# Properties

 Label 2-320-4.3-c2-0-14 Degree $2$ Conductor $320$ Sign $-1$ Analytic cond. $8.71936$ Root an. cond. $2.95285$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.80i·3-s − 2.23·5-s − 8.50i·7-s − 5.47·9-s − 1.79i·11-s − 0.472·13-s + 8.50i·15-s − 23.8·17-s + 9.40i·19-s − 32.3·21-s − 16.1i·23-s + 5.00·25-s − 13.4i·27-s − 6.94·29-s + 47.4i·31-s + ⋯
 L(s)  = 1 − 1.26i·3-s − 0.447·5-s − 1.21i·7-s − 0.608·9-s − 0.163i·11-s − 0.0363·13-s + 0.567i·15-s − 1.40·17-s + 0.494i·19-s − 1.54·21-s − 0.700i·23-s + 0.200·25-s − 0.497i·27-s − 0.239·29-s + 1.53i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$320$$    =    $$2^{6} \cdot 5$$ Sign: $-1$ Analytic conductor: $$8.71936$$ Root analytic conductor: $$2.95285$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{320} (191, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 320,\ (\ :1),\ -1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$-0.969692i$$ $$L(\frac12)$$ $$\approx$$ $$-0.969692i$$ $$L(2)$$ 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 + 2.23T$$
good3 $$1 + 3.80iT - 9T^{2}$$
7 $$1 + 8.50iT - 49T^{2}$$
11 $$1 + 1.79iT - 121T^{2}$$
13 $$1 + 0.472T + 169T^{2}$$
17 $$1 + 23.8T + 289T^{2}$$
19 $$1 - 9.40iT - 361T^{2}$$
23 $$1 + 16.1iT - 529T^{2}$$
29 $$1 + 6.94T + 841T^{2}$$
31 $$1 - 47.4iT - 961T^{2}$$
37 $$1 + 26.3T + 1.36e3T^{2}$$
41 $$1 + 41.4T + 1.68e3T^{2}$$
43 $$1 - 2.00iT - 1.84e3T^{2}$$
47 $$1 + 35.3iT - 2.20e3T^{2}$$
53 $$1 - 21.6T + 2.80e3T^{2}$$
59 $$1 + 73.8iT - 3.48e3T^{2}$$
61 $$1 - 26.1T + 3.72e3T^{2}$$
67 $$1 + 88.8iT - 4.48e3T^{2}$$
71 $$1 + 39.4iT - 5.04e3T^{2}$$
73 $$1 - 137.T + 5.32e3T^{2}$$
79 $$1 + 113. iT - 6.24e3T^{2}$$
83 $$1 + 21.2iT - 6.88e3T^{2}$$
89 $$1 - 67.4T + 7.92e3T^{2}$$
97 $$1 + 39.1T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$