# Properties

 Label 2-48e2-8.3-c2-0-20 Degree $2$ Conductor $2304$ Sign $-0.707 - 0.707i$ 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 + 8i·5-s − 10i·13-s + 16·17-s − 39·25-s + 40i·29-s + 70i·37-s + 80·41-s + 49·49-s + 56i·53-s − 22i·61-s + 80·65-s − 110·73-s + 128i·85-s − 160·89-s − 130·97-s + ⋯
 L(s)  = 1 + 1.60i·5-s − 0.769i·13-s + 0.941·17-s − 1.56·25-s + 1.37i·29-s + 1.89i·37-s + 1.95·41-s + 0.999·49-s + 1.05i·53-s − 0.360i·61-s + 1.23·65-s − 1.50·73-s + 1.50i·85-s − 1.79·89-s − 1.34·97-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, 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: $$62.7794$$ Root analytic conductor: $$7.92334$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{2304} (127, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2304,\ (\ :1),\ -0.707 - 0.707i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.629627728$$ $$L(\frac12)$$ $$\approx$$ $$1.629627728$$ $$L(2)$$ 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 - 8iT - 25T^{2}$$
7 $$1 - 49T^{2}$$
11 $$1 + 121T^{2}$$
13 $$1 + 10iT - 169T^{2}$$
17 $$1 - 16T + 289T^{2}$$
19 $$1 + 361T^{2}$$
23 $$1 - 529T^{2}$$
29 $$1 - 40iT - 841T^{2}$$
31 $$1 - 961T^{2}$$
37 $$1 - 70iT - 1.36e3T^{2}$$
41 $$1 - 80T + 1.68e3T^{2}$$
43 $$1 + 1.84e3T^{2}$$
47 $$1 - 2.20e3T^{2}$$
53 $$1 - 56iT - 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 + 110T + 5.32e3T^{2}$$
79 $$1 - 6.24e3T^{2}$$
83 $$1 + 6.88e3T^{2}$$
89 $$1 + 160T + 7.92e3T^{2}$$
97 $$1 + 130T + 9.40e3T^{2}$$
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$$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

−9.203115553953619061146510685632, −8.153787549790755160341835904355, −7.49631161009048434862140078490, −6.86598541253982223175523954476, −6.05515020503578341482657294307, −5.34692039867896409792298054887, −4.12110078672504490633348312926, −3.11297863139009321289180509152, −2.71827680301516099413511440193, −1.24912961235045319114499090181, 0.42743982385974766447887423793, 1.38334480358141592952942846253, 2.46254032131779528597604432431, 3.96658397178248202279254696339, 4.39039331770081936004751706489, 5.47037191862805957776267867435, 5.89076253622260548545836568733, 7.14026767403011227624318095523, 7.891538340471918946697573102329, 8.580882273943320243697807363457