# Properties

 Label 2-570-285.62-c1-0-13 Degree $2$ Conductor $570$ Sign $0.830 - 0.556i$ Analytic cond. $4.55147$ Root an. cond. $2.13341$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.422 + 0.906i)2-s + (−1.61 + 0.629i)3-s + (−0.642 − 0.766i)4-s + (−1.65 + 1.49i)5-s + (0.111 − 1.72i)6-s + (0.539 − 2.01i)7-s + (0.965 − 0.258i)8-s + (2.20 − 2.03i)9-s + (−0.657 − 2.13i)10-s + (−3.78 − 2.18i)11-s + (1.51 + 0.831i)12-s + (3.22 + 2.25i)13-s + (1.59 + 1.33i)14-s + (1.73 − 3.46i)15-s + (−0.173 + 0.984i)16-s + (2.50 − 5.38i)17-s + ⋯
 L(s)  = 1 + (−0.298 + 0.640i)2-s + (−0.931 + 0.363i)3-s + (−0.321 − 0.383i)4-s + (−0.741 + 0.670i)5-s + (0.0454 − 0.705i)6-s + (0.203 − 0.760i)7-s + (0.341 − 0.0915i)8-s + (0.735 − 0.677i)9-s + (−0.207 − 0.675i)10-s + (−1.14 − 0.659i)11-s + (0.438 + 0.239i)12-s + (0.894 + 0.626i)13-s + (0.426 + 0.358i)14-s + (0.447 − 0.894i)15-s + (−0.0434 + 0.246i)16-s + (0.608 − 1.30i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$570$$    =    $$2 \cdot 3 \cdot 5 \cdot 19$$ Sign: $0.830 - 0.556i$ Analytic conductor: $$4.55147$$ Root analytic conductor: $$2.13341$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{570} (347, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 570,\ (\ :1/2),\ 0.830 - 0.556i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.685420 + 0.208473i$$ $$L(\frac12)$$ $$\approx$$ $$0.685420 + 0.208473i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.422 - 0.906i)T$$
3 $$1 + (1.61 - 0.629i)T$$
5 $$1 + (1.65 - 1.49i)T$$
19 $$1 + (3.26 - 2.88i)T$$
good7 $$1 + (-0.539 + 2.01i)T + (-6.06 - 3.5i)T^{2}$$
11 $$1 + (3.78 + 2.18i)T + (5.5 + 9.52i)T^{2}$$
13 $$1 + (-3.22 - 2.25i)T + (4.44 + 12.2i)T^{2}$$
17 $$1 + (-2.50 + 5.38i)T + (-10.9 - 13.0i)T^{2}$$
23 $$1 + (-5.77 - 0.504i)T + (22.6 + 3.99i)T^{2}$$
29 $$1 + (-3.16 - 1.15i)T + (22.2 + 18.6i)T^{2}$$
31 $$1 + (-2.67 - 4.63i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-1.42 - 1.42i)T + 37iT^{2}$$
41 $$1 + (2.40 + 0.424i)T + (38.5 + 14.0i)T^{2}$$
43 $$1 + (-2.38 + 0.208i)T + (42.3 - 7.46i)T^{2}$$
47 $$1 + (-2.85 + 1.33i)T + (30.2 - 36.0i)T^{2}$$
53 $$1 + (-12.3 - 1.08i)T + (52.1 + 9.20i)T^{2}$$
59 $$1 + (4.57 - 1.66i)T + (45.1 - 37.9i)T^{2}$$
61 $$1 + (-6.07 + 5.09i)T + (10.5 - 60.0i)T^{2}$$
67 $$1 + (6.48 + 13.9i)T + (-43.0 + 51.3i)T^{2}$$
71 $$1 + (-7.74 + 9.22i)T + (-12.3 - 69.9i)T^{2}$$
73 $$1 + (-2.89 - 4.12i)T + (-24.9 + 68.5i)T^{2}$$
79 $$1 + (2.39 + 0.421i)T + (74.2 + 27.0i)T^{2}$$
83 $$1 + (-0.621 + 2.31i)T + (-71.8 - 41.5i)T^{2}$$
89 $$1 + (1.17 + 6.64i)T + (-83.6 + 30.4i)T^{2}$$
97 $$1 + (-10.3 - 4.80i)T + (62.3 + 74.3i)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

−10.68451097894842717706819390599, −10.29118987383595408824468310965, −8.992872523870493822233503924044, −7.964651778856381030160579808240, −7.13969649132568622770740353663, −6.43781786321621533168837327592, −5.32953737066130880948298723173, −4.40241810346137927350214674423, −3.27961058055456471263576441282, −0.73642084100324430828372888357, 0.941022071518293648774373375522, 2.44300359820318226527007371247, 4.06950222095477120924685083552, 5.05915734064662371298233483751, 5.86321313320180184722013954095, 7.25447220827740433317292954190, 8.180053207938286078970629566401, 8.710954810178844158440224133500, 10.10309407524385419047850767536, 10.77452541210348731247599610454