# Properties

 Label 2-675-5.4-c3-0-34 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 − 0.258i·2-s + 7.93·4-s − 14.5i·7-s − 4.12i·8-s + 49.2·11-s + 72.1i·13-s − 3.75·14-s + 62.3·16-s + 118. i·17-s − 123.·19-s − 12.7i·22-s + 91.4i·23-s + 18.6·26-s − 115. i·28-s + 174.·29-s + ⋯
 L(s)  = 1 − 0.0914i·2-s + 0.991·4-s − 0.783i·7-s − 0.182i·8-s + 1.35·11-s + 1.53i·13-s − 0.0716·14-s + 0.974·16-s + 1.68i·17-s − 1.48·19-s − 0.123i·22-s + 0.829i·23-s + 0.140·26-s − 0.777i·28-s + 1.11·29-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

 $$L(2)$$ $$\approx$$ $$2.775819968$$ $$L(\frac12)$$ $$\approx$$ $$2.775819968$$ $$L(\frac{5}{2})$$ 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 + 0.258iT - 8T^{2}$$
7 $$1 + 14.5iT - 343T^{2}$$
11 $$1 - 49.2T + 1.33e3T^{2}$$
13 $$1 - 72.1iT - 2.19e3T^{2}$$
17 $$1 - 118. iT - 4.91e3T^{2}$$
19 $$1 + 123.T + 6.85e3T^{2}$$
23 $$1 - 91.4iT - 1.21e4T^{2}$$
29 $$1 - 174.T + 2.43e4T^{2}$$
31 $$1 + 46.2T + 2.97e4T^{2}$$
37 $$1 + 154. iT - 5.06e4T^{2}$$
41 $$1 - 364.T + 6.89e4T^{2}$$
43 $$1 - 125. iT - 7.95e4T^{2}$$
47 $$1 + 221. iT - 1.03e5T^{2}$$
53 $$1 - 13.6iT - 1.48e5T^{2}$$
59 $$1 - 239.T + 2.05e5T^{2}$$
61 $$1 + 54.5T + 2.26e5T^{2}$$
67 $$1 - 76.0iT - 3.00e5T^{2}$$
71 $$1 + 728.T + 3.57e5T^{2}$$
73 $$1 + 501. iT - 3.89e5T^{2}$$
79 $$1 + 397.T + 4.93e5T^{2}$$
83 $$1 - 1.36e3iT - 5.71e5T^{2}$$
89 $$1 - 1.46e3T + 7.04e5T^{2}$$
97 $$1 + 335. iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$