# Properties

 Label 2-2736-76.75-c1-0-4 Degree $2$ Conductor $2736$ Sign $-0.114 - 0.993i$ Analytic cond. $21.8470$ Root an. cond. $4.67408$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.37·5-s − 2.52i·7-s + 2.52i·11-s − 1.58i·13-s + 0.372·17-s + (−4 − 1.73i)19-s + 1.87i·23-s + 0.627·25-s − 3.16i·29-s + 2.74·31-s + 5.98i·35-s − 1.58i·37-s + 6.92i·41-s + 0.644i·43-s + 0.939i·47-s + ⋯
 L(s)  = 1 − 1.06·5-s − 0.954i·7-s + 0.761i·11-s − 0.439i·13-s + 0.0902·17-s + (−0.917 − 0.397i)19-s + 0.391i·23-s + 0.125·25-s − 0.588i·29-s + 0.492·31-s + 1.01i·35-s − 0.260i·37-s + 1.08i·41-s + 0.0983i·43-s + 0.137i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2736$$    =    $$2^{4} \cdot 3^{2} \cdot 19$$ Sign: $-0.114 - 0.993i$ Analytic conductor: $$21.8470$$ Root analytic conductor: $$4.67408$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2736} (2431, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2736,\ (\ :1/2),\ -0.114 - 0.993i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6285900663$$ $$L(\frac12)$$ $$\approx$$ $$0.6285900663$$ $$L(\frac{3}{2})$$ 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$$
19 $$1 + (4 + 1.73i)T$$
good5 $$1 + 2.37T + 5T^{2}$$
7 $$1 + 2.52iT - 7T^{2}$$
11 $$1 - 2.52iT - 11T^{2}$$
13 $$1 + 1.58iT - 13T^{2}$$
17 $$1 - 0.372T + 17T^{2}$$
23 $$1 - 1.87iT - 23T^{2}$$
29 $$1 + 3.16iT - 29T^{2}$$
31 $$1 - 2.74T + 31T^{2}$$
37 $$1 + 1.58iT - 37T^{2}$$
41 $$1 - 6.92iT - 41T^{2}$$
43 $$1 - 0.644iT - 43T^{2}$$
47 $$1 - 0.939iT - 47T^{2}$$
53 $$1 - 10.0iT - 53T^{2}$$
59 $$1 + 4T + 59T^{2}$$
61 $$1 + 0.372T + 61T^{2}$$
67 $$1 + 13.4T + 67T^{2}$$
71 $$1 + 4T + 71T^{2}$$
73 $$1 - 13.1T + 73T^{2}$$
79 $$1 - 6.74T + 79T^{2}$$
83 $$1 - 3.46iT - 83T^{2}$$
89 $$1 - 13.2iT - 89T^{2}$$
97 $$1 - 13.2iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$