# Properties

 Label 2-392-392.109-c1-0-34 Degree $2$ Conductor $392$ Sign $0.591 + 0.806i$ Analytic cond. $3.13013$ Root an. cond. $1.76921$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.19 + 0.762i)2-s + (−0.209 − 1.39i)3-s + (0.836 − 1.81i)4-s + (1.73 + 0.679i)5-s + (1.31 + 1.49i)6-s + (−0.592 − 2.57i)7-s + (0.389 + 2.80i)8-s + (0.971 − 0.299i)9-s + (−2.58 + 0.511i)10-s + (−0.224 + 0.728i)11-s + (−2.70 − 0.783i)12-s + (1.74 − 0.398i)13-s + (2.67 + 2.61i)14-s + (0.582 − 2.55i)15-s + (−2.60 − 3.03i)16-s + (4.21 − 2.87i)17-s + ⋯
 L(s)  = 1 + (−0.842 + 0.539i)2-s + (−0.121 − 0.803i)3-s + (0.418 − 0.908i)4-s + (0.774 + 0.303i)5-s + (0.535 + 0.611i)6-s + (−0.223 − 0.974i)7-s + (0.137 + 0.990i)8-s + (0.323 − 0.0999i)9-s + (−0.815 + 0.161i)10-s + (−0.0677 + 0.219i)11-s + (−0.780 − 0.226i)12-s + (0.483 − 0.110i)13-s + (0.714 + 0.700i)14-s + (0.150 − 0.659i)15-s + (−0.650 − 0.759i)16-s + (1.02 − 0.697i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$392$$    =    $$2^{3} \cdot 7^{2}$$ Sign: $0.591 + 0.806i$ Analytic conductor: $$3.13013$$ Root analytic conductor: $$1.76921$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{392} (109, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 392,\ (\ :1/2),\ 0.591 + 0.806i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.894829 - 0.453668i$$ $$L(\frac12)$$ $$\approx$$ $$0.894829 - 0.453668i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.19 - 0.762i)T$$
7 $$1 + (0.592 + 2.57i)T$$
good3 $$1 + (0.209 + 1.39i)T + (-2.86 + 0.884i)T^{2}$$
5 $$1 + (-1.73 - 0.679i)T + (3.66 + 3.40i)T^{2}$$
11 $$1 + (0.224 - 0.728i)T + (-9.08 - 6.19i)T^{2}$$
13 $$1 + (-1.74 + 0.398i)T + (11.7 - 5.64i)T^{2}$$
17 $$1 + (-4.21 + 2.87i)T + (6.21 - 15.8i)T^{2}$$
19 $$1 + (7.23 - 4.17i)T + (9.5 - 16.4i)T^{2}$$
23 $$1 + (0.115 + 0.0785i)T + (8.40 + 21.4i)T^{2}$$
29 $$1 + (-2.76 + 5.74i)T + (-18.0 - 22.6i)T^{2}$$
31 $$1 + (-3.93 + 6.81i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-4.03 + 0.302i)T + (36.5 - 5.51i)T^{2}$$
41 $$1 + (-2.31 - 2.90i)T + (-9.12 + 39.9i)T^{2}$$
43 $$1 + (-6.71 - 5.35i)T + (9.56 + 41.9i)T^{2}$$
47 $$1 + (-4.15 + 3.85i)T + (3.51 - 46.8i)T^{2}$$
53 $$1 + (7.02 + 0.526i)T + (52.4 + 7.89i)T^{2}$$
59 $$1 + (7.74 - 3.03i)T + (43.2 - 40.1i)T^{2}$$
61 $$1 + (7.47 - 0.559i)T + (60.3 - 9.09i)T^{2}$$
67 $$1 + (-6.85 - 3.95i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + (-2.31 + 1.11i)T + (44.2 - 55.5i)T^{2}$$
73 $$1 + (-0.0710 - 0.0659i)T + (5.45 + 72.7i)T^{2}$$
79 $$1 + (2.79 + 4.84i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (-9.51 - 2.17i)T + (74.7 + 36.0i)T^{2}$$
89 $$1 + (0.741 - 0.228i)T + (73.5 - 50.1i)T^{2}$$
97 $$1 + 12.3T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$