# Properties

 Label 2-105-105.104-c3-0-42 Degree $2$ Conductor $105$ Sign $-0.974 + 0.224i$ 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 + 2.24·2-s + (−1.06 − 5.08i)3-s − 2.96·4-s + (−8.51 − 7.24i)5-s + (−2.39 − 11.4i)6-s + (12.2 + 13.8i)7-s − 24.6·8-s + (−24.7 + 10.8i)9-s + (−19.1 − 16.2i)10-s − 25.7i·11-s + (3.16 + 15.0i)12-s − 68.2·13-s + (27.5 + 31.1i)14-s + (−27.7 + 51.0i)15-s − 31.5·16-s − 30.6i·17-s + ⋯
 L(s)  = 1 + 0.793·2-s + (−0.205 − 0.978i)3-s − 0.370·4-s + (−0.761 − 0.648i)5-s + (−0.162 − 0.776i)6-s + (0.662 + 0.748i)7-s − 1.08·8-s + (−0.915 + 0.401i)9-s + (−0.604 − 0.514i)10-s − 0.705i·11-s + (0.0760 + 0.362i)12-s − 1.45·13-s + (0.526 + 0.594i)14-s + (−0.478 + 0.878i)15-s − 0.492·16-s − 0.437i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $-0.974 + 0.224i$ 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: $$(2,\ 105,\ (\ :3/2),\ -0.974 + 0.224i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.100028 - 0.880467i$$ $$L(\frac12)$$ $$\approx$$ $$0.100028 - 0.880467i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (1.06 + 5.08i)T$$
5 $$1 + (8.51 + 7.24i)T$$
7 $$1 + (-12.2 - 13.8i)T$$
good2 $$1 - 2.24T + 8T^{2}$$
11 $$1 + 25.7iT - 1.33e3T^{2}$$
13 $$1 + 68.2T + 2.19e3T^{2}$$
17 $$1 + 30.6iT - 4.91e3T^{2}$$
19 $$1 + 109. iT - 6.85e3T^{2}$$
23 $$1 - 152.T + 1.21e4T^{2}$$
29 $$1 + 191. iT - 2.43e4T^{2}$$
31 $$1 + 16.4iT - 2.97e4T^{2}$$
37 $$1 + 81.9iT - 5.06e4T^{2}$$
41 $$1 + 372.T + 6.89e4T^{2}$$
43 $$1 + 192. iT - 7.95e4T^{2}$$
47 $$1 - 0.366iT - 1.03e5T^{2}$$
53 $$1 + 5.95T + 1.48e5T^{2}$$
59 $$1 - 198.T + 2.05e5T^{2}$$
61 $$1 + 83.5iT - 2.26e5T^{2}$$
67 $$1 - 1.08e3iT - 3.00e5T^{2}$$
71 $$1 + 773. iT - 3.57e5T^{2}$$
73 $$1 - 448.T + 3.89e5T^{2}$$
79 $$1 + 1.27e3T + 4.93e5T^{2}$$
83 $$1 + 429. iT - 5.71e5T^{2}$$
89 $$1 + 5.29T + 7.04e5T^{2}$$
97 $$1 + 435.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$