# Properties

 Label 2-585-117.103-c1-0-19 Degree $2$ Conductor $585$ Sign $0.740 - 0.671i$ 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 + (−1.71 + 0.988i)2-s + (−1.13 + 1.31i)3-s + (0.954 − 1.65i)4-s + (−0.866 − 0.5i)5-s + (0.642 − 3.36i)6-s + (−3.18 + 1.84i)7-s − 0.179i·8-s + (−0.436 − 2.96i)9-s + 1.97·10-s + (1.59 − 0.921i)11-s + (1.08 + 3.12i)12-s + (−3.58 − 0.388i)13-s + (3.64 − 6.30i)14-s + (1.63 − 0.569i)15-s + (2.08 + 3.61i)16-s + 0.179·17-s + ⋯
 L(s)  = 1 + (−1.21 + 0.699i)2-s + (−0.653 + 0.756i)3-s + (0.477 − 0.826i)4-s + (−0.387 − 0.223i)5-s + (0.262 − 1.37i)6-s + (−1.20 + 0.695i)7-s − 0.0635i·8-s + (−0.145 − 0.989i)9-s + 0.625·10-s + (0.481 − 0.277i)11-s + (0.313 + 0.901i)12-s + (−0.994 − 0.107i)13-s + (0.972 − 1.68i)14-s + (0.422 − 0.146i)15-s + (0.521 + 0.903i)16-s + 0.0436·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$585$$    =    $$3^{2} \cdot 5 \cdot 13$$ Sign: $0.740 - 0.671i$ Analytic conductor: $$4.67124$$ Root analytic conductor: $$2.16130$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{585} (571, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 585,\ (\ :1/2),\ 0.740 - 0.671i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.316190 + 0.121972i$$ $$L(\frac12)$$ $$\approx$$ $$0.316190 + 0.121972i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (1.13 - 1.31i)T$$
5 $$1 + (0.866 + 0.5i)T$$
13 $$1 + (3.58 + 0.388i)T$$
good2 $$1 + (1.71 - 0.988i)T + (1 - 1.73i)T^{2}$$
7 $$1 + (3.18 - 1.84i)T + (3.5 - 6.06i)T^{2}$$
11 $$1 + (-1.59 + 0.921i)T + (5.5 - 9.52i)T^{2}$$
17 $$1 - 0.179T + 17T^{2}$$
19 $$1 + 0.475iT - 19T^{2}$$
23 $$1 + (-3.15 + 5.46i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (0.00188 + 0.00326i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (1.31 + 0.758i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 - 5.39iT - 37T^{2}$$
41 $$1 + (-1.53 - 0.885i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (-1.10 - 1.90i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (-7.86 + 4.54i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + 9.08T + 53T^{2}$$
59 $$1 + (-3.80 - 2.19i)T + (29.5 + 51.0i)T^{2}$$
61 $$1 + (-3.04 - 5.26i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-10.1 - 5.84i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 - 12.0iT - 71T^{2}$$
73 $$1 - 4.31iT - 73T^{2}$$
79 $$1 + (5.42 + 9.40i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (-15.1 + 8.75i)T + (41.5 - 71.8i)T^{2}$$
89 $$1 + 2.06iT - 89T^{2}$$
97 $$1 + (-9.54 + 5.50i)T + (48.5 - 84.0i)T^{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

−10.39295346269882226121066792284, −9.744937214713654914534144801538, −9.100932603974091615277794382709, −8.457278546649196059545141593210, −7.13556261298571204762397923230, −6.46975538378085513654539890692, −5.57952253943143179827681674687, −4.30544512372909210684478605979, −3.04732296305035054747933837424, −0.49414581134291501142146685065, 0.76067142099262618999539032514, 2.22348014350513760672150747047, 3.52882518285558458926860821186, 5.10796358458104325722313619453, 6.42460869023461930667134308749, 7.29017435328097942736432825962, 7.76980818742203950784460218823, 9.167337746340582452760572475696, 9.745316518999899053833173401603, 10.64261885342177826444294016115