# Properties

 Label 2-48e2-3.2-c2-0-6 Degree $2$ Conductor $2304$ Sign $-0.577 + 0.816i$ Analytic cond. $62.7794$ Root an. cond. $7.92334$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 7.34i·5-s − 10.3·7-s + 8.48i·11-s + 10.3·13-s + 21.2i·17-s − 20·19-s + 14.6i·23-s − 29·25-s + 36.7i·29-s − 51.9·31-s − 76.3i·35-s + 41.5·37-s − 72.1i·41-s + 40·43-s + 73.4i·47-s + ⋯
 L(s)  = 1 + 1.46i·5-s − 1.48·7-s + 0.771i·11-s + 0.799·13-s + 1.24i·17-s − 1.05·19-s + 0.638i·23-s − 1.15·25-s + 1.26i·29-s − 1.67·31-s − 2.18i·35-s + 1.12·37-s − 1.75i·41-s + 0.930·43-s + 1.56i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2304$$    =    $$2^{8} \cdot 3^{2}$$ Sign: $-0.577 + 0.816i$ Analytic conductor: $$62.7794$$ Root analytic conductor: $$7.92334$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{2304} (1025, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2304,\ (\ :1),\ -0.577 + 0.816i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.5838312627$$ $$L(\frac12)$$ $$\approx$$ $$0.5838312627$$ $$L(2)$$ 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 - 7.34iT - 25T^{2}$$
7 $$1 + 10.3T + 49T^{2}$$
11 $$1 - 8.48iT - 121T^{2}$$
13 $$1 - 10.3T + 169T^{2}$$
17 $$1 - 21.2iT - 289T^{2}$$
19 $$1 + 20T + 361T^{2}$$
23 $$1 - 14.6iT - 529T^{2}$$
29 $$1 - 36.7iT - 841T^{2}$$
31 $$1 + 51.9T + 961T^{2}$$
37 $$1 - 41.5T + 1.36e3T^{2}$$
41 $$1 + 72.1iT - 1.68e3T^{2}$$
43 $$1 - 40T + 1.84e3T^{2}$$
47 $$1 - 73.4iT - 2.20e3T^{2}$$
53 $$1 - 36.7iT - 2.80e3T^{2}$$
59 $$1 - 33.9iT - 3.48e3T^{2}$$
61 $$1 + 3.72e3T^{2}$$
67 $$1 + 100T + 4.48e3T^{2}$$
71 $$1 - 73.4iT - 5.04e3T^{2}$$
73 $$1 + 20T + 5.32e3T^{2}$$
79 $$1 - 51.9T + 6.24e3T^{2}$$
83 $$1 + 127. iT - 6.88e3T^{2}$$
89 $$1 + 12.7iT - 7.92e3T^{2}$$
97 $$1 - 40T + 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

−9.348868545547980897639963466082, −8.721400181636294095761019571848, −7.40489781973582862997289246242, −7.10034933104836193753999043321, −6.09288373282015450709020734316, −5.93155442862215245174799442768, −4.19395666683100672097590801666, −3.55404649740844485069620985038, −2.80799604569383465595993447916, −1.74839264346501482224174085955, 0.17406383151243383918410930347, 0.874446351895503344352321540752, 2.36341095923150852238131432571, 3.45674327454576437847480472955, 4.23580639065801223972631129085, 5.16293701910768687090895482061, 6.05946304455884958806522186364, 6.51762177426436259539589015234, 7.71081379798575330030611226725, 8.484572850364494500426227134952