# Properties

 Degree 2 Conductor $3^{2} \cdot 7 \cdot 127$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.42·2-s + 3.89·4-s − 4.13·5-s + 7-s − 4.59·8-s + 10.0·10-s + 4.39·11-s + 0.702·13-s − 2.42·14-s + 3.37·16-s + 0.979·17-s − 7.50·19-s − 16.0·20-s − 10.6·22-s + 4.28·23-s + 12.0·25-s − 1.70·26-s + 3.89·28-s − 8.97·29-s − 0.872·31-s + 0.999·32-s − 2.37·34-s − 4.13·35-s + 6.83·37-s + 18.2·38-s + 19.0·40-s + 4.36·41-s + ⋯
 L(s)  = 1 − 1.71·2-s + 1.94·4-s − 1.84·5-s + 0.377·7-s − 1.62·8-s + 3.17·10-s + 1.32·11-s + 0.194·13-s − 0.648·14-s + 0.844·16-s + 0.237·17-s − 1.72·19-s − 3.59·20-s − 2.27·22-s + 0.893·23-s + 2.41·25-s − 0.334·26-s + 0.735·28-s − 1.66·29-s − 0.156·31-s + 0.176·32-s − 0.407·34-s − 0.698·35-s + 1.12·37-s + 2.95·38-s + 3.00·40-s + 0.681·41-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8001$$    =    $$3^{2} \cdot 7 \cdot 127$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{8001} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 8001,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $0.4930520818$ $L(\frac12)$ $\approx$ $0.4930520818$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;7,\;127\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;7,\;127\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1$$
7 $$1 - T$$
127 $$1 - T$$
good2 $$1 + 2.42T + 2T^{2}$$
5 $$1 + 4.13T + 5T^{2}$$
11 $$1 - 4.39T + 11T^{2}$$
13 $$1 - 0.702T + 13T^{2}$$
17 $$1 - 0.979T + 17T^{2}$$
19 $$1 + 7.50T + 19T^{2}$$
23 $$1 - 4.28T + 23T^{2}$$
29 $$1 + 8.97T + 29T^{2}$$
31 $$1 + 0.872T + 31T^{2}$$
37 $$1 - 6.83T + 37T^{2}$$
41 $$1 - 4.36T + 41T^{2}$$
43 $$1 + 6.34T + 43T^{2}$$
47 $$1 + 2.85T + 47T^{2}$$
53 $$1 - 5.93T + 53T^{2}$$
59 $$1 + 5.83T + 59T^{2}$$
61 $$1 - 0.431T + 61T^{2}$$
67 $$1 - 1.64T + 67T^{2}$$
71 $$1 - 15.5T + 71T^{2}$$
73 $$1 + 3.79T + 73T^{2}$$
79 $$1 - 14.6T + 79T^{2}$$
83 $$1 - 5.45T + 83T^{2}$$
89 $$1 - 12.1T + 89T^{2}$$
97 $$1 - 9.29T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}