# Properties

 Label 2-2496-13.12-c1-0-44 Degree $2$ Conductor $2496$ Sign $-0.277 + 0.960i$ Analytic cond. $19.9306$ Root an. cond. $4.46437$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3-s − 3.46i·7-s + 9-s + 3.46i·11-s + (1 − 3.46i)13-s − 6·17-s − 3.46i·19-s − 3.46i·21-s + 5·25-s + 27-s − 6·29-s − 3.46i·31-s + 3.46i·33-s − 6.92i·37-s + (1 − 3.46i)39-s + ⋯
 L(s)  = 1 + 0.577·3-s − 1.30i·7-s + 0.333·9-s + 1.04i·11-s + (0.277 − 0.960i)13-s − 1.45·17-s − 0.794i·19-s − 0.755i·21-s + 25-s + 0.192·27-s − 1.11·29-s − 0.622i·31-s + 0.603i·33-s − 1.13i·37-s + (0.160 − 0.554i)39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2496$$    =    $$2^{6} \cdot 3 \cdot 13$$ Sign: $-0.277 + 0.960i$ Analytic conductor: $$19.9306$$ Root analytic conductor: $$4.46437$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2496} (961, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2496,\ (\ :1/2),\ -0.277 + 0.960i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.688111609$$ $$L(\frac12)$$ $$\approx$$ $$1.688111609$$ $$L(\frac{3}{2})$$ 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 - T$$
13 $$1 + (-1 + 3.46i)T$$
good5 $$1 - 5T^{2}$$
7 $$1 + 3.46iT - 7T^{2}$$
11 $$1 - 3.46iT - 11T^{2}$$
17 $$1 + 6T + 17T^{2}$$
19 $$1 + 3.46iT - 19T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 + 3.46iT - 31T^{2}$$
37 $$1 + 6.92iT - 37T^{2}$$
41 $$1 + 6.92iT - 41T^{2}$$
43 $$1 - 4T + 43T^{2}$$
47 $$1 - 3.46iT - 47T^{2}$$
53 $$1 + 6T + 53T^{2}$$
59 $$1 + 10.3iT - 59T^{2}$$
61 $$1 - 2T + 61T^{2}$$
67 $$1 - 10.3iT - 67T^{2}$$
71 $$1 - 3.46iT - 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 + 8T + 79T^{2}$$
83 $$1 - 3.46iT - 83T^{2}$$
89 $$1 + 6.92iT - 89T^{2}$$
97 $$1 + 13.8iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$