s − 1.80·36-s − 1.80·37-s + 3.24·38-s − 2.80·40-s + 1.24·43-s − 45-s − 0.801·48-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$71$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : $\chi_{71} (70, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 71,\ (\ :0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $0.2523313857$ $L(\frac12)$ $\approx$ $0.2523313857$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 71$, $$F_p(T)$$ is a polynomial of degree 2. If $p = 71$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad71 $$1 - T$$
good2 $$1 + 1.80T + T^{2}$$
3 $$1 + 0.445T + T^{2}$$
5 $$1 - 1.24T + T^{2}$$
7 $$1 - T^{2}$$
11 $$1 - T^{2}$$
13 $$1 - T^{2}$$
17 $$1 - T^{2}$$
19 $$1 + 1.80T + T^{2}$$
23 $$1 - T^{2}$$
29 $$1 + 0.445T + T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + 1.80T + T^{2}$$
41 $$1 - T^{2}$$
43 $$1 - 1.24T + T^{2}$$
47 $$1 - T^{2}$$
53 $$1 - T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 - T^{2}$$
73 $$1 - 1.24T + T^{2}$$
79 $$1 - 1.24T + T^{2}$$
83 $$1 + 1.80T + T^{2}$$
89 $$1 + 0.445T + T^{2}$$
97 $$1 - T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}