# Properties

 Degree $2$ Conductor $6300$ Sign $0.447 - 0.894i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·7-s + 11-s − 2i·13-s − 6·19-s − i·23-s + 29-s − 2·31-s + 7i·37-s + 8·41-s − i·43-s − 2i·47-s − 49-s + 14i·53-s + 10·59-s + 3i·67-s + ⋯
 L(s)  = 1 + 0.377i·7-s + 0.301·11-s − 0.554i·13-s − 1.37·19-s − 0.208i·23-s + 0.185·29-s − 0.359·31-s + 1.15i·37-s + 1.24·41-s − 0.152i·43-s − 0.291i·47-s − 0.142·49-s + 1.92i·53-s + 1.30·59-s + 0.366i·67-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$6300$$    =    $$2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Sign: $0.447 - 0.894i$ Motivic weight: $$1$$ Character: $\chi_{6300} (6049, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 6300,\ (\ :1/2),\ 0.447 - 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.563705878$$ $$L(\frac12)$$ $$\approx$$ $$1.563705878$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1$$
7 $$1 - iT$$
good11 $$1 - T + 11T^{2}$$
13 $$1 + 2iT - 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 + 6T + 19T^{2}$$
23 $$1 + iT - 23T^{2}$$
29 $$1 - T + 29T^{2}$$
31 $$1 + 2T + 31T^{2}$$
37 $$1 - 7iT - 37T^{2}$$
41 $$1 - 8T + 41T^{2}$$
43 $$1 + iT - 43T^{2}$$
47 $$1 + 2iT - 47T^{2}$$
53 $$1 - 14iT - 53T^{2}$$
59 $$1 - 10T + 59T^{2}$$
61 $$1 + 61T^{2}$$
67 $$1 - 3iT - 67T^{2}$$
71 $$1 - 9T + 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 + T + 79T^{2}$$
83 $$1 + 2iT - 83T^{2}$$
89 $$1 - 2T + 89T^{2}$$
97 $$1 - 10iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$