# Properties

 Label 2-320-5.3-c2-0-17 Degree $2$ Conductor $320$ Sign $0.850 + 0.525i$ 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 + (2 + 2i)3-s − 5i·5-s + (2 − 2i)7-s − i·9-s + 8·11-s + (−3 − 3i)13-s + (10 − 10i)15-s + (7 − 7i)17-s − 20i·19-s + 8·21-s + (−2 − 2i)23-s − 25·25-s + (20 − 20i)27-s + 40i·29-s + 52·31-s + ⋯
 L(s)  = 1 + (0.666 + 0.666i)3-s − i·5-s + (0.285 − 0.285i)7-s − 0.111i·9-s + 0.727·11-s + (−0.230 − 0.230i)13-s + (0.666 − 0.666i)15-s + (0.411 − 0.411i)17-s − 1.05i·19-s + 0.380·21-s + (−0.0869 − 0.0869i)23-s − 25-s + (0.740 − 0.740i)27-s + 1.37i·29-s + 1.67·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$320$$    =    $$2^{6} \cdot 5$$ Sign: $0.850 + 0.525i$ Analytic conductor: $$8.71936$$ Root analytic conductor: $$2.95285$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{320} (193, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 320,\ (\ :1),\ 0.850 + 0.525i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$2.04539 - 0.581055i$$ $$L(\frac12)$$ $$\approx$$ $$2.04539 - 0.581055i$$ $$L(2)$$ 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 + 5iT$$
good3 $$1 + (-2 - 2i)T + 9iT^{2}$$
7 $$1 + (-2 + 2i)T - 49iT^{2}$$
11 $$1 - 8T + 121T^{2}$$
13 $$1 + (3 + 3i)T + 169iT^{2}$$
17 $$1 + (-7 + 7i)T - 289iT^{2}$$
19 $$1 + 20iT - 361T^{2}$$
23 $$1 + (2 + 2i)T + 529iT^{2}$$
29 $$1 - 40iT - 841T^{2}$$
31 $$1 - 52T + 961T^{2}$$
37 $$1 + (-3 + 3i)T - 1.36e3iT^{2}$$
41 $$1 + 8T + 1.68e3T^{2}$$
43 $$1 + (-42 - 42i)T + 1.84e3iT^{2}$$
47 $$1 + (18 - 18i)T - 2.20e3iT^{2}$$
53 $$1 + (53 + 53i)T + 2.80e3iT^{2}$$
59 $$1 + 20iT - 3.48e3T^{2}$$
61 $$1 - 48T + 3.72e3T^{2}$$
67 $$1 + (62 - 62i)T - 4.48e3iT^{2}$$
71 $$1 + 28T + 5.04e3T^{2}$$
73 $$1 + (47 + 47i)T + 5.32e3iT^{2}$$
79 $$1 - 6.24e3T^{2}$$
83 $$1 + (18 + 18i)T + 6.88e3iT^{2}$$
89 $$1 + 80iT - 7.92e3T^{2}$$
97 $$1 + (63 - 63i)T - 9.40e3iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$