# Properties

 Label 2-48e2-24.5-c2-0-10 Degree $2$ Conductor $2304$ Sign $-0.985 - 0.169i$ Analytic cond. $62.7794$ Root an. cond. $7.92334$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.41·5-s + 24i·13-s + 32.5i·17-s − 23·25-s + 1.41·29-s − 70i·37-s − 69.2i·41-s − 49·49-s − 103.·53-s − 22i·61-s + 33.9i·65-s + 96·73-s + 46i·85-s + 168. i·89-s − 144·97-s + ⋯
 L(s)  = 1 + 0.282·5-s + 1.84i·13-s + 1.91i·17-s − 0.920·25-s + 0.0487·29-s − 1.89i·37-s − 1.69i·41-s − 0.999·49-s − 1.94·53-s − 0.360i·61-s + 0.522i·65-s + 1.31·73-s + 0.541i·85-s + 1.89i·89-s − 1.48·97-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2304$$    =    $$2^{8} \cdot 3^{2}$$ Sign: $-0.985 - 0.169i$ Analytic conductor: $$62.7794$$ Root analytic conductor: $$7.92334$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{2304} (2177, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2304,\ (\ :1),\ -0.985 - 0.169i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.6972998128$$ $$L(\frac12)$$ $$\approx$$ $$0.6972998128$$ $$L(2)$$ 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 - 1.41T + 25T^{2}$$
7 $$1 + 49T^{2}$$
11 $$1 + 121T^{2}$$
13 $$1 - 24iT - 169T^{2}$$
17 $$1 - 32.5iT - 289T^{2}$$
19 $$1 - 361T^{2}$$
23 $$1 - 529T^{2}$$
29 $$1 - 1.41T + 841T^{2}$$
31 $$1 + 961T^{2}$$
37 $$1 + 70iT - 1.36e3T^{2}$$
41 $$1 + 69.2iT - 1.68e3T^{2}$$
43 $$1 - 1.84e3T^{2}$$
47 $$1 - 2.20e3T^{2}$$
53 $$1 + 103.T + 2.80e3T^{2}$$
59 $$1 + 3.48e3T^{2}$$
61 $$1 + 22iT - 3.72e3T^{2}$$
67 $$1 - 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 - 96T + 5.32e3T^{2}$$
79 $$1 + 6.24e3T^{2}$$
83 $$1 + 6.88e3T^{2}$$
89 $$1 - 168. iT - 7.92e3T^{2}$$
97 $$1 + 144T + 9.40e3T^{2}$$
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−9.254085497229185929248265822851, −8.468251183867451756198536950410, −7.69273795052874336591117886594, −6.72525936889767904456656062311, −6.18218578877772472100235833310, −5.31985803890213966103242703215, −4.17156508503479391566961883679, −3.73368506844486739178244947320, −2.18194333067412668526138470071, −1.60670615511753637354622513128, 0.16319160761093751898636623169, 1.31365068187300220348025650684, 2.76277005879972610911122811248, 3.22188906562215573384401901285, 4.64865654372599771548220061868, 5.23448687442913221682203019954, 6.08263385305934633375010398408, 6.90424808053979964882234791919, 7.898450500653818807549465135643, 8.202365239962739368080389969130