# Properties

 Label 2-2028-13.4-c1-0-24 Degree $2$ Conductor $2028$ Sign $-0.997 + 0.0771i$ Analytic cond. $16.1936$ Root an. cond. $4.02413$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 − 0.866i)3-s − 4i·5-s + (1.73 − i)7-s + (−0.499 − 0.866i)9-s + (−3.46 − 2i)11-s + (−3.46 − 2i)15-s + (1 + 1.73i)17-s + (−1.73 + i)19-s − 1.99i·21-s − 11·25-s − 0.999·27-s + (3 − 5.19i)29-s − 10i·31-s + (−3.46 + 1.99i)33-s + (−4 − 6.92i)35-s + ⋯
 L(s)  = 1 + (0.288 − 0.499i)3-s − 1.78i·5-s + (0.654 − 0.377i)7-s + (−0.166 − 0.288i)9-s + (−1.04 − 0.603i)11-s + (−0.894 − 0.516i)15-s + (0.242 + 0.420i)17-s + (−0.397 + 0.229i)19-s − 0.436i·21-s − 2.20·25-s − 0.192·27-s + (0.557 − 0.964i)29-s − 1.79i·31-s + (−0.603 + 0.348i)33-s + (−0.676 − 1.17i)35-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0771i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0771i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2028$$    =    $$2^{2} \cdot 3 \cdot 13^{2}$$ Sign: $-0.997 + 0.0771i$ Analytic conductor: $$16.1936$$ Root analytic conductor: $$4.02413$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2028} (1837, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2028,\ (\ :1/2),\ -0.997 + 0.0771i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.520115856$$ $$L(\frac12)$$ $$\approx$$ $$1.520115856$$ $$L(\frac{3}{2})$$ 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 + (-0.5 + 0.866i)T$$
13 $$1$$
good5 $$1 + 4iT - 5T^{2}$$
7 $$1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2}$$
11 $$1 + (3.46 + 2i)T + (5.5 + 9.52i)T^{2}$$
17 $$1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (1.73 - i)T + (9.5 - 16.4i)T^{2}$$
23 $$1 + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + 10iT - 31T^{2}$$
37 $$1 + (-8.66 - 5i)T + (18.5 + 32.0i)T^{2}$$
41 $$1 + (6.92 + 4i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 - 4iT - 47T^{2}$$
53 $$1 + 10T + 53T^{2}$$
59 $$1 + (-6.92 + 4i)T + (29.5 - 51.0i)T^{2}$$
61 $$1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (1.73 + i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + (-13.8 + 8i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 - 10iT - 73T^{2}$$
79 $$1 + 16T + 79T^{2}$$
83 $$1 - 83T^{2}$$
89 $$1 + (3.46 + 2i)T + (44.5 + 77.0i)T^{2}$$
97 $$1 + (1.73 - i)T + (48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$