# Properties

 Label 2-48e2-24.5-c2-0-52 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 − 8·7-s + 11.3·11-s + 8i·13-s + 12.7i·17-s − 32i·19-s + 33.9i·23-s − 23·25-s − 43.8·29-s + 40·31-s − 11.3·35-s − 26i·37-s − 66.4i·41-s + 16i·43-s + 11.3i·47-s + ⋯
 L(s)  = 1 + 0.282·5-s − 1.14·7-s + 1.02·11-s + 0.615i·13-s + 0.748i·17-s − 1.68i·19-s + 1.47i·23-s − 0.920·25-s − 1.51·29-s + 1.29·31-s − 0.323·35-s − 0.702i·37-s − 1.62i·41-s + 0.372i·43-s + 0.240i·47-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

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.03667752542$$ $$L(\frac12)$$ $$\approx$$ $$0.03667752542$$ $$L(2)$$ 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 + 8T + 49T^{2}$$
11 $$1 - 11.3T + 121T^{2}$$
13 $$1 - 8iT - 169T^{2}$$
17 $$1 - 12.7iT - 289T^{2}$$
19 $$1 + 32iT - 361T^{2}$$
23 $$1 - 33.9iT - 529T^{2}$$
29 $$1 + 43.8T + 841T^{2}$$
31 $$1 - 40T + 961T^{2}$$
37 $$1 + 26iT - 1.36e3T^{2}$$
41 $$1 + 66.4iT - 1.68e3T^{2}$$
43 $$1 - 16iT - 1.84e3T^{2}$$
47 $$1 - 11.3iT - 2.20e3T^{2}$$
53 $$1 - 32.5T + 2.80e3T^{2}$$
59 $$1 - 22.6T + 3.48e3T^{2}$$
61 $$1 - 54iT - 3.72e3T^{2}$$
67 $$1 - 80iT - 4.48e3T^{2}$$
71 $$1 + 79.1iT - 5.04e3T^{2}$$
73 $$1 + 96T + 5.32e3T^{2}$$
79 $$1 + 104T + 6.24e3T^{2}$$
83 $$1 - 101.T + 6.88e3T^{2}$$
89 $$1 + 77.7iT - 7.92e3T^{2}$$
97 $$1 + 80T + 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

−8.757419179723430416722830087848, −7.47208755448719434494998410274, −6.89873717083227407722932990951, −6.14647719220171940046115242553, −5.49153526887748362746881151699, −4.19511532519424435870255524045, −3.63843157890471106415011584868, −2.52443740819015770592049993572, −1.43393750777722667586831938109, −0.008980643404592978051554967899, 1.29518052050737481884793050177, 2.55977495382256667386514216578, 3.47850624896089051184017132241, 4.21997817161554947594306570506, 5.39816563331625518987754171549, 6.25373823218466162766484865047, 6.60458126636050941415000544718, 7.70341303724271353838874022206, 8.416759049353437500978055393875, 9.369773671871922505369854058771