# Properties

 Label 2-675-5.4-c3-0-38 Degree $2$ Conductor $675$ Sign $0.894 - 0.447i$ Analytic cond. $39.8262$ Root an. cond. $6.31080$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4.45i·2-s − 11.8·4-s − 5.08i·7-s − 17.3i·8-s − 58.3·11-s + 21.2i·13-s + 22.6·14-s − 17.8·16-s − 68.8i·17-s + 40.8·19-s − 259. i·22-s + 144. i·23-s − 94.5·26-s + 60.3i·28-s − 220.·29-s + ⋯
 L(s)  = 1 + 1.57i·2-s − 1.48·4-s − 0.274i·7-s − 0.765i·8-s − 1.59·11-s + 0.452i·13-s + 0.432·14-s − 0.278·16-s − 0.982i·17-s + 0.492·19-s − 2.51i·22-s + 1.30i·23-s − 0.713·26-s + 0.407i·28-s − 1.40·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{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: $$675$$    =    $$3^{3} \cdot 5^{2}$$ Sign: $0.894 - 0.447i$ Analytic conductor: $$39.8262$$ Root analytic conductor: $$6.31080$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{675} (649, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 675,\ (\ :3/2),\ 0.894 - 0.447i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.9943677373$$ $$L(\frac12)$$ $$\approx$$ $$0.9943677373$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
good2 $$1 - 4.45iT - 8T^{2}$$
7 $$1 + 5.08iT - 343T^{2}$$
11 $$1 + 58.3T + 1.33e3T^{2}$$
13 $$1 - 21.2iT - 2.19e3T^{2}$$
17 $$1 + 68.8iT - 4.91e3T^{2}$$
19 $$1 - 40.8T + 6.85e3T^{2}$$
23 $$1 - 144. iT - 1.21e4T^{2}$$
29 $$1 + 220.T + 2.43e4T^{2}$$
31 $$1 - 291.T + 2.97e4T^{2}$$
37 $$1 + 260. iT - 5.06e4T^{2}$$
41 $$1 - 169.T + 6.89e4T^{2}$$
43 $$1 + 438. iT - 7.95e4T^{2}$$
47 $$1 + 255. iT - 1.03e5T^{2}$$
53 $$1 + 214. iT - 1.48e5T^{2}$$
59 $$1 - 331.T + 2.05e5T^{2}$$
61 $$1 - 54.9T + 2.26e5T^{2}$$
67 $$1 + 758. iT - 3.00e5T^{2}$$
71 $$1 - 904.T + 3.57e5T^{2}$$
73 $$1 - 866. iT - 3.89e5T^{2}$$
79 $$1 + 206.T + 4.93e5T^{2}$$
83 $$1 - 463. iT - 5.71e5T^{2}$$
89 $$1 - 601.T + 7.04e5T^{2}$$
97 $$1 + 229. iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$