# Properties

 Degree 2 Conductor $2^{3} \cdot 7 \cdot 11 \cdot 13$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.00·3-s + 2.49·5-s − 7-s − 1.98·9-s + 11-s + 13-s − 2.52·15-s − 4.53·17-s − 0.365·19-s + 1.00·21-s + 2.18·23-s + 1.23·25-s + 5.02·27-s + 7.70·29-s − 6.21·31-s − 1.00·33-s − 2.49·35-s − 6.41·37-s − 1.00·39-s − 0.881·41-s − 2.18·43-s − 4.94·45-s + 13.5·47-s + 49-s + 4.57·51-s − 5.98·53-s + 2.49·55-s + ⋯
 L(s)  = 1 − 0.582·3-s + 1.11·5-s − 0.377·7-s − 0.660·9-s + 0.301·11-s + 0.277·13-s − 0.651·15-s − 1.09·17-s − 0.0839·19-s + 0.220·21-s + 0.455·23-s + 0.247·25-s + 0.967·27-s + 1.43·29-s − 1.11·31-s − 0.175·33-s − 0.422·35-s − 1.05·37-s − 0.161·39-s − 0.137·41-s − 0.332·43-s − 0.737·45-s + 1.97·47-s + 0.142·49-s + 0.641·51-s − 0.821·53-s + 0.336·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8008$$    =    $$2^{3} \cdot 7 \cdot 11 \cdot 13$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{8008} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 8008,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;7,\;11,\;13\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
7 $$1 + T$$
11 $$1 - T$$
13 $$1 - T$$
good3 $$1 + 1.00T + 3T^{2}$$
5 $$1 - 2.49T + 5T^{2}$$
17 $$1 + 4.53T + 17T^{2}$$
19 $$1 + 0.365T + 19T^{2}$$
23 $$1 - 2.18T + 23T^{2}$$
29 $$1 - 7.70T + 29T^{2}$$
31 $$1 + 6.21T + 31T^{2}$$
37 $$1 + 6.41T + 37T^{2}$$
41 $$1 + 0.881T + 41T^{2}$$
43 $$1 + 2.18T + 43T^{2}$$
47 $$1 - 13.5T + 47T^{2}$$
53 $$1 + 5.98T + 53T^{2}$$
59 $$1 - 11.1T + 59T^{2}$$
61 $$1 + 4.84T + 61T^{2}$$
67 $$1 + 0.801T + 67T^{2}$$
71 $$1 - 10.5T + 71T^{2}$$
73 $$1 + 15.0T + 73T^{2}$$
79 $$1 + 10.8T + 79T^{2}$$
83 $$1 + 11.4T + 83T^{2}$$
89 $$1 - 2.15T + 89T^{2}$$
97 $$1 + 4.92T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}