# Properties

 Label 2-1350-5.4-c3-0-54 Degree $2$ Conductor $1350$ Sign $0.894 + 0.447i$ Analytic cond. $79.6525$ Root an. cond. $8.92482$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2i·2-s − 4·4-s − 23i·7-s − 8i·8-s + 30·11-s − 34i·13-s + 46·14-s + 16·16-s + 42i·17-s + 139·19-s + 60i·22-s + 192i·23-s + 68·26-s + 92i·28-s − 234·29-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.5·4-s − 1.24i·7-s − 0.353i·8-s + 0.822·11-s − 0.725i·13-s + 0.878·14-s + 0.250·16-s + 0.599i·17-s + 1.67·19-s + 0.581i·22-s + 1.74i·23-s + 0.512·26-s + 0.620i·28-s − 1.49·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1350$$    =    $$2 \cdot 3^{3} \cdot 5^{2}$$ Sign: $0.894 + 0.447i$ Analytic conductor: $$79.6525$$ Root analytic conductor: $$8.92482$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1350} (649, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1350,\ (\ :3/2),\ 0.894 + 0.447i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.948077068$$ $$L(\frac12)$$ $$\approx$$ $$1.948077068$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 2iT$$
3 $$1$$
5 $$1$$
good7 $$1 + 23iT - 343T^{2}$$
11 $$1 - 30T + 1.33e3T^{2}$$
13 $$1 + 34iT - 2.19e3T^{2}$$
17 $$1 - 42iT - 4.91e3T^{2}$$
19 $$1 - 139T + 6.85e3T^{2}$$
23 $$1 - 192iT - 1.21e4T^{2}$$
29 $$1 + 234T + 2.43e4T^{2}$$
31 $$1 + 55T + 2.97e4T^{2}$$
37 $$1 + 191iT - 5.06e4T^{2}$$
41 $$1 - 138T + 6.89e4T^{2}$$
43 $$1 - 53iT - 7.95e4T^{2}$$
47 $$1 + 366iT - 1.03e5T^{2}$$
53 $$1 + 330iT - 1.48e5T^{2}$$
59 $$1 - 396T + 2.05e5T^{2}$$
61 $$1 - 23T + 2.26e5T^{2}$$
67 $$1 + 452iT - 3.00e5T^{2}$$
71 $$1 - 204T + 3.57e5T^{2}$$
73 $$1 + 691iT - 3.89e5T^{2}$$
79 $$1 - 709T + 4.93e5T^{2}$$
83 $$1 - 1.09e3iT - 5.71e5T^{2}$$
89 $$1 - 816T + 7.04e5T^{2}$$
97 $$1 + 905iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$