# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7$ Sign $0.894 + 0.447i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + i·7-s + 6·11-s − 2i·13-s + 4·19-s − 6i·23-s + 6·29-s + 8·31-s + 2i·37-s − 12·41-s + 4i·43-s − 12i·47-s − 49-s − 6i·53-s − 10·61-s + 8i·67-s + ⋯
 L(s)  = 1 + 0.377i·7-s + 1.80·11-s − 0.554i·13-s + 0.917·19-s − 1.25i·23-s + 1.11·29-s + 1.43·31-s + 0.328i·37-s − 1.87·41-s + 0.609i·43-s − 1.75i·47-s − 0.142·49-s − 0.824i·53-s − 1.28·61-s + 0.977i·67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$6300$$    =    $$2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $0.894 + 0.447i$ motivic weight = $$1$$ character : $\chi_{6300} (6049, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 6300,\ (\ :1/2),\ 0.894 + 0.447i)$$ $$L(1)$$ $$\approx$$ $$2.399372177$$ $$L(\frac12)$$ $$\approx$$ $$2.399372177$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1$$
7 $$1 - iT$$
good11 $$1 - 6T + 11T^{2}$$
13 $$1 + 2iT - 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 - 4T + 19T^{2}$$
23 $$1 + 6iT - 23T^{2}$$
29 $$1 - 6T + 29T^{2}$$
31 $$1 - 8T + 31T^{2}$$
37 $$1 - 2iT - 37T^{2}$$
41 $$1 + 12T + 41T^{2}$$
43 $$1 - 4iT - 43T^{2}$$
47 $$1 + 12iT - 47T^{2}$$
53 $$1 + 6iT - 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 + 10T + 61T^{2}$$
67 $$1 - 8iT - 67T^{2}$$
71 $$1 + 6T + 71T^{2}$$
73 $$1 - 10iT - 73T^{2}$$
79 $$1 - 4T + 79T^{2}$$
83 $$1 + 12iT - 83T^{2}$$
89 $$1 - 12T + 89T^{2}$$
97 $$1 + 10iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}