# Properties

 Label 2-585-13.12-c1-0-16 Degree $2$ Conductor $585$ Sign $1$ Analytic cond. $4.67124$ Root an. cond. $2.16130$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 2i·2-s − 2·4-s − i·5-s − 3.60i·7-s + 2·10-s − 3i·11-s − 3.60i·13-s + 7.21·14-s − 4·16-s − 3.60·17-s − 7.21i·19-s + 2i·20-s + 6·22-s + 3.60·23-s − 25-s + 7.21·26-s + ⋯
 L(s)  = 1 + 1.41i·2-s − 4-s − 0.447i·5-s − 1.36i·7-s + 0.632·10-s − 0.904i·11-s − 0.999i·13-s + 1.92·14-s − 16-s − 0.874·17-s − 1.65i·19-s + 0.447i·20-s + 1.27·22-s + 0.751·23-s − 0.200·25-s + 1.41·26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$585$$    =    $$3^{2} \cdot 5 \cdot 13$$ Sign: $1$ Analytic conductor: $$4.67124$$ Root analytic conductor: $$2.16130$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{585} (181, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 585,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.23602$$ $$L(\frac12)$$ $$\approx$$ $$1.23602$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1 + iT$$
13 $$1 + 3.60iT$$
good2 $$1 - 2iT - 2T^{2}$$
7 $$1 + 3.60iT - 7T^{2}$$
11 $$1 + 3iT - 11T^{2}$$
17 $$1 + 3.60T + 17T^{2}$$
19 $$1 + 7.21iT - 19T^{2}$$
23 $$1 - 3.60T + 23T^{2}$$
29 $$1 - 7.21T + 29T^{2}$$
31 $$1 - 7.21iT - 31T^{2}$$
37 $$1 + 3.60iT - 37T^{2}$$
41 $$1 - 11iT - 41T^{2}$$
43 $$1 + 4T + 43T^{2}$$
47 $$1 - 4iT - 47T^{2}$$
53 $$1 + 10.8T + 53T^{2}$$
59 $$1 + 12iT - 59T^{2}$$
61 $$1 - 13T + 61T^{2}$$
67 $$1 - 67T^{2}$$
71 $$1 + 5iT - 71T^{2}$$
73 $$1 - 7.21iT - 73T^{2}$$
79 $$1 - 13T + 79T^{2}$$
83 $$1 - 6iT - 83T^{2}$$
89 $$1 - 3iT - 89T^{2}$$
97 $$1 + 3.60iT - 97T^{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

−10.83011117654957591658900061588, −9.557616754549003930218114164368, −8.551689569591561994460434535049, −8.021412537110836645874120258115, −6.94391595049713855404006931951, −6.48528091341689320842539822025, −5.15547989339809802879574550380, −4.56971035817774309661104182808, −3.06659558239422310515133068402, −0.70107886146530611510706438638, 1.85551754050527559636569970945, 2.50548538284112622547594275756, 3.76840870973028695816046339357, 4.79420707122433485941673250125, 6.13476236443244883773583525109, 7.03708597777763116281173647207, 8.440582805946293332288588797354, 9.264104021263137205625715809473, 9.951921666342017309018157034326, 10.76244194943917901207514640954