# Properties

 Label 2-48e2-8.5-c1-0-35 Degree $2$ Conductor $2304$ Sign $-0.707 - 0.707i$ Analytic cond. $18.3975$ Root an. cond. $4.28923$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2i·5-s − 4·7-s − 2i·11-s − 2i·13-s + 2·17-s + 2i·19-s − 4·23-s + 25-s − 6i·29-s + 8i·35-s + 10i·37-s − 6·41-s − 6i·43-s − 8·47-s + 9·49-s + ⋯
 L(s)  = 1 − 0.894i·5-s − 1.51·7-s − 0.603i·11-s − 0.554i·13-s + 0.485·17-s + 0.458i·19-s − 0.834·23-s + 0.200·25-s − 1.11i·29-s + 1.35i·35-s + 1.64i·37-s − 0.937·41-s − 0.914i·43-s − 1.16·47-s + 1.28·49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2304$$    =    $$2^{8} \cdot 3^{2}$$ Sign: $-0.707 - 0.707i$ Analytic conductor: $$18.3975$$ Root analytic conductor: $$4.28923$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2304} (1153, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$1$$ Selberg data: $$(2,\ 2304,\ (\ :1/2),\ -0.707 - 0.707i)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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$$
good5 $$1 + 2iT - 5T^{2}$$
7 $$1 + 4T + 7T^{2}$$
11 $$1 + 2iT - 11T^{2}$$
13 $$1 + 2iT - 13T^{2}$$
17 $$1 - 2T + 17T^{2}$$
19 $$1 - 2iT - 19T^{2}$$
23 $$1 + 4T + 23T^{2}$$
29 $$1 + 6iT - 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 10iT - 37T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 + 6iT - 43T^{2}$$
47 $$1 + 8T + 47T^{2}$$
53 $$1 - 6iT - 53T^{2}$$
59 $$1 - 14iT - 59T^{2}$$
61 $$1 + 2iT - 61T^{2}$$
67 $$1 - 10iT - 67T^{2}$$
71 $$1 + 12T + 71T^{2}$$
73 $$1 + 14T + 73T^{2}$$
79 $$1 - 8T + 79T^{2}$$
83 $$1 - 6iT - 83T^{2}$$
89 $$1 + 2T + 89T^{2}$$
97 $$1 + 2T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$