# Properties

 Label 2-336-12.11-c3-0-26 Degree $2$ Conductor $336$ Sign $-0.907 + 0.420i$ Analytic cond. $19.8246$ Root an. cond. $4.45248$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2.18 − 4.71i)3-s + 14.5i·5-s + 7i·7-s + (−17.4 + 20.6i)9-s + 31.0·11-s − 55.7·13-s + (68.7 − 31.8i)15-s − 86.4i·17-s − 96.8i·19-s + (32.9 − 15.3i)21-s − 115.·23-s − 87.7·25-s + (135. + 37.0i)27-s − 49.1i·29-s + 12.5i·31-s + ⋯
 L(s)  = 1 + (−0.420 − 0.907i)3-s + 1.30i·5-s + 0.377i·7-s + (−0.645 + 0.763i)9-s + 0.850·11-s − 1.18·13-s + (1.18 − 0.549i)15-s − 1.23i·17-s − 1.16i·19-s + (0.342 − 0.159i)21-s − 1.04·23-s − 0.701·25-s + (0.964 + 0.264i)27-s − 0.314i·29-s + 0.0728i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$336$$    =    $$2^{4} \cdot 3 \cdot 7$$ Sign: $-0.907 + 0.420i$ Analytic conductor: $$19.8246$$ Root analytic conductor: $$4.45248$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{336} (239, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 336,\ (\ :3/2),\ -0.907 + 0.420i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.3590024833$$ $$L(\frac12)$$ $$\approx$$ $$0.3590024833$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (2.18 + 4.71i)T$$
7 $$1 - 7iT$$
good5 $$1 - 14.5iT - 125T^{2}$$
11 $$1 - 31.0T + 1.33e3T^{2}$$
13 $$1 + 55.7T + 2.19e3T^{2}$$
17 $$1 + 86.4iT - 4.91e3T^{2}$$
19 $$1 + 96.8iT - 6.85e3T^{2}$$
23 $$1 + 115.T + 1.21e4T^{2}$$
29 $$1 + 49.1iT - 2.43e4T^{2}$$
31 $$1 - 12.5iT - 2.97e4T^{2}$$
37 $$1 + 296.T + 5.06e4T^{2}$$
41 $$1 + 213. iT - 6.89e4T^{2}$$
43 $$1 + 165. iT - 7.95e4T^{2}$$
47 $$1 + 426.T + 1.03e5T^{2}$$
53 $$1 - 460. iT - 1.48e5T^{2}$$
59 $$1 - 686.T + 2.05e5T^{2}$$
61 $$1 + 583.T + 2.26e5T^{2}$$
67 $$1 + 589. iT - 3.00e5T^{2}$$
71 $$1 + 766.T + 3.57e5T^{2}$$
73 $$1 + 904.T + 3.89e5T^{2}$$
79 $$1 + 459. iT - 4.93e5T^{2}$$
83 $$1 - 1.11e3T + 5.71e5T^{2}$$
89 $$1 - 119. iT - 7.04e5T^{2}$$
97 $$1 + 331.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$