# Properties

 Label 2-1872-13.4-c1-0-18 Degree $2$ Conductor $1872$ Sign $0.964 + 0.265i$ Analytic cond. $14.9479$ Root an. cond. $3.86626$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.73i·5-s + (−2.5 − 2.59i)13-s + (−1.5 − 2.59i)17-s + (3 − 1.73i)19-s + (3 − 5.19i)23-s + 2.00·25-s + (1.5 − 2.59i)29-s − 3.46i·31-s + (7.5 + 4.33i)37-s + (4.5 + 2.59i)41-s + (4 + 6.92i)43-s − 3.46i·47-s + (−3.5 + 6.06i)49-s + 3·53-s + (6 − 3.46i)59-s + ⋯
 L(s)  = 1 + 0.774i·5-s + (−0.693 − 0.720i)13-s + (−0.363 − 0.630i)17-s + (0.688 − 0.397i)19-s + (0.625 − 1.08i)23-s + 0.400·25-s + (0.278 − 0.482i)29-s − 0.622i·31-s + (1.23 + 0.711i)37-s + (0.702 + 0.405i)41-s + (0.609 + 1.05i)43-s − 0.505i·47-s + (−0.5 + 0.866i)49-s + 0.412·53-s + (0.781 − 0.450i)59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1872$$    =    $$2^{4} \cdot 3^{2} \cdot 13$$ Sign: $0.964 + 0.265i$ Analytic conductor: $$14.9479$$ Root analytic conductor: $$3.86626$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1872} (433, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1872,\ (\ :1/2),\ 0.964 + 0.265i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.649925025$$ $$L(\frac12)$$ $$\approx$$ $$1.649925025$$ $$L(\frac{3}{2})$$ 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$$
13 $$1 + (2.5 + 2.59i)T$$
good5 $$1 - 1.73iT - 5T^{2}$$
7 $$1 + (3.5 - 6.06i)T^{2}$$
11 $$1 + (5.5 + 9.52i)T^{2}$$
17 $$1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2}$$
23 $$1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + 3.46iT - 31T^{2}$$
37 $$1 + (-7.5 - 4.33i)T + (18.5 + 32.0i)T^{2}$$
41 $$1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + 3.46iT - 47T^{2}$$
53 $$1 - 3T + 53T^{2}$$
59 $$1 + (-6 + 3.46i)T + (29.5 - 51.0i)T^{2}$$
61 $$1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (3 + 1.73i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + (-3 + 1.73i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 - 1.73iT - 73T^{2}$$
79 $$1 + 4T + 79T^{2}$$
83 $$1 - 13.8iT - 83T^{2}$$
89 $$1 + (-6 - 3.46i)T + (44.5 + 77.0i)T^{2}$$
97 $$1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$