# Properties

 Label 2-336-7.2-c5-0-2 Degree $2$ Conductor $336$ Sign $-0.643 + 0.765i$ Analytic cond. $53.8889$ Root an. cond. $7.34091$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (4.5 + 7.79i)3-s + (−11.8 + 20.5i)5-s + (−30.6 + 125. i)7-s + (−40.5 + 70.1i)9-s + (232. + 403. i)11-s − 1.01e3·13-s − 213.·15-s + (−280. − 486. i)17-s + (−693. + 1.20e3i)19-s + (−1.11e3 + 327. i)21-s + (2.05e3 − 3.56e3i)23-s + (1.28e3 + 2.21e3i)25-s − 729·27-s − 2.38e3·29-s + (1.47e3 + 2.55e3i)31-s + ⋯
 L(s)  = 1 + (0.288 + 0.499i)3-s + (−0.212 + 0.367i)5-s + (−0.236 + 0.971i)7-s + (−0.166 + 0.288i)9-s + (0.580 + 1.00i)11-s − 1.67·13-s − 0.245·15-s + (−0.235 − 0.408i)17-s + (−0.440 + 0.763i)19-s + (−0.554 + 0.162i)21-s + (0.810 − 1.40i)23-s + (0.409 + 0.709i)25-s − 0.192·27-s − 0.525·29-s + (0.275 + 0.477i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$336$$    =    $$2^{4} \cdot 3 \cdot 7$$ Sign: $-0.643 + 0.765i$ Analytic conductor: $$53.8889$$ Root analytic conductor: $$7.34091$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{336} (289, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 336,\ (\ :5/2),\ -0.643 + 0.765i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.4975474005$$ $$L(\frac12)$$ $$\approx$$ $$0.4975474005$$ $$L(\frac{7}{2})$$ 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 + (-4.5 - 7.79i)T$$
7 $$1 + (30.6 - 125. i)T$$
good5 $$1 + (11.8 - 20.5i)T + (-1.56e3 - 2.70e3i)T^{2}$$
11 $$1 + (-232. - 403. i)T + (-8.05e4 + 1.39e5i)T^{2}$$
13 $$1 + 1.01e3T + 3.71e5T^{2}$$
17 $$1 + (280. + 486. i)T + (-7.09e5 + 1.22e6i)T^{2}$$
19 $$1 + (693. - 1.20e3i)T + (-1.23e6 - 2.14e6i)T^{2}$$
23 $$1 + (-2.05e3 + 3.56e3i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 + 2.38e3T + 2.05e7T^{2}$$
31 $$1 + (-1.47e3 - 2.55e3i)T + (-1.43e7 + 2.47e7i)T^{2}$$
37 $$1 + (-4.95e3 + 8.58e3i)T + (-3.46e7 - 6.00e7i)T^{2}$$
41 $$1 + 4.47e3T + 1.15e8T^{2}$$
43 $$1 + 5.18e3T + 1.47e8T^{2}$$
47 $$1 + (1.56e3 - 2.70e3i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 + (570. + 988. i)T + (-2.09e8 + 3.62e8i)T^{2}$$
59 $$1 + (-1.37e4 - 2.38e4i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (-1.05e4 + 1.82e4i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (2.77e4 + 4.81e4i)T + (-6.75e8 + 1.16e9i)T^{2}$$
71 $$1 - 6.07e3T + 1.80e9T^{2}$$
73 $$1 + (-8.38e3 - 1.45e4i)T + (-1.03e9 + 1.79e9i)T^{2}$$
79 $$1 + (2.42e3 - 4.19e3i)T + (-1.53e9 - 2.66e9i)T^{2}$$
83 $$1 + 6.01e4T + 3.93e9T^{2}$$
89 $$1 + (-3.12e4 + 5.41e4i)T + (-2.79e9 - 4.83e9i)T^{2}$$
97 $$1 + 6.36e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$