# Properties

 Label 2-378-3.2-c4-0-21 Degree $2$ Conductor $378$ Sign $i$ Analytic cond. $39.0738$ Root an. cond. $6.25090$ Motivic weight $4$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.82i·2-s − 8.00·4-s − 3.12i·5-s + 18.5·7-s + 22.6i·8-s − 8.83·10-s + 141. i·11-s − 237.·13-s − 52.3i·14-s + 64.0·16-s − 394. i·17-s + 535.·19-s + 24.9i·20-s + 399.·22-s − 487. i·23-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.500·4-s − 0.124i·5-s + 0.377·7-s + 0.353i·8-s − 0.0883·10-s + 1.16i·11-s − 1.40·13-s − 0.267i·14-s + 0.250·16-s − 1.36i·17-s + 1.48·19-s + 0.0624i·20-s + 0.825·22-s − 0.922i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$378$$    =    $$2 \cdot 3^{3} \cdot 7$$ Sign: $i$ Analytic conductor: $$39.0738$$ Root analytic conductor: $$6.25090$$ Motivic weight: $$4$$ Rational: no Arithmetic: yes Character: $\chi_{378} (323, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 378,\ (\ :2),\ i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$1.735166221$$ $$L(\frac12)$$ $$\approx$$ $$1.735166221$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 2.82iT$$
3 $$1$$
7 $$1 - 18.5T$$
good5 $$1 + 3.12iT - 625T^{2}$$
11 $$1 - 141. iT - 1.46e4T^{2}$$
13 $$1 + 237.T + 2.85e4T^{2}$$
17 $$1 + 394. iT - 8.35e4T^{2}$$
19 $$1 - 535.T + 1.30e5T^{2}$$
23 $$1 + 487. iT - 2.79e5T^{2}$$
29 $$1 - 1.20e3iT - 7.07e5T^{2}$$
31 $$1 - 296.T + 9.23e5T^{2}$$
37 $$1 + 380.T + 1.87e6T^{2}$$
41 $$1 - 73.6iT - 2.82e6T^{2}$$
43 $$1 - 2.91e3T + 3.41e6T^{2}$$
47 $$1 + 2.84e3iT - 4.87e6T^{2}$$
53 $$1 + 2.12e3iT - 7.89e6T^{2}$$
59 $$1 + 5.48e3iT - 1.21e7T^{2}$$
61 $$1 + 5.03e3T + 1.38e7T^{2}$$
67 $$1 - 7.34e3T + 2.01e7T^{2}$$
71 $$1 + 8.60e3iT - 2.54e7T^{2}$$
73 $$1 - 2.16e3T + 2.83e7T^{2}$$
79 $$1 + 5.79e3T + 3.89e7T^{2}$$
83 $$1 + 2.41e3iT - 4.74e7T^{2}$$
89 $$1 + 1.37e4iT - 6.27e7T^{2}$$
97 $$1 - 1.06e4T + 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$