# Properties

 Label 2-2016-7.4-c1-0-6 Degree $2$ Conductor $2016$ Sign $0.895 - 0.444i$ Analytic cond. $16.0978$ Root an. cond. $4.01221$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.5 − 2.59i)5-s − 2.64·7-s + (−1.32 + 2.29i)11-s − 4·13-s + (0.5 − 0.866i)17-s + (3.96 + 6.87i)19-s + (−1.32 − 2.29i)23-s + (−2 + 3.46i)25-s + 4·29-s + (1.32 − 2.29i)31-s + (3.96 + 6.87i)35-s + (2.5 + 4.33i)37-s − 8·41-s + 10.5·43-s + (1.32 + 2.29i)47-s + ⋯
 L(s)  = 1 + (−0.670 − 1.16i)5-s − 0.999·7-s + (−0.398 + 0.690i)11-s − 1.10·13-s + (0.121 − 0.210i)17-s + (0.910 + 1.57i)19-s + (−0.275 − 0.477i)23-s + (−0.400 + 0.692i)25-s + 0.742·29-s + (0.237 − 0.411i)31-s + (0.670 + 1.16i)35-s + (0.410 + 0.711i)37-s − 1.24·41-s + 1.61·43-s + (0.192 + 0.334i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2016$$    =    $$2^{5} \cdot 3^{2} \cdot 7$$ Sign: $0.895 - 0.444i$ Analytic conductor: $$16.0978$$ Root analytic conductor: $$4.01221$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2016} (865, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2016,\ (\ :1/2),\ 0.895 - 0.444i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9250146640$$ $$L(\frac12)$$ $$\approx$$ $$0.9250146640$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1 + 2.64T$$
good5 $$1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (1.32 - 2.29i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + 4T + 13T^{2}$$
17 $$1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-3.96 - 6.87i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (1.32 + 2.29i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 - 4T + 29T^{2}$$
31 $$1 + (-1.32 + 2.29i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + 8T + 41T^{2}$$
43 $$1 - 10.5T + 43T^{2}$$
47 $$1 + (-1.32 - 2.29i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-3.5 + 6.06i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-1.32 + 2.29i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (1.32 - 2.29i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + (-4.5 + 7.79i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-1.32 - 2.29i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 10.5T + 83T^{2}$$
89 $$1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 8T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$