# 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 − 1.40·3-s + 3.37·5-s − 3.92·7-s − 1.02·9-s + 1.19·11-s + 1.76·13-s − 4.73·15-s − 17-s − 5.97·19-s + 5.51·21-s − 1.06·23-s + 6.38·25-s + 5.65·27-s + 5.79·29-s + 0.924·31-s − 1.67·33-s − 13.2·35-s − 6.65·37-s − 2.48·39-s + 8.89·41-s + 8.28·43-s − 3.46·45-s − 1.68·47-s + 8.40·49-s + 1.40·51-s + 3.41·53-s + 4.03·55-s + ⋯
 L(s)  = 1 − 0.810·3-s + 1.50·5-s − 1.48·7-s − 0.342·9-s + 0.360·11-s + 0.490·13-s − 1.22·15-s − 0.242·17-s − 1.36·19-s + 1.20·21-s − 0.221·23-s + 1.27·25-s + 1.08·27-s + 1.07·29-s + 0.165·31-s − 0.292·33-s − 2.23·35-s − 1.09·37-s − 0.397·39-s + 1.38·41-s + 1.26·43-s − 0.517·45-s − 0.245·47-s + 1.20·49-s + 0.196·51-s + 0.469·53-s + 0.543·55-s + ⋯

## Functional equation

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

## 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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
17 $$1 + T$$
59 $$1 + T$$
good3 $$1 + 1.40T + 3T^{2}$$
5 $$1 - 3.37T + 5T^{2}$$
7 $$1 + 3.92T + 7T^{2}$$
11 $$1 - 1.19T + 11T^{2}$$
13 $$1 - 1.76T + 13T^{2}$$
19 $$1 + 5.97T + 19T^{2}$$
23 $$1 + 1.06T + 23T^{2}$$
29 $$1 - 5.79T + 29T^{2}$$
31 $$1 - 0.924T + 31T^{2}$$
37 $$1 + 6.65T + 37T^{2}$$
41 $$1 - 8.89T + 41T^{2}$$
43 $$1 - 8.28T + 43T^{2}$$
47 $$1 + 1.68T + 47T^{2}$$
53 $$1 - 3.41T + 53T^{2}$$
61 $$1 + 6.47T + 61T^{2}$$
67 $$1 + 1.63T + 67T^{2}$$
71 $$1 + 11.1T + 71T^{2}$$
73 $$1 - 12.4T + 73T^{2}$$
79 $$1 + 11.1T + 79T^{2}$$
83 $$1 - 9.00T + 83T^{2}$$
89 $$1 + 11.8T + 89T^{2}$$
97 $$1 + 17.1T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}