Properties

 Label 2-3920-980.799-c0-0-0 Degree $2$ Conductor $3920$ Sign $0.981 + 0.191i$ Analytic cond. $1.95633$ Root an. cond. $1.39869$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (0.974 − 1.22i)3-s + (−0.623 + 0.781i)5-s + (−0.433 + 0.900i)7-s + (−0.321 − 1.40i)9-s + (0.347 + 1.52i)15-s + (0.678 + 1.40i)21-s + (1.40 + 0.678i)23-s + (−0.222 − 0.974i)25-s + (−0.626 − 0.301i)27-s + (1.62 − 0.781i)29-s + (−0.433 − 0.900i)35-s + (0.277 − 0.347i)41-s + (1.21 + 1.52i)43-s + (1.30 + 0.626i)45-s + (−0.347 + 1.52i)47-s + ⋯
 L(s)  = 1 + (0.974 − 1.22i)3-s + (−0.623 + 0.781i)5-s + (−0.433 + 0.900i)7-s + (−0.321 − 1.40i)9-s + (0.347 + 1.52i)15-s + (0.678 + 1.40i)21-s + (1.40 + 0.678i)23-s + (−0.222 − 0.974i)25-s + (−0.626 − 0.301i)27-s + (1.62 − 0.781i)29-s + (−0.433 − 0.900i)35-s + (0.277 − 0.347i)41-s + (1.21 + 1.52i)43-s + (1.30 + 0.626i)45-s + (−0.347 + 1.52i)47-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$3920$$    =    $$2^{4} \cdot 5 \cdot 7^{2}$$ Sign: $0.981 + 0.191i$ Analytic conductor: $$1.95633$$ Root analytic conductor: $$1.39869$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3920} (799, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3920,\ (\ :0),\ 0.981 + 0.191i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.581777741$$ $$L(\frac12)$$ $$\approx$$ $$1.581777741$$ $$L(1)$$ 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 + (0.623 - 0.781i)T$$
7 $$1 + (0.433 - 0.900i)T$$
good3 $$1 + (-0.974 + 1.22i)T + (-0.222 - 0.974i)T^{2}$$
11 $$1 + (0.900 + 0.433i)T^{2}$$
13 $$1 + (0.900 + 0.433i)T^{2}$$
17 $$1 + (-0.623 + 0.781i)T^{2}$$
19 $$1 - T^{2}$$
23 $$1 + (-1.40 - 0.678i)T + (0.623 + 0.781i)T^{2}$$
29 $$1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + (-0.623 + 0.781i)T^{2}$$
41 $$1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2}$$
43 $$1 + (-1.21 - 1.52i)T + (-0.222 + 0.974i)T^{2}$$
47 $$1 + (0.347 - 1.52i)T + (-0.900 - 0.433i)T^{2}$$
53 $$1 + (-0.623 - 0.781i)T^{2}$$
59 $$1 + (0.222 - 0.974i)T^{2}$$
61 $$1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2}$$
67 $$1 + T^{2}$$
71 $$1 + (-0.623 - 0.781i)T^{2}$$
73 $$1 + (0.900 - 0.433i)T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + (0.193 + 0.846i)T + (-0.900 + 0.433i)T^{2}$$
89 $$1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2}$$
97 $$1 - T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$