# Properties

 Degree 2 Conductor $2^{4} \cdot 503$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.78·3-s − 0.701·5-s + 2.02·7-s + 0.187·9-s + 0.626·11-s − 2.93·13-s + 1.25·15-s − 2.71·17-s + 1.11·19-s − 3.62·21-s + 0.412·23-s − 4.50·25-s + 5.02·27-s + 6.46·29-s + 4.14·31-s − 1.11·33-s − 1.42·35-s + 2.98·37-s + 5.24·39-s − 0.135·41-s + 0.861·43-s − 0.131·45-s − 1.67·47-s − 2.87·49-s + 4.83·51-s − 8.51·53-s − 0.439·55-s + ⋯
 L(s)  = 1 − 1.03·3-s − 0.313·5-s + 0.767·7-s + 0.0624·9-s + 0.189·11-s − 0.814·13-s + 0.323·15-s − 0.657·17-s + 0.254·19-s − 0.790·21-s + 0.0859·23-s − 0.901·25-s + 0.966·27-s + 1.20·29-s + 0.743·31-s − 0.194·33-s − 0.240·35-s + 0.490·37-s + 0.839·39-s − 0.0211·41-s + 0.131·43-s − 0.0195·45-s − 0.244·47-s − 0.411·49-s + 0.677·51-s − 1.16·53-s − 0.0592·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8048$$    =    $$2^{4} \cdot 503$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{8048} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 8048,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;503\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
503 $$1 - T$$
good3 $$1 + 1.78T + 3T^{2}$$
5 $$1 + 0.701T + 5T^{2}$$
7 $$1 - 2.02T + 7T^{2}$$
11 $$1 - 0.626T + 11T^{2}$$
13 $$1 + 2.93T + 13T^{2}$$
17 $$1 + 2.71T + 17T^{2}$$
19 $$1 - 1.11T + 19T^{2}$$
23 $$1 - 0.412T + 23T^{2}$$
29 $$1 - 6.46T + 29T^{2}$$
31 $$1 - 4.14T + 31T^{2}$$
37 $$1 - 2.98T + 37T^{2}$$
41 $$1 + 0.135T + 41T^{2}$$
43 $$1 - 0.861T + 43T^{2}$$
47 $$1 + 1.67T + 47T^{2}$$
53 $$1 + 8.51T + 53T^{2}$$
59 $$1 - 0.406T + 59T^{2}$$
61 $$1 + 8.53T + 61T^{2}$$
67 $$1 - 13.0T + 67T^{2}$$
71 $$1 + 2.78T + 71T^{2}$$
73 $$1 + 2.11T + 73T^{2}$$
79 $$1 - 0.317T + 79T^{2}$$
83 $$1 - 5.05T + 83T^{2}$$
89 $$1 - 16.4T + 89T^{2}$$
97 $$1 - 7.68T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}