# Properties

 Degree $2$ Conductor $2352$ Sign $-0.386 + 0.922i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.499 − 0.866i)9-s + (2.5 − 4.33i)11-s + 0.999·15-s + (−2 + 3.46i)17-s + (−4 − 6.92i)19-s + (−2 − 3.46i)23-s + (2 − 3.46i)25-s − 0.999·27-s − 5·29-s + (−1.5 + 2.59i)31-s + (−2.5 − 4.33i)33-s + (2 + 3.46i)37-s − 2·43-s + ⋯
 L(s)  = 1 + (0.288 − 0.499i)3-s + (0.223 + 0.387i)5-s + (−0.166 − 0.288i)9-s + (0.753 − 1.30i)11-s + 0.258·15-s + (−0.485 + 0.840i)17-s + (−0.917 − 1.58i)19-s + (−0.417 − 0.722i)23-s + (0.400 − 0.692i)25-s − 0.192·27-s − 0.928·29-s + (−0.269 + 0.466i)31-s + (−0.435 − 0.753i)33-s + (0.328 + 0.569i)37-s − 0.304·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2352$$    =    $$2^{4} \cdot 3 \cdot 7^{2}$$ Sign: $-0.386 + 0.922i$ Motivic weight: $$1$$ Character: $\chi_{2352} (1537, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2352,\ (\ :1/2),\ -0.386 + 0.922i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.583094808$$ $$L(\frac12)$$ $$\approx$$ $$1.583094808$$ $$L(\frac{3}{2})$$ 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 + (-0.5 + 0.866i)T$$
7 $$1$$
good5 $$1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + 13T^{2}$$
17 $$1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + 5T + 29T^{2}$$
31 $$1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + 41T^{2}$$
43 $$1 + 2T + 43T^{2}$$
47 $$1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-5.5 + 9.52i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 2T + 71T^{2}$$
73 $$1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + 7T + 83T^{2}$$
89 $$1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 7T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$