# Properties

 Label 2-336-7.3-c6-0-6 Degree $2$ Conductor $336$ Sign $0.169 - 0.985i$ Analytic cond. $77.2981$ Root an. cond. $8.79193$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−13.5 − 7.79i)3-s + (71.9 − 41.5i)5-s + (−77.0 − 334. i)7-s + (121.5 + 210. i)9-s + (−221. + 383. i)11-s + 696. i·13-s − 1.29e3·15-s + (−5.44e3 − 3.14e3i)17-s + (2.32e3 − 1.34e3i)19-s + (−1.56e3 + 5.11e3i)21-s + (7.88e3 + 1.36e4i)23-s + (−4.36e3 + 7.55e3i)25-s − 3.78e3i·27-s − 2.32e4·29-s + (−4.11e4 − 2.37e4i)31-s + ⋯
 L(s)  = 1 + (−0.5 − 0.288i)3-s + (0.575 − 0.332i)5-s + (−0.224 − 0.974i)7-s + (0.166 + 0.288i)9-s + (−0.166 + 0.287i)11-s + 0.317i·13-s − 0.383·15-s + (−1.10 − 0.640i)17-s + (0.339 − 0.195i)19-s + (−0.168 + 0.552i)21-s + (0.648 + 1.12i)23-s + (−0.279 + 0.483i)25-s − 0.192i·27-s − 0.954·29-s + (−1.38 − 0.798i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$336$$    =    $$2^{4} \cdot 3 \cdot 7$$ Sign: $0.169 - 0.985i$ Analytic conductor: $$77.2981$$ Root analytic conductor: $$8.79193$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{336} (241, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 336,\ (\ :3),\ 0.169 - 0.985i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.7310018877$$ $$L(\frac12)$$ $$\approx$$ $$0.7310018877$$ $$L(4)$$ 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 + (13.5 + 7.79i)T$$
7 $$1 + (77.0 + 334. i)T$$
good5 $$1 + (-71.9 + 41.5i)T + (7.81e3 - 1.35e4i)T^{2}$$
11 $$1 + (221. - 383. i)T + (-8.85e5 - 1.53e6i)T^{2}$$
13 $$1 - 696. iT - 4.82e6T^{2}$$
17 $$1 + (5.44e3 + 3.14e3i)T + (1.20e7 + 2.09e7i)T^{2}$$
19 $$1 + (-2.32e3 + 1.34e3i)T + (2.35e7 - 4.07e7i)T^{2}$$
23 $$1 + (-7.88e3 - 1.36e4i)T + (-7.40e7 + 1.28e8i)T^{2}$$
29 $$1 + 2.32e4T + 5.94e8T^{2}$$
31 $$1 + (4.11e4 + 2.37e4i)T + (4.43e8 + 7.68e8i)T^{2}$$
37 $$1 + (-5.07e3 - 8.79e3i)T + (-1.28e9 + 2.22e9i)T^{2}$$
41 $$1 - 3.81e4iT - 4.75e9T^{2}$$
43 $$1 - 1.51e5T + 6.32e9T^{2}$$
47 $$1 + (-4.35e4 + 2.51e4i)T + (5.38e9 - 9.33e9i)T^{2}$$
53 $$1 + (-9.97e4 + 1.72e5i)T + (-1.10e10 - 1.91e10i)T^{2}$$
59 $$1 + (3.35e5 + 1.93e5i)T + (2.10e10 + 3.65e10i)T^{2}$$
61 $$1 + (1.02e4 - 5.89e3i)T + (2.57e10 - 4.46e10i)T^{2}$$
67 $$1 + (1.92e5 - 3.32e5i)T + (-4.52e10 - 7.83e10i)T^{2}$$
71 $$1 + 1.56e5T + 1.28e11T^{2}$$
73 $$1 + (-3.25e5 - 1.87e5i)T + (7.56e10 + 1.31e11i)T^{2}$$
79 $$1 + (-1.65e4 - 2.87e4i)T + (-1.21e11 + 2.10e11i)T^{2}$$
83 $$1 - 9.84e5iT - 3.26e11T^{2}$$
89 $$1 + (2.27e5 - 1.31e5i)T + (2.48e11 - 4.30e11i)T^{2}$$
97 $$1 - 5.75e5iT - 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$