# Properties

 Label 2-378-7.2-c3-0-10 Degree $2$ Conductor $378$ Sign $0.836 - 0.548i$ Analytic cond. $22.3027$ Root an. cond. $4.72257$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (1.18 − 2.05i)5-s + (9.15 + 16.0i)7-s − 7.99·8-s + (−2.37 − 4.11i)10-s + (19.5 + 33.8i)11-s − 24.4·13-s + (37.0 + 0.232i)14-s + (−8 + 13.8i)16-s + (−23.8 − 41.3i)17-s + (−59.8 + 103. i)19-s − 9.49·20-s + 78.2·22-s + (−76.6 + 132. i)23-s + ⋯
 L(s)  = 1 + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.106 − 0.183i)5-s + (0.494 + 0.869i)7-s − 0.353·8-s + (−0.0750 − 0.130i)10-s + (0.535 + 0.928i)11-s − 0.521·13-s + (0.707 + 0.00444i)14-s + (−0.125 + 0.216i)16-s + (−0.340 − 0.589i)17-s + (−0.722 + 1.25i)19-s − 0.106·20-s + 0.757·22-s + (−0.694 + 1.20i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$378$$    =    $$2 \cdot 3^{3} \cdot 7$$ Sign: $0.836 - 0.548i$ Analytic conductor: $$22.3027$$ Root analytic conductor: $$4.72257$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{378} (163, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 378,\ (\ :3/2),\ 0.836 - 0.548i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.991744505$$ $$L(\frac12)$$ $$\approx$$ $$1.991744505$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1 + 1.73i)T$$
3 $$1$$
7 $$1 + (-9.15 - 16.0i)T$$
good5 $$1 + (-1.18 + 2.05i)T + (-62.5 - 108. i)T^{2}$$
11 $$1 + (-19.5 - 33.8i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + 24.4T + 2.19e3T^{2}$$
17 $$1 + (23.8 + 41.3i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (59.8 - 103. i)T + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (76.6 - 132. i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 215.T + 2.43e4T^{2}$$
31 $$1 + (-29.6 - 51.3i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (-104. + 181. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 - 415.T + 6.89e4T^{2}$$
43 $$1 + 452.T + 7.95e4T^{2}$$
47 $$1 + (-114. + 198. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (-220. - 382. i)T + (-7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-362. - 627. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-170. + 295. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (125. + 217. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 - 209.T + 3.57e5T^{2}$$
73 $$1 + (60.9 + 105. i)T + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (399. - 692. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 - 116.T + 5.71e5T^{2}$$
89 $$1 + (-183. + 317. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + 1.04e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$