# Properties

 Label 2-2352-28.19-c1-0-4 Degree $2$ Conductor $2352$ Sign $0.832 - 0.553i$ Analytic cond. $18.7808$ Root an. cond. $4.33368$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)3-s + (−1.5 − 0.866i)5-s + (−0.499 + 0.866i)9-s + (−1.5 + 0.866i)11-s + 1.73i·15-s + (−3 + 1.73i)17-s + (1 − 1.73i)19-s + (−1 − 1.73i)25-s + 0.999·27-s + 9·29-s + (−2.5 − 4.33i)31-s + (1.5 + 0.866i)33-s + (−5 + 8.66i)37-s + 10.3i·41-s − 3.46i·43-s + ⋯
 L(s)  = 1 + (−0.288 − 0.499i)3-s + (−0.670 − 0.387i)5-s + (−0.166 + 0.288i)9-s + (−0.452 + 0.261i)11-s + 0.447i·15-s + (−0.727 + 0.420i)17-s + (0.229 − 0.397i)19-s + (−0.200 − 0.346i)25-s + 0.192·27-s + 1.67·29-s + (−0.449 − 0.777i)31-s + (0.261 + 0.150i)33-s + (−0.821 + 1.42i)37-s + 1.62i·41-s − 0.528i·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2352$$    =    $$2^{4} \cdot 3 \cdot 7^{2}$$ Sign: $0.832 - 0.553i$ Analytic conductor: $$18.7808$$ Root analytic conductor: $$4.33368$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2352} (607, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2352,\ (\ :1/2),\ 0.832 - 0.553i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9232449533$$ $$L(\frac12)$$ $$\approx$$ $$0.9232449533$$ $$L(\frac{3}{2})$$ 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 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2}$$
11 $$1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 + (3 - 1.73i)T + (8.5 - 14.7i)T^{2}$$
19 $$1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (11.5 + 19.9i)T^{2}$$
29 $$1 - 9T + 29T^{2}$$
31 $$1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 - 10.3iT - 41T^{2}$$
43 $$1 + 3.46iT - 43T^{2}$$
47 $$1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (30.5 + 52.8i)T^{2}$$
67 $$1 + (-12 + 6.92i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + 13.8iT - 71T^{2}$$
73 $$1 + (-6 + 3.46i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (4.5 + 2.59i)T + (39.5 + 68.4i)T^{2}$$
83 $$1 - 3T + 83T^{2}$$
89 $$1 + (-3 - 1.73i)T + (44.5 + 77.0i)T^{2}$$
97 $$1 - 19.0iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$