# Properties

 Label 2-1110-185.117-c1-0-16 Degree $2$ Conductor $1110$ Sign $0.229 + 0.973i$ Analytic cond. $8.86339$ Root an. cond. $2.97714$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + (−0.707 − 0.707i)3-s + 4-s + (1.21 + 1.87i)5-s + (0.707 + 0.707i)6-s + (−2.24 − 2.24i)7-s − 8-s + 1.00i·9-s + (−1.21 − 1.87i)10-s + 5.65i·11-s + (−0.707 − 0.707i)12-s − 4.81·13-s + (2.24 + 2.24i)14-s + (0.466 − 2.18i)15-s + 16-s − 7.58i·17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (0.543 + 0.839i)5-s + (0.288 + 0.288i)6-s + (−0.849 − 0.849i)7-s − 0.353·8-s + 0.333i·9-s + (−0.384 − 0.593i)10-s + 1.70i·11-s + (−0.204 − 0.204i)12-s − 1.33·13-s + (0.600 + 0.600i)14-s + (0.120 − 0.564i)15-s + 0.250·16-s − 1.83i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1110$$    =    $$2 \cdot 3 \cdot 5 \cdot 37$$ Sign: $0.229 + 0.973i$ Analytic conductor: $$8.86339$$ Root analytic conductor: $$2.97714$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1110} (487, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1110,\ (\ :1/2),\ 0.229 + 0.973i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.7252836472$$ $$L(\frac12)$$ $$\approx$$ $$0.7252836472$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + (0.707 + 0.707i)T$$
5 $$1 + (-1.21 - 1.87i)T$$
37 $$1 + (-4.42 + 4.17i)T$$
good7 $$1 + (2.24 + 2.24i)T + 7iT^{2}$$
11 $$1 - 5.65iT - 11T^{2}$$
13 $$1 + 4.81T + 13T^{2}$$
17 $$1 + 7.58iT - 17T^{2}$$
19 $$1 + (-3.73 + 3.73i)T - 19iT^{2}$$
23 $$1 - 4.23T + 23T^{2}$$
29 $$1 + (3.26 + 3.26i)T + 29iT^{2}$$
31 $$1 + (-1.98 + 1.98i)T - 31iT^{2}$$
41 $$1 + 2.43iT - 41T^{2}$$
43 $$1 + 3.34T + 43T^{2}$$
47 $$1 + (-6.04 - 6.04i)T + 47iT^{2}$$
53 $$1 + (-8.02 + 8.02i)T - 53iT^{2}$$
59 $$1 + (-9.84 + 9.84i)T - 59iT^{2}$$
61 $$1 + (1.11 - 1.11i)T - 61iT^{2}$$
67 $$1 + (6.91 - 6.91i)T - 67iT^{2}$$
71 $$1 + 6.12T + 71T^{2}$$
73 $$1 + (-2.58 - 2.58i)T + 73iT^{2}$$
79 $$1 + (-2.85 + 2.85i)T - 79iT^{2}$$
83 $$1 + (-0.906 + 0.906i)T - 83iT^{2}$$
89 $$1 + (10.1 + 10.1i)T + 89iT^{2}$$
97 $$1 + 2.87iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$