# Properties

 Label 2-105-105.104-c3-0-13 Degree $2$ Conductor $105$ Sign $0.860 - 0.509i$ 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 + (−2.64 − 4.47i)3-s − 8·4-s + 11.1i·5-s + 18.5·7-s + (−13.0 + 23.6i)9-s − 11.8i·11-s + (21.1 + 35.7i)12-s + 84.6·13-s + (50.0 − 29.5i)15-s + 64·16-s + 102. i·17-s − 89.4i·20-s + (−49.0 − 82.8i)21-s − 125.·25-s + (140. − 4.47i)27-s − 148.·28-s + ⋯
 L(s)  = 1 + (−0.509 − 0.860i)3-s − 4-s + 0.999i·5-s + 0.999·7-s + (−0.481 + 0.876i)9-s − 0.324i·11-s + (0.509 + 0.860i)12-s + 1.80·13-s + (0.860 − 0.509i)15-s + 16-s + 1.46i·17-s − 0.999i·20-s + (−0.509 − 0.860i)21-s − 1.00·25-s + (0.999 − 0.0318i)27-s − 0.999·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $0.860 - 0.509i$ 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: $$(2,\ 105,\ (\ :3/2),\ 0.860 - 0.509i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.08372 + 0.296564i$$ $$L(\frac12)$$ $$\approx$$ $$1.08372 + 0.296564i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (2.64 + 4.47i)T$$
5 $$1 - 11.1iT$$
7 $$1 - 18.5T$$
good2 $$1 + 8T^{2}$$
11 $$1 + 11.8iT - 1.33e3T^{2}$$
13 $$1 - 84.6T + 2.19e3T^{2}$$
17 $$1 - 102. iT - 4.91e3T^{2}$$
19 $$1 - 6.85e3T^{2}$$
23 $$1 + 1.21e4T^{2}$$
29 $$1 - 307. iT - 2.43e4T^{2}$$
31 $$1 - 2.97e4T^{2}$$
37 $$1 - 5.06e4T^{2}$$
41 $$1 + 6.89e4T^{2}$$
43 $$1 - 7.95e4T^{2}$$
47 $$1 + 178. iT - 1.03e5T^{2}$$
53 $$1 + 1.48e5T^{2}$$
59 $$1 + 2.05e5T^{2}$$
61 $$1 - 2.26e5T^{2}$$
67 $$1 - 3.00e5T^{2}$$
71 $$1 + 863. iT - 3.57e5T^{2}$$
73 $$1 - 1.13e3T + 3.89e5T^{2}$$
79 $$1 - 236T + 4.93e5T^{2}$$
83 $$1 - 1.51e3iT - 5.71e5T^{2}$$
89 $$1 + 7.04e5T^{2}$$
97 $$1 + 963.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$