# Properties

 Label 2-105-105.104-c3-0-5 Degree $2$ Conductor $105$ Sign $0.665 - 0.746i$ Analytic cond. $6.19520$ Root an. cond. $2.48901$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.22·2-s + (−4.91 − 1.67i)3-s − 6.50·4-s + (−9.89 − 5.21i)5-s + (6.01 + 2.04i)6-s + (−1.55 − 18.4i)7-s + 17.7·8-s + (21.4 + 16.4i)9-s + (12.0 + 6.37i)10-s + 54.5i·11-s + (32.0 + 10.8i)12-s − 24.4·13-s + (1.90 + 22.5i)14-s + (39.9 + 42.1i)15-s + 30.3·16-s + 36.0i·17-s + ⋯
 L(s)  = 1 − 0.432·2-s + (−0.946 − 0.322i)3-s − 0.813·4-s + (−0.884 − 0.466i)5-s + (0.409 + 0.139i)6-s + (−0.0841 − 0.996i)7-s + 0.783·8-s + (0.792 + 0.609i)9-s + (0.382 + 0.201i)10-s + 1.49i·11-s + (0.769 + 0.261i)12-s − 0.522·13-s + (0.0363 + 0.430i)14-s + (0.687 + 0.726i)15-s + 0.474·16-s + 0.513i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $0.665 - 0.746i$ Analytic conductor: $$6.19520$$ Root analytic conductor: $$2.48901$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{105} (104, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 105,\ (\ :3/2),\ 0.665 - 0.746i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.352496 + 0.157878i$$ $$L(\frac12)$$ $$\approx$$ $$0.352496 + 0.157878i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (4.91 + 1.67i)T$$
5 $$1 + (9.89 + 5.21i)T$$
7 $$1 + (1.55 + 18.4i)T$$
good2 $$1 + 1.22T + 8T^{2}$$
11 $$1 - 54.5iT - 1.33e3T^{2}$$
13 $$1 + 24.4T + 2.19e3T^{2}$$
17 $$1 - 36.0iT - 4.91e3T^{2}$$
19 $$1 + 99.5iT - 6.85e3T^{2}$$
23 $$1 - 97.3T + 1.21e4T^{2}$$
29 $$1 - 14.1iT - 2.43e4T^{2}$$
31 $$1 - 186. iT - 2.97e4T^{2}$$
37 $$1 + 199. iT - 5.06e4T^{2}$$
41 $$1 - 313.T + 6.89e4T^{2}$$
43 $$1 - 30.0iT - 7.95e4T^{2}$$
47 $$1 - 514. iT - 1.03e5T^{2}$$
53 $$1 + 451.T + 1.48e5T^{2}$$
59 $$1 - 355.T + 2.05e5T^{2}$$
61 $$1 - 656. iT - 2.26e5T^{2}$$
67 $$1 - 73.7iT - 3.00e5T^{2}$$
71 $$1 - 611. iT - 3.57e5T^{2}$$
73 $$1 + 424.T + 3.89e5T^{2}$$
79 $$1 - 814.T + 4.93e5T^{2}$$
83 $$1 - 585. iT - 5.71e5T^{2}$$
89 $$1 + 732.T + 7.04e5T^{2}$$
97 $$1 - 272.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$