# Properties

 Label 2-48e2-16.5-c1-0-18 Degree $2$ Conductor $2304$ Sign $0.991 + 0.130i$ Analytic cond. $18.3975$ Root an. cond. $4.28923$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.378i·7-s + (−2.46 + 2.46i)13-s + (2.44 − 2.44i)19-s − 5i·25-s + 10.1·31-s + (1.53 + 1.53i)37-s + (7.34 + 7.34i)43-s + 6.85·49-s + (−10.4 + 10.4i)61-s + (11.3 − 11.3i)67-s − 13.8i·73-s + 9.41·79-s + (0.933 + 0.933i)91-s + 13.8·97-s − 11.6i·103-s + ⋯
 L(s)  = 1 − 0.143i·7-s + (−0.683 + 0.683i)13-s + (0.561 − 0.561i)19-s − i·25-s + 1.82·31-s + (0.252 + 0.252i)37-s + (1.12 + 1.12i)43-s + 0.979·49-s + (−1.33 + 1.33i)61-s + (1.38 − 1.38i)67-s − 1.62i·73-s + 1.05·79-s + (0.0978 + 0.0978i)91-s + 1.40·97-s − 1.15i·103-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2304$$    =    $$2^{8} \cdot 3^{2}$$ Sign: $0.991 + 0.130i$ Analytic conductor: $$18.3975$$ Root analytic conductor: $$4.28923$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2304} (1729, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2304,\ (\ :1/2),\ 0.991 + 0.130i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.717444341$$ $$L(\frac12)$$ $$\approx$$ $$1.717444341$$ $$L(\frac{3}{2})$$ 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 + 5iT^{2}$$
7 $$1 + 0.378iT - 7T^{2}$$
11 $$1 + 11iT^{2}$$
13 $$1 + (2.46 - 2.46i)T - 13iT^{2}$$
17 $$1 + 17T^{2}$$
19 $$1 + (-2.44 + 2.44i)T - 19iT^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 - 29iT^{2}$$
31 $$1 - 10.1T + 31T^{2}$$
37 $$1 + (-1.53 - 1.53i)T + 37iT^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 + (-7.34 - 7.34i)T + 43iT^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + 53iT^{2}$$
59 $$1 + 59iT^{2}$$
61 $$1 + (10.4 - 10.4i)T - 61iT^{2}$$
67 $$1 + (-11.3 + 11.3i)T - 67iT^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 + 13.8iT - 73T^{2}$$
79 $$1 - 9.41T + 79T^{2}$$
83 $$1 - 83iT^{2}$$
89 $$1 - 89T^{2}$$
97 $$1 - 13.8T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$