# Properties

 Label 2-105-105.104-c3-0-18 Degree $2$ Conductor $105$ Sign $0.337 + 0.941i$ 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 − 3.23·2-s + (−3.88 − 3.45i)3-s + 2.45·4-s + (8.12 + 7.68i)5-s + (12.5 + 11.1i)6-s + (−17.9 + 4.72i)7-s + 17.9·8-s + (3.15 + 26.8i)9-s + (−26.2 − 24.8i)10-s − 0.605i·11-s + (−9.52 − 8.47i)12-s + 12.8·13-s + (57.8 − 15.2i)14-s + (−5.01 − 57.8i)15-s − 77.6·16-s − 117. i·17-s + ⋯
 L(s)  = 1 − 1.14·2-s + (−0.747 − 0.664i)3-s + 0.306·4-s + (0.726 + 0.687i)5-s + (0.854 + 0.759i)6-s + (−0.966 + 0.255i)7-s + 0.792·8-s + (0.116 + 0.993i)9-s + (−0.830 − 0.785i)10-s − 0.0165i·11-s + (−0.229 − 0.203i)12-s + 0.273·13-s + (1.10 − 0.291i)14-s + (−0.0863 − 0.996i)15-s − 1.21·16-s − 1.68i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $0.337 + 0.941i$ 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: $$(2,\ 105,\ (\ :3/2),\ 0.337 + 0.941i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.441314 - 0.310505i$$ $$L(\frac12)$$ $$\approx$$ $$0.441314 - 0.310505i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (3.88 + 3.45i)T$$
5 $$1 + (-8.12 - 7.68i)T$$
7 $$1 + (17.9 - 4.72i)T$$
good2 $$1 + 3.23T + 8T^{2}$$
11 $$1 + 0.605iT - 1.33e3T^{2}$$
13 $$1 - 12.8T + 2.19e3T^{2}$$
17 $$1 + 117. iT - 4.91e3T^{2}$$
19 $$1 + 98.5iT - 6.85e3T^{2}$$
23 $$1 - 136.T + 1.21e4T^{2}$$
29 $$1 + 77.5iT - 2.43e4T^{2}$$
31 $$1 + 131. iT - 2.97e4T^{2}$$
37 $$1 - 260. iT - 5.06e4T^{2}$$
41 $$1 + 58.0T + 6.89e4T^{2}$$
43 $$1 + 519. iT - 7.95e4T^{2}$$
47 $$1 - 104. iT - 1.03e5T^{2}$$
53 $$1 - 550.T + 1.48e5T^{2}$$
59 $$1 - 498.T + 2.05e5T^{2}$$
61 $$1 - 172. iT - 2.26e5T^{2}$$
67 $$1 + 622. iT - 3.00e5T^{2}$$
71 $$1 + 151. iT - 3.57e5T^{2}$$
73 $$1 - 242.T + 3.89e5T^{2}$$
79 $$1 - 94.6T + 4.93e5T^{2}$$
83 $$1 + 779. iT - 5.71e5T^{2}$$
89 $$1 - 1.00e3T + 7.04e5T^{2}$$
97 $$1 + 1.12e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$