# Properties

 Degree 2 Conductor $2^{3} \cdot 17 \cdot 59$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.725·3-s − 2.56·5-s − 0.934·7-s − 2.47·9-s + 0.446·11-s + 5.91·13-s + 1.85·15-s − 17-s − 3.85·19-s + 0.677·21-s − 7.09·23-s + 1.56·25-s + 3.97·27-s + 9.38·29-s − 8.41·31-s − 0.324·33-s + 2.39·35-s + 5.65·37-s − 4.29·39-s + 8.92·41-s + 9.40·43-s + 6.33·45-s + 9.71·47-s − 6.12·49-s + 0.725·51-s − 4.68·53-s − 1.14·55-s + ⋯
 L(s)  = 1 − 0.418·3-s − 1.14·5-s − 0.353·7-s − 0.824·9-s + 0.134·11-s + 1.64·13-s + 0.479·15-s − 0.242·17-s − 0.883·19-s + 0.147·21-s − 1.48·23-s + 0.312·25-s + 0.764·27-s + 1.74·29-s − 1.51·31-s − 0.0564·33-s + 0.404·35-s + 0.930·37-s − 0.687·39-s + 1.39·41-s + 1.43·43-s + 0.944·45-s + 1.41·47-s − 0.875·49-s + 0.101·51-s − 0.643·53-s − 0.154·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8024$$    =    $$2^{3} \cdot 17 \cdot 59$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{8024} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 8024,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;17,\;59\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;17,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
17 $$1 + T$$
59 $$1 + T$$
good3 $$1 + 0.725T + 3T^{2}$$
5 $$1 + 2.56T + 5T^{2}$$
7 $$1 + 0.934T + 7T^{2}$$
11 $$1 - 0.446T + 11T^{2}$$
13 $$1 - 5.91T + 13T^{2}$$
19 $$1 + 3.85T + 19T^{2}$$
23 $$1 + 7.09T + 23T^{2}$$
29 $$1 - 9.38T + 29T^{2}$$
31 $$1 + 8.41T + 31T^{2}$$
37 $$1 - 5.65T + 37T^{2}$$
41 $$1 - 8.92T + 41T^{2}$$
43 $$1 - 9.40T + 43T^{2}$$
47 $$1 - 9.71T + 47T^{2}$$
53 $$1 + 4.68T + 53T^{2}$$
61 $$1 - 3.41T + 61T^{2}$$
67 $$1 - 1.82T + 67T^{2}$$
71 $$1 - 0.929T + 71T^{2}$$
73 $$1 + 12.9T + 73T^{2}$$
79 $$1 - 10.8T + 79T^{2}$$
83 $$1 + 13.5T + 83T^{2}$$
89 $$1 - 8.98T + 89T^{2}$$
97 $$1 + 0.541T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}