# Properties

 Degree $2$ Conductor $1512$ Sign $0.984 + 0.174i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.366 + 0.633i)5-s + (2.12 + 1.57i)7-s + (1.92 − 3.33i)11-s − 4.87·13-s + (3.09 − 5.36i)17-s + (−1.39 − 2.41i)19-s + (3.69 + 6.40i)23-s + (2.23 − 3.86i)25-s + 9.78·29-s + (3.15 − 5.46i)31-s + (−0.216 + 1.92i)35-s + (2.82 + 4.88i)37-s + 1.50·41-s − 12.1·43-s + (−2.95 − 5.12i)47-s + ⋯
 L(s)  = 1 + (0.163 + 0.283i)5-s + (0.804 + 0.593i)7-s + (0.580 − 1.00i)11-s − 1.35·13-s + (0.751 − 1.30i)17-s + (−0.320 − 0.555i)19-s + (0.771 + 1.33i)23-s + (0.446 − 0.773i)25-s + 1.81·29-s + (0.566 − 0.982i)31-s + (−0.0365 + 0.325i)35-s + (0.463 + 0.803i)37-s + 0.234·41-s − 1.85·43-s + (−0.431 − 0.747i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1512$$    =    $$2^{3} \cdot 3^{3} \cdot 7$$ Sign: $0.984 + 0.174i$ Motivic weight: $$1$$ Character: $\chi_{1512} (865, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1512,\ (\ :1/2),\ 0.984 + 0.174i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.948348686$$ $$L(\frac12)$$ $$\approx$$ $$1.948348686$$ $$L(\frac{3}{2})$$ 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$$
7 $$1 + (-2.12 - 1.57i)T$$
good5 $$1 + (-0.366 - 0.633i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-1.92 + 3.33i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + 4.87T + 13T^{2}$$
17 $$1 + (-3.09 + 5.36i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (1.39 + 2.41i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-3.69 - 6.40i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 - 9.78T + 29T^{2}$$
31 $$1 + (-3.15 + 5.46i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-2.82 - 4.88i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 1.50T + 41T^{2}$$
43 $$1 + 12.1T + 43T^{2}$$
47 $$1 + (2.95 + 5.12i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (6.43 - 11.1i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-3.19 + 5.52i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-6.29 - 10.9i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (0.834 - 1.44i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 - 13.3T + 71T^{2}$$
73 $$1 + (-0.720 + 1.24i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (3.55 + 6.15i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 7.98T + 83T^{2}$$
89 $$1 + (-3.62 - 6.27i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 - 12.1T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$