# Properties

 Label 2-336-12.11-c3-0-22 Degree $2$ Conductor $336$ Sign $0.0571 + 0.998i$ 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 + (−2.33 + 4.64i)3-s + 12.0i·5-s − 7i·7-s + (−16.0 − 21.6i)9-s − 33.9·11-s − 13.8·13-s + (−55.7 − 28.0i)15-s + 28.7i·17-s − 75.7i·19-s + (32.4 + 16.3i)21-s − 104.·23-s − 19.1·25-s + (138. − 23.9i)27-s + 242. i·29-s − 316. i·31-s + ⋯
 L(s)  = 1 + (−0.449 + 0.893i)3-s + 1.07i·5-s − 0.377i·7-s + (−0.595 − 0.803i)9-s − 0.931·11-s − 0.295·13-s + (−0.959 − 0.482i)15-s + 0.409i·17-s − 0.914i·19-s + (0.337 + 0.169i)21-s − 0.945·23-s − 0.153·25-s + (0.985 − 0.170i)27-s + 1.55i·29-s − 1.83i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0571 + 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0571 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$336$$    =    $$2^{4} \cdot 3 \cdot 7$$ Sign: $0.0571 + 0.998i$ 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: $$(2,\ 336,\ (\ :3/2),\ 0.0571 + 0.998i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.2921501467$$ $$L(\frac12)$$ $$\approx$$ $$0.2921501467$$ $$L(\frac{5}{2})$$ 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 + (2.33 - 4.64i)T$$
7 $$1 + 7iT$$
good5 $$1 - 12.0iT - 125T^{2}$$
11 $$1 + 33.9T + 1.33e3T^{2}$$
13 $$1 + 13.8T + 2.19e3T^{2}$$
17 $$1 - 28.7iT - 4.91e3T^{2}$$
19 $$1 + 75.7iT - 6.85e3T^{2}$$
23 $$1 + 104.T + 1.21e4T^{2}$$
29 $$1 - 242. iT - 2.43e4T^{2}$$
31 $$1 + 316. iT - 2.97e4T^{2}$$
37 $$1 - 303.T + 5.06e4T^{2}$$
41 $$1 + 382. iT - 6.89e4T^{2}$$
43 $$1 + 12.0iT - 7.95e4T^{2}$$
47 $$1 + 314.T + 1.03e5T^{2}$$
53 $$1 + 418. iT - 1.48e5T^{2}$$
59 $$1 + 477.T + 2.05e5T^{2}$$
61 $$1 - 112.T + 2.26e5T^{2}$$
67 $$1 + 180. iT - 3.00e5T^{2}$$
71 $$1 + 25.4T + 3.57e5T^{2}$$
73 $$1 + 349.T + 3.89e5T^{2}$$
79 $$1 + 452. iT - 4.93e5T^{2}$$
83 $$1 - 1.08e3T + 5.71e5T^{2}$$
89 $$1 - 816. iT - 7.04e5T^{2}$$
97 $$1 + 1.07e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$