# Properties

 Label 2-105-21.17-c3-0-18 Degree $2$ Conductor $105$ Sign $0.962 + 0.270i$ 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 + (−3.06 + 1.76i)2-s + (−0.433 + 5.17i)3-s + (2.25 − 3.91i)4-s + (−2.5 − 4.33i)5-s + (−7.83 − 16.6i)6-s + (5.77 − 17.5i)7-s − 12.3i·8-s + (−26.6 − 4.48i)9-s + (15.3 + 8.84i)10-s + (3.05 + 1.76i)11-s + (19.2 + 13.3i)12-s − 14.5i·13-s + (13.4 + 64.1i)14-s + (23.5 − 11.0i)15-s + (39.8 + 69.0i)16-s + (30.9 − 53.5i)17-s + ⋯
 L(s)  = 1 + (−1.08 + 0.625i)2-s + (−0.0834 + 0.996i)3-s + (0.282 − 0.488i)4-s + (−0.223 − 0.387i)5-s + (−0.532 − 1.13i)6-s + (0.311 − 0.950i)7-s − 0.544i·8-s + (−0.986 − 0.166i)9-s + (0.484 + 0.279i)10-s + (0.0837 + 0.0483i)11-s + (0.463 + 0.322i)12-s − 0.310i·13-s + (0.256 + 1.22i)14-s + (0.404 − 0.190i)15-s + (0.622 + 1.07i)16-s + (0.441 − 0.764i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $0.962 + 0.270i$ Analytic conductor: $$6.19520$$ Root analytic conductor: $$2.48901$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{105} (101, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 105,\ (\ :3/2),\ 0.962 + 0.270i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.596183 - 0.0822087i$$ $$L(\frac12)$$ $$\approx$$ $$0.596183 - 0.0822087i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (0.433 - 5.17i)T$$
5 $$1 + (2.5 + 4.33i)T$$
7 $$1 + (-5.77 + 17.5i)T$$
good2 $$1 + (3.06 - 1.76i)T + (4 - 6.92i)T^{2}$$
11 $$1 + (-3.05 - 1.76i)T + (665.5 + 1.15e3i)T^{2}$$
13 $$1 + 14.5iT - 2.19e3T^{2}$$
17 $$1 + (-30.9 + 53.5i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (39.8 - 23.0i)T + (3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-99.5 + 57.4i)T + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + 127. iT - 2.43e4T^{2}$$
31 $$1 + (-183. - 106. i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (142. + 246. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 328.T + 6.89e4T^{2}$$
43 $$1 + 108.T + 7.95e4T^{2}$$
47 $$1 + (194. + 337. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (-514. - 297. i)T + (7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-275. + 477. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-411. + 237. i)T + (1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-208. + 360. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 399. iT - 3.57e5T^{2}$$
73 $$1 + (134. + 77.6i)T + (1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (586. + 1.01e3i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 - 4.43T + 5.71e5T^{2}$$
89 $$1 + (-505. - 875. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 + 27.5iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$