# Properties

 Label 2-3744-39.29-c0-0-1 Degree $2$ Conductor $3744$ Sign $0.935 - 0.352i$ Analytic cond. $1.86849$ Root an. cond. $1.36693$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.517i·5-s + (0.866 − 0.5i)13-s + (0.448 + 0.258i)17-s + 0.732·25-s + (−0.448 + 0.258i)29-s + (−0.5 − 0.866i)37-s + (0.448 − 0.258i)41-s + (0.5 + 0.866i)49-s + 1.93i·53-s + (0.5 − 0.866i)61-s + (0.258 + 0.448i)65-s + 1.73·73-s + (−0.133 + 0.232i)85-s + (1.22 − 0.707i)89-s + (−1.67 + 0.965i)101-s + ⋯
 L(s)  = 1 + 0.517i·5-s + (0.866 − 0.5i)13-s + (0.448 + 0.258i)17-s + 0.732·25-s + (−0.448 + 0.258i)29-s + (−0.5 − 0.866i)37-s + (0.448 − 0.258i)41-s + (0.5 + 0.866i)49-s + 1.93i·53-s + (0.5 − 0.866i)61-s + (0.258 + 0.448i)65-s + 1.73·73-s + (−0.133 + 0.232i)85-s + (1.22 − 0.707i)89-s + (−1.67 + 0.965i)101-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3744$$    =    $$2^{5} \cdot 3^{2} \cdot 13$$ Sign: $0.935 - 0.352i$ Analytic conductor: $$1.86849$$ Root analytic conductor: $$1.36693$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3744} (3617, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3744,\ (\ :0),\ 0.935 - 0.352i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.367721578$$ $$L(\frac12)$$ $$\approx$$ $$1.367721578$$ $$L(1)$$ 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 + (-0.866 + 0.5i)T$$
good5 $$1 - 0.517iT - T^{2}$$
7 $$1 + (-0.5 - 0.866i)T^{2}$$
11 $$1 + (0.5 - 0.866i)T^{2}$$
17 $$1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2}$$
19 $$1 + (-0.5 - 0.866i)T^{2}$$
23 $$1 + (0.5 - 0.866i)T^{2}$$
29 $$1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2}$$
31 $$1 + T^{2}$$
37 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
41 $$1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2}$$
43 $$1 + (-0.5 - 0.866i)T^{2}$$
47 $$1 - T^{2}$$
53 $$1 - 1.93iT - T^{2}$$
59 $$1 + (0.5 + 0.866i)T^{2}$$
61 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (-0.5 + 0.866i)T^{2}$$
71 $$1 + (0.5 + 0.866i)T^{2}$$
73 $$1 - 1.73T + T^{2}$$
79 $$1 + T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}$$
97 $$1 + (-0.5 - 0.866i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$