# Properties

 Degree 2 Conductor $2 \cdot 19 \cdot 211$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s − 3.22·3-s + 4-s + 3.63·5-s − 3.22·6-s − 0.166·7-s + 8-s + 7.38·9-s + 3.63·10-s + 2.07·11-s − 3.22·12-s + 5.70·13-s − 0.166·14-s − 11.7·15-s + 16-s − 7.23·17-s + 7.38·18-s − 19-s + 3.63·20-s + 0.536·21-s + 2.07·22-s − 0.228·23-s − 3.22·24-s + 8.22·25-s + 5.70·26-s − 14.1·27-s − 0.166·28-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1.86·3-s + 0.5·4-s + 1.62·5-s − 1.31·6-s − 0.0628·7-s + 0.353·8-s + 2.46·9-s + 1.14·10-s + 0.624·11-s − 0.930·12-s + 1.58·13-s − 0.0444·14-s − 3.02·15-s + 0.250·16-s − 1.75·17-s + 1.74·18-s − 0.229·19-s + 0.813·20-s + 0.116·21-s + 0.441·22-s − 0.0476·23-s − 0.657·24-s + 1.64·25-s + 1.11·26-s − 2.71·27-s − 0.0314·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8018$$    =    $$2 \cdot 19 \cdot 211$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{8018} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 8018,\ (\ :1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$2.844005307$$ $$L(\frac12)$$ $$\approx$$ $$2.844005307$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;19,\;211\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;19,\;211\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - T$$
19 $$1 + T$$
211 $$1 + T$$
good3 $$1 + 3.22T + 3T^{2}$$
5 $$1 - 3.63T + 5T^{2}$$
7 $$1 + 0.166T + 7T^{2}$$
11 $$1 - 2.07T + 11T^{2}$$
13 $$1 - 5.70T + 13T^{2}$$
17 $$1 + 7.23T + 17T^{2}$$
23 $$1 + 0.228T + 23T^{2}$$
29 $$1 - 5.83T + 29T^{2}$$
31 $$1 + 6.23T + 31T^{2}$$
37 $$1 - 5.72T + 37T^{2}$$
41 $$1 - 7.52T + 41T^{2}$$
43 $$1 - 6.52T + 43T^{2}$$
47 $$1 - 12.3T + 47T^{2}$$
53 $$1 + 7.72T + 53T^{2}$$
59 $$1 - 0.482T + 59T^{2}$$
61 $$1 - 6.01T + 61T^{2}$$
67 $$1 - 3.78T + 67T^{2}$$
71 $$1 - 9.35T + 71T^{2}$$
73 $$1 + 5.89T + 73T^{2}$$
79 $$1 + 12.8T + 79T^{2}$$
83 $$1 + 8.79T + 83T^{2}$$
89 $$1 - 18.4T + 89T^{2}$$
97 $$1 + 0.277T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}