# Properties

 Degree 2 Conductor $2^{3} \cdot 3 \cdot 5^{2}$ Sign $0.766 + 0.641i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.192 + 1.40i)2-s − 3-s + (−1.92 + 0.540i)4-s + (−0.192 − 1.40i)6-s − 0.0802i·7-s + (−1.12 − 2.59i)8-s + 9-s − 2.41i·11-s + (1.92 − 0.540i)12-s − 5.26·13-s + (0.112 − 0.0154i)14-s + (3.41 − 2.08i)16-s + 0.255i·17-s + (0.192 + 1.40i)18-s − 6.95i·19-s + ⋯
 L(s)  = 1 + (0.136 + 0.990i)2-s − 0.577·3-s + (−0.962 + 0.270i)4-s + (−0.0786 − 0.571i)6-s − 0.0303i·7-s + (−0.398 − 0.917i)8-s + 0.333·9-s − 0.728i·11-s + (0.555 − 0.155i)12-s − 1.46·13-s + (0.0300 − 0.00413i)14-s + (0.854 − 0.520i)16-s + 0.0620i·17-s + (0.0454 + 0.330i)18-s − 1.59i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$600$$    =    $$2^{3} \cdot 3 \cdot 5^{2}$$ $$\varepsilon$$ = $0.766 + 0.641i$ motivic weight = $$1$$ character : $\chi_{600} (349, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 600,\ (\ :1/2),\ 0.766 + 0.641i)$$ $$L(1)$$ $$\approx$$ $$0.634681 - 0.230579i$$ $$L(\frac12)$$ $$\approx$$ $$0.634681 - 0.230579i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-0.192 - 1.40i)T$$
3 $$1 + T$$
5 $$1$$
good7 $$1 + 0.0802iT - 7T^{2}$$
11 $$1 + 2.41iT - 11T^{2}$$
13 $$1 + 5.26T + 13T^{2}$$
17 $$1 - 0.255iT - 17T^{2}$$
19 $$1 + 6.95iT - 19T^{2}$$
23 $$1 + 1.64iT - 23T^{2}$$
29 $$1 + 4.51iT - 29T^{2}$$
31 $$1 - 8.29T + 31T^{2}$$
37 $$1 - 2.67T + 37T^{2}$$
41 $$1 + 8.11T + 41T^{2}$$
43 $$1 - 4.08T + 43T^{2}$$
47 $$1 + 5.70iT - 47T^{2}$$
53 $$1 + 11.5T + 53T^{2}$$
59 $$1 + 12.6iT - 59T^{2}$$
61 $$1 - 11.9iT - 61T^{2}$$
67 $$1 + 7.27T + 67T^{2}$$
71 $$1 + 11.3T + 71T^{2}$$
73 $$1 + 12.0iT - 73T^{2}$$
79 $$1 + 5.50T + 79T^{2}$$
83 $$1 - 9.20T + 83T^{2}$$
89 $$1 + 11.9T + 89T^{2}$$
97 $$1 + 8.50iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}