Properties

 Degree $2$ Conductor $320$ Sign $0.441 - 0.897i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 10.4·3-s + (−17.9 − 52.9i)5-s + 86.7i·7-s − 134.·9-s + 258. i·11-s + 465.·13-s + (−187. − 551. i)15-s − 2.02e3i·17-s + 1.05e3i·19-s + 904. i·21-s + 3.99e3i·23-s + (−2.47e3 + 1.90e3i)25-s − 3.93e3·27-s + 3.24e3i·29-s + 991.·31-s + ⋯
 L(s)  = 1 + 0.668·3-s + (−0.321 − 0.946i)5-s + 0.669i·7-s − 0.552·9-s + 0.644i·11-s + 0.763·13-s + (−0.215 − 0.633i)15-s − 1.70i·17-s + 0.667i·19-s + 0.447i·21-s + 1.57i·23-s + (−0.792 + 0.609i)25-s − 1.03·27-s + 0.715i·29-s + 0.185·31-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$320$$    =    $$2^{6} \cdot 5$$ Sign: $0.441 - 0.897i$ Motivic weight: $$5$$ Character: $\chi_{320} (289, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 320,\ (\ :5/2),\ 0.441 - 0.897i)$$

Particular Values

 $$L(3)$$ $$\approx$$ $$1.867154611$$ $$L(\frac12)$$ $$\approx$$ $$1.867154611$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + (17.9 + 52.9i)T$$
good3 $$1 - 10.4T + 243T^{2}$$
7 $$1 - 86.7iT - 1.68e4T^{2}$$
11 $$1 - 258. iT - 1.61e5T^{2}$$
13 $$1 - 465.T + 3.71e5T^{2}$$
17 $$1 + 2.02e3iT - 1.41e6T^{2}$$
19 $$1 - 1.05e3iT - 2.47e6T^{2}$$
23 $$1 - 3.99e3iT - 6.43e6T^{2}$$
29 $$1 - 3.24e3iT - 2.05e7T^{2}$$
31 $$1 - 991.T + 2.86e7T^{2}$$
37 $$1 - 8.82e3T + 6.93e7T^{2}$$
41 $$1 - 5.63e3T + 1.15e8T^{2}$$
43 $$1 - 1.77e3T + 1.47e8T^{2}$$
47 $$1 - 1.77e4iT - 2.29e8T^{2}$$
53 $$1 + 2.59e4T + 4.18e8T^{2}$$
59 $$1 - 1.81e4iT - 7.14e8T^{2}$$
61 $$1 - 3.54e4iT - 8.44e8T^{2}$$
67 $$1 - 4.20e4T + 1.35e9T^{2}$$
71 $$1 - 7.13e4T + 1.80e9T^{2}$$
73 $$1 - 3.25e3iT - 2.07e9T^{2}$$
79 $$1 - 8.42e4T + 3.07e9T^{2}$$
83 $$1 + 7.90e4T + 3.93e9T^{2}$$
89 $$1 + 6.51e4T + 5.58e9T^{2}$$
97 $$1 + 3.39e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$