Properties

 Label 2-105-21.17-c3-0-17 Degree $2$ Conductor $105$ Sign $0.913 - 0.406i$ 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 + (0.155 − 0.0897i)2-s + (5.17 − 0.514i)3-s + (−3.98 + 6.90i)4-s + (−2.5 − 4.33i)5-s + (0.757 − 0.544i)6-s + (18.1 − 3.54i)7-s + 2.86i·8-s + (26.4 − 5.31i)9-s + (−0.777 − 0.448i)10-s + (42.8 + 24.7i)11-s + (−17.0 + 37.7i)12-s + 62.5i·13-s + (2.50 − 2.18i)14-s + (−15.1 − 21.1i)15-s + (−31.6 − 54.7i)16-s + (19.4 − 33.6i)17-s + ⋯
 L(s)  = 1 + (0.0549 − 0.0317i)2-s + (0.995 − 0.0989i)3-s + (−0.497 + 0.862i)4-s + (−0.223 − 0.387i)5-s + (0.0515 − 0.0370i)6-s + (0.981 − 0.191i)7-s + 0.126i·8-s + (0.980 − 0.196i)9-s + (−0.0245 − 0.0141i)10-s + (1.17 + 0.677i)11-s + (−0.410 + 0.907i)12-s + 1.33i·13-s + (0.0478 − 0.0416i)14-s + (−0.260 − 0.363i)15-s + (−0.493 − 0.855i)16-s + (0.277 − 0.480i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $0.913 - 0.406i$ 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: $$(2,\ 105,\ (\ :3/2),\ 0.913 - 0.406i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$2.13710 + 0.453903i$$ $$L(\frac12)$$ $$\approx$$ $$2.13710 + 0.453903i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-5.17 + 0.514i)T$$
5 $$1 + (2.5 + 4.33i)T$$
7 $$1 + (-18.1 + 3.54i)T$$
good2 $$1 + (-0.155 + 0.0897i)T + (4 - 6.92i)T^{2}$$
11 $$1 + (-42.8 - 24.7i)T + (665.5 + 1.15e3i)T^{2}$$
13 $$1 - 62.5iT - 2.19e3T^{2}$$
17 $$1 + (-19.4 + 33.6i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (-14.1 + 8.18i)T + (3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (125. - 72.6i)T + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + 246. iT - 2.43e4T^{2}$$
31 $$1 + (-128. - 74.0i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (174. + 301. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 429.T + 6.89e4T^{2}$$
43 $$1 - 73.5T + 7.95e4T^{2}$$
47 $$1 + (124. + 214. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (263. + 152. i)T + (7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (336. - 582. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (279. - 161. i)T + (1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (103. - 179. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 717. iT - 3.57e5T^{2}$$
73 $$1 + (-84.3 - 48.7i)T + (1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-450. - 779. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 - 90.9T + 5.71e5T^{2}$$
89 $$1 + (-328. - 568. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 + 1.46e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$