# Properties

 Label 2-336-21.17-c3-0-44 Degree $2$ Conductor $336$ Sign $-0.999 - 0.0303i$ 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 + (3.47 − 3.86i)3-s + (0.623 + 1.08i)5-s + (−10.0 − 15.5i)7-s + (−2.84 − 26.8i)9-s + (−35.2 − 20.3i)11-s + 19.5i·13-s + (6.34 + 1.34i)15-s + (−52.3 + 90.6i)17-s + (−35.0 + 20.2i)19-s + (−95.0 − 15.0i)21-s + (−69.6 + 40.2i)23-s + (61.7 − 106. i)25-s + (−113. − 82.3i)27-s − 211. i·29-s + (86.6 + 50.0i)31-s + ⋯
 L(s)  = 1 + (0.668 − 0.743i)3-s + (0.0557 + 0.0966i)5-s + (−0.544 − 0.838i)7-s + (−0.105 − 0.994i)9-s + (−0.965 − 0.557i)11-s + 0.418i·13-s + (0.109 + 0.0231i)15-s + (−0.746 + 1.29i)17-s + (−0.423 + 0.244i)19-s + (−0.987 − 0.156i)21-s + (−0.631 + 0.364i)23-s + (0.493 − 0.855i)25-s + (−0.809 − 0.586i)27-s − 1.35i·29-s + (0.501 + 0.289i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0303i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0303i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$336$$    =    $$2^{4} \cdot 3 \cdot 7$$ Sign: $-0.999 - 0.0303i$ Analytic conductor: $$19.8246$$ Root analytic conductor: $$4.45248$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{336} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 336,\ (\ :3/2),\ -0.999 - 0.0303i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.8741562594$$ $$L(\frac12)$$ $$\approx$$ $$0.8741562594$$ $$L(\frac{5}{2})$$ 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 + (-3.47 + 3.86i)T$$
7 $$1 + (10.0 + 15.5i)T$$
good5 $$1 + (-0.623 - 1.08i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (35.2 + 20.3i)T + (665.5 + 1.15e3i)T^{2}$$
13 $$1 - 19.5iT - 2.19e3T^{2}$$
17 $$1 + (52.3 - 90.6i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (35.0 - 20.2i)T + (3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (69.6 - 40.2i)T + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + 211. iT - 2.43e4T^{2}$$
31 $$1 + (-86.6 - 50.0i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (-94.9 - 164. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 186.T + 6.89e4T^{2}$$
43 $$1 + 158.T + 7.95e4T^{2}$$
47 $$1 + (179. + 310. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (366. + 211. i)T + (7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (312. - 541. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-699. + 403. i)T + (1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-149. + 258. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 455. iT - 3.57e5T^{2}$$
73 $$1 + (434. + 250. i)T + (1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (30.9 + 53.6i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 - 73.1T + 5.71e5T^{2}$$
89 $$1 + (-57.3 - 99.3i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 + 1.41e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$