# Properties

 Label 2-1110-111.68-c1-0-13 Degree $2$ Conductor $1110$ Sign $0.888 - 0.458i$ 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 + (−0.707 + 0.707i)2-s + (−1.08 + 1.35i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (−0.189 − 1.72i)6-s − 3.77·7-s + (0.707 + 0.707i)8-s + (−0.651 − 2.92i)9-s + 1.00·10-s + 1.81·11-s + (1.35 + 1.08i)12-s + (−3.90 + 3.90i)13-s + (2.67 − 2.67i)14-s + (1.72 − 0.189i)15-s − 1.00·16-s + (−1.04 − 1.04i)17-s + ⋯
 L(s)  = 1 + (−0.499 + 0.499i)2-s + (−0.625 + 0.780i)3-s − 0.500i·4-s + (−0.316 − 0.316i)5-s + (−0.0772 − 0.702i)6-s − 1.42·7-s + (0.250 + 0.250i)8-s + (−0.217 − 0.976i)9-s + 0.316·10-s + 0.547·11-s + (0.390 + 0.312i)12-s + (−1.08 + 1.08i)13-s + (0.713 − 0.713i)14-s + (0.444 − 0.0488i)15-s − 0.250·16-s + (−0.253 − 0.253i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1110$$    =    $$2 \cdot 3 \cdot 5 \cdot 37$$ Sign: $0.888 - 0.458i$ Analytic conductor: $$8.86339$$ Root analytic conductor: $$2.97714$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1110} (401, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1110,\ (\ :1/2),\ 0.888 - 0.458i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5215866783$$ $$L(\frac12)$$ $$\approx$$ $$0.5215866783$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.707 - 0.707i)T$$
3 $$1 + (1.08 - 1.35i)T$$
5 $$1 + (0.707 + 0.707i)T$$
37 $$1 + (-5.83 + 1.70i)T$$
good7 $$1 + 3.77T + 7T^{2}$$
11 $$1 - 1.81T + 11T^{2}$$
13 $$1 + (3.90 - 3.90i)T - 13iT^{2}$$
17 $$1 + (1.04 + 1.04i)T + 17iT^{2}$$
19 $$1 + (1.01 - 1.01i)T - 19iT^{2}$$
23 $$1 + (5.79 + 5.79i)T + 23iT^{2}$$
29 $$1 + (-3.80 + 3.80i)T - 29iT^{2}$$
31 $$1 + (-4.46 - 4.46i)T + 31iT^{2}$$
41 $$1 - 9.74T + 41T^{2}$$
43 $$1 + (-6.17 + 6.17i)T - 43iT^{2}$$
47 $$1 - 7.59iT - 47T^{2}$$
53 $$1 + 1.46iT - 53T^{2}$$
59 $$1 + (3.27 + 3.27i)T + 59iT^{2}$$
61 $$1 + (0.251 + 0.251i)T + 61iT^{2}$$
67 $$1 + 1.79iT - 67T^{2}$$
71 $$1 - 14.5iT - 71T^{2}$$
73 $$1 + 7.13iT - 73T^{2}$$
79 $$1 + (8.05 - 8.05i)T - 79iT^{2}$$
83 $$1 - 7.15iT - 83T^{2}$$
89 $$1 + (-7.47 + 7.47i)T - 89iT^{2}$$
97 $$1 + (-3.22 + 3.22i)T - 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$