# Properties

 Degree $2$ Conductor $294$ Sign $1$ Motivic weight $5$ Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 4·2-s + 9·3-s + 16·4-s + 66·5-s + 36·6-s + 64·8-s + 81·9-s + 264·10-s − 60·11-s + 144·12-s + 658·13-s + 594·15-s + 256·16-s + 414·17-s + 324·18-s − 956·19-s + 1.05e3·20-s − 240·22-s + 600·23-s + 576·24-s + 1.23e3·25-s + 2.63e3·26-s + 729·27-s + 5.57e3·29-s + 2.37e3·30-s + 3.59e3·31-s + 1.02e3·32-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.18·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.834·10-s − 0.149·11-s + 0.288·12-s + 1.07·13-s + 0.681·15-s + 1/4·16-s + 0.347·17-s + 0.235·18-s − 0.607·19-s + 0.590·20-s − 0.105·22-s + 0.236·23-s + 0.204·24-s + 0.393·25-s + 0.763·26-s + 0.192·27-s + 1.23·29-s + 0.481·30-s + 0.671·31-s + 0.176·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$294$$    =    $$2 \cdot 3 \cdot 7^{2}$$ Sign: $1$ Motivic weight: $$5$$ Character: $\chi_{294} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 294,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$5.424389917$$ $$L(\frac12)$$ $$\approx$$ $$5.424389917$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - p^{2} T$$
3 $$1 - p^{2} T$$
7 $$1$$
good5 $$1 - 66 T + p^{5} T^{2}$$
11 $$1 + 60 T + p^{5} T^{2}$$
13 $$1 - 658 T + p^{5} T^{2}$$
17 $$1 - 414 T + p^{5} T^{2}$$
19 $$1 + 956 T + p^{5} T^{2}$$
23 $$1 - 600 T + p^{5} T^{2}$$
29 $$1 - 5574 T + p^{5} T^{2}$$
31 $$1 - 3592 T + p^{5} T^{2}$$
37 $$1 + 8458 T + p^{5} T^{2}$$
41 $$1 + 19194 T + p^{5} T^{2}$$
43 $$1 - 13316 T + p^{5} T^{2}$$
47 $$1 - 19680 T + p^{5} T^{2}$$
53 $$1 + 31266 T + p^{5} T^{2}$$
59 $$1 + 26340 T + p^{5} T^{2}$$
61 $$1 - 31090 T + p^{5} T^{2}$$
67 $$1 + 16804 T + p^{5} T^{2}$$
71 $$1 - 6120 T + p^{5} T^{2}$$
73 $$1 - 25558 T + p^{5} T^{2}$$
79 $$1 - 74408 T + p^{5} T^{2}$$
83 $$1 - 6468 T + p^{5} T^{2}$$
89 $$1 - 32742 T + p^{5} T^{2}$$
97 $$1 + 166082 T + p^{5} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$