# Properties

 Label 2-336-21.17-c3-0-35 Degree $2$ Conductor $336$ Sign $0.633 + 0.773i$ 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 + (2.24 + 4.68i)3-s + (−5.80 − 10.0i)5-s + (18.4 − 2.09i)7-s + (−16.9 + 21.0i)9-s + (−15.5 − 8.95i)11-s − 62.4i·13-s + (34.1 − 49.7i)15-s + (10.7 − 18.5i)17-s + (−9.50 + 5.48i)19-s + (51.0 + 81.5i)21-s + (59.8 − 34.5i)23-s + (−4.82 + 8.35i)25-s + (−136. − 32.3i)27-s − 265. i·29-s + (−8.85 − 5.11i)31-s + ⋯
 L(s)  = 1 + (0.431 + 0.902i)3-s + (−0.518 − 0.898i)5-s + (0.993 − 0.112i)7-s + (−0.627 + 0.778i)9-s + (−0.425 − 0.245i)11-s − 1.33i·13-s + (0.587 − 0.855i)15-s + (0.152 − 0.264i)17-s + (−0.114 + 0.0662i)19-s + (0.530 + 0.847i)21-s + (0.542 − 0.313i)23-s + (−0.0385 + 0.0668i)25-s + (−0.972 − 0.230i)27-s − 1.70i·29-s + (−0.0513 − 0.0296i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$336$$    =    $$2^{4} \cdot 3 \cdot 7$$ Sign: $0.633 + 0.773i$ 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: $$(2,\ 336,\ (\ :3/2),\ 0.633 + 0.773i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.818243804$$ $$L(\frac12)$$ $$\approx$$ $$1.818243804$$ $$L(\frac{5}{2})$$ 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 + (-2.24 - 4.68i)T$$
7 $$1 + (-18.4 + 2.09i)T$$
good5 $$1 + (5.80 + 10.0i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (15.5 + 8.95i)T + (665.5 + 1.15e3i)T^{2}$$
13 $$1 + 62.4iT - 2.19e3T^{2}$$
17 $$1 + (-10.7 + 18.5i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (9.50 - 5.48i)T + (3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-59.8 + 34.5i)T + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + 265. iT - 2.43e4T^{2}$$
31 $$1 + (8.85 + 5.11i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (20.8 + 36.0i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 31.0T + 6.89e4T^{2}$$
43 $$1 - 224.T + 7.95e4T^{2}$$
47 $$1 + (-81.8 - 141. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (-456. - 263. i)T + (7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-205. + 356. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-223. + 129. i)T + (1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-161. + 280. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 45.4iT - 3.57e5T^{2}$$
73 $$1 + (486. + 281. i)T + (1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-144. - 250. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + 448.T + 5.71e5T^{2}$$
89 $$1 + (280. + 486. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 214. iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$