# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5^{2}$ Sign $0.788 - 0.615i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.587 + 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (1.86 − 1.23i)5-s + (0.309 + 0.951i)6-s − 2.70i·7-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (2.09 + 0.786i)10-s + (−4.54 + 3.30i)11-s + (−0.587 + 0.809i)12-s + (−2.84 + 3.91i)13-s + (2.19 − 1.59i)14-s + (2.15 − 0.593i)15-s + (−0.809 − 0.587i)16-s + (0.994 − 0.323i)17-s + ⋯
 L(s)  = 1 + (0.415 + 0.572i)2-s + (0.549 + 0.178i)3-s + (−0.154 + 0.475i)4-s + (0.834 − 0.550i)5-s + (0.126 + 0.388i)6-s − 1.02i·7-s + (−0.336 + 0.109i)8-s + (0.269 + 0.195i)9-s + (0.661 + 0.248i)10-s + (−1.37 + 0.996i)11-s + (−0.169 + 0.233i)12-s + (−0.789 + 1.08i)13-s + (0.585 − 0.425i)14-s + (0.556 − 0.153i)15-s + (−0.202 − 0.146i)16-s + (0.241 − 0.0783i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$150$$    =    $$2 \cdot 3 \cdot 5^{2}$$ $$\varepsilon$$ = $0.788 - 0.615i$ motivic weight = $$1$$ character : $\chi_{150} (139, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 150,\ (\ :1/2),\ 0.788 - 0.615i)$ $L(1)$ $\approx$ $1.52551 + 0.525218i$ $L(\frac12)$ $\approx$ $1.52551 + 0.525218i$ $L(\frac{3}{2})$ 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 + (-0.587 - 0.809i)T$$
3 $$1 + (-0.951 - 0.309i)T$$
5 $$1 + (-1.86 + 1.23i)T$$
good7 $$1 + 2.70iT - 7T^{2}$$
11 $$1 + (4.54 - 3.30i)T + (3.39 - 10.4i)T^{2}$$
13 $$1 + (2.84 - 3.91i)T + (-4.01 - 12.3i)T^{2}$$
17 $$1 + (-0.994 + 0.323i)T + (13.7 - 9.99i)T^{2}$$
19 $$1 + (2.59 + 7.97i)T + (-15.3 + 11.1i)T^{2}$$
23 $$1 + (-3.11 - 4.28i)T + (-7.10 + 21.8i)T^{2}$$
29 $$1 + (1.29 - 3.98i)T + (-23.4 - 17.0i)T^{2}$$
31 $$1 + (1.72 + 5.30i)T + (-25.0 + 18.2i)T^{2}$$
37 $$1 + (-1.75 + 2.42i)T + (-11.4 - 35.1i)T^{2}$$
41 $$1 + (-1.27 - 0.927i)T + (12.6 + 38.9i)T^{2}$$
43 $$1 - 3.29iT - 43T^{2}$$
47 $$1 + (-2.92 - 0.949i)T + (38.0 + 27.6i)T^{2}$$
53 $$1 + (-5.87 - 1.90i)T + (42.8 + 31.1i)T^{2}$$
59 $$1 + (11.4 + 8.33i)T + (18.2 + 56.1i)T^{2}$$
61 $$1 + (0.218 - 0.159i)T + (18.8 - 58.0i)T^{2}$$
67 $$1 + (-6.00 + 1.94i)T + (54.2 - 39.3i)T^{2}$$
71 $$1 + (1.40 - 4.31i)T + (-57.4 - 41.7i)T^{2}$$
73 $$1 + (-4.79 - 6.60i)T + (-22.5 + 69.4i)T^{2}$$
79 $$1 + (3.85 - 11.8i)T + (-63.9 - 46.4i)T^{2}$$
83 $$1 + (-0.127 + 0.0415i)T + (67.1 - 48.7i)T^{2}$$
89 $$1 + (-9.53 + 6.93i)T + (27.5 - 84.6i)T^{2}$$
97 $$1 + (-2.51 - 0.815i)T + (78.4 + 57.0i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}