# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5^{2}$ Sign $-0.212 - 0.977i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.891 + 0.453i)2-s + (−1.37 + 1.05i)3-s + (0.587 + 0.809i)4-s + (−1.04 + 1.97i)5-s + (−1.70 + 0.313i)6-s + (0.462 + 0.462i)7-s + (0.156 + 0.987i)8-s + (0.784 − 2.89i)9-s + (−1.82 + 1.28i)10-s + (−2.73 + 0.888i)11-s + (−1.66 − 0.494i)12-s + (1.97 + 3.86i)13-s + (0.202 + 0.621i)14-s + (−0.641 − 3.81i)15-s + (−0.309 + 0.951i)16-s + (5.75 − 0.911i)17-s + ⋯
 L(s)  = 1 + (0.630 + 0.321i)2-s + (−0.794 + 0.607i)3-s + (0.293 + 0.404i)4-s + (−0.467 + 0.883i)5-s + (−0.695 + 0.127i)6-s + (0.174 + 0.174i)7-s + (0.0553 + 0.349i)8-s + (0.261 − 0.965i)9-s + (−0.578 + 0.406i)10-s + (−0.824 + 0.267i)11-s + (−0.479 − 0.142i)12-s + (0.546 + 1.07i)13-s + (0.0539 + 0.166i)14-s + (−0.165 − 0.986i)15-s + (−0.0772 + 0.237i)16-s + (1.39 − 0.221i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$150$$    =    $$2 \cdot 3 \cdot 5^{2}$$ $$\varepsilon$$ = $-0.212 - 0.977i$ motivic weight = $$1$$ character : $\chi_{150} (17, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 150,\ (\ :1/2),\ -0.212 - 0.977i)$ $L(1)$ $\approx$ $0.725142 + 0.899315i$ $L(\frac12)$ $\approx$ $0.725142 + 0.899315i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;5\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + (-0.891 - 0.453i)T$$
3 $$1 + (1.37 - 1.05i)T$$
5 $$1 + (1.04 - 1.97i)T$$
good7 $$1 + (-0.462 - 0.462i)T + 7iT^{2}$$
11 $$1 + (2.73 - 0.888i)T + (8.89 - 6.46i)T^{2}$$
13 $$1 + (-1.97 - 3.86i)T + (-7.64 + 10.5i)T^{2}$$
17 $$1 + (-5.75 + 0.911i)T + (16.1 - 5.25i)T^{2}$$
19 $$1 + (-4.28 + 5.89i)T + (-5.87 - 18.0i)T^{2}$$
23 $$1 + (-0.316 + 0.622i)T + (-13.5 - 18.6i)T^{2}$$
29 $$1 + (-2.60 + 1.89i)T + (8.96 - 27.5i)T^{2}$$
31 $$1 + (4.82 + 3.50i)T + (9.57 + 29.4i)T^{2}$$
37 $$1 + (1.93 - 0.988i)T + (21.7 - 29.9i)T^{2}$$
41 $$1 + (-6.34 - 2.06i)T + (33.1 + 24.0i)T^{2}$$
43 $$1 + (5.08 - 5.08i)T - 43iT^{2}$$
47 $$1 + (0.474 - 2.99i)T + (-44.6 - 14.5i)T^{2}$$
53 $$1 + (-2.23 - 0.353i)T + (50.4 + 16.3i)T^{2}$$
59 $$1 + (2.66 - 8.19i)T + (-47.7 - 34.6i)T^{2}$$
61 $$1 + (-2.77 - 8.55i)T + (-49.3 + 35.8i)T^{2}$$
67 $$1 + (2.22 + 14.0i)T + (-63.7 + 20.7i)T^{2}$$
71 $$1 + (7.15 + 9.84i)T + (-21.9 + 67.5i)T^{2}$$
73 $$1 + (-0.498 - 0.254i)T + (42.9 + 59.0i)T^{2}$$
79 $$1 + (-6.00 - 8.25i)T + (-24.4 + 75.1i)T^{2}$$
83 $$1 + (0.742 + 4.68i)T + (-78.9 + 25.6i)T^{2}$$
89 $$1 + (1.78 + 5.49i)T + (-72.0 + 52.3i)T^{2}$$
97 $$1 + (-5.26 - 0.833i)T + (92.2 + 29.9i)T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}