Properties

Label 2-570-285.62-c1-0-14
Degree $2$
Conductor $570$
Sign $-0.658 - 0.752i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.422 + 0.906i)2-s + (−0.573 + 1.63i)3-s + (−0.642 − 0.766i)4-s + (1.58 + 1.57i)5-s + (−1.23 − 1.21i)6-s + (0.599 − 2.23i)7-s + (0.965 − 0.258i)8-s + (−2.34 − 1.87i)9-s + (−2.09 + 0.771i)10-s + (4.47 + 2.58i)11-s + (1.62 − 0.610i)12-s + (4.74 + 3.32i)13-s + (1.77 + 1.48i)14-s + (−3.48 + 1.68i)15-s + (−0.173 + 0.984i)16-s + (−1.73 + 3.71i)17-s + ⋯
L(s)  = 1  + (−0.298 + 0.640i)2-s + (−0.331 + 0.943i)3-s + (−0.321 − 0.383i)4-s + (0.709 + 0.704i)5-s + (−0.505 − 0.494i)6-s + (0.226 − 0.845i)7-s + (0.341 − 0.0915i)8-s + (−0.780 − 0.625i)9-s + (−0.663 + 0.243i)10-s + (1.35 + 0.779i)11-s + (0.467 − 0.176i)12-s + (1.31 + 0.922i)13-s + (0.473 + 0.397i)14-s + (−0.900 + 0.435i)15-s + (−0.0434 + 0.246i)16-s + (−0.419 + 0.900i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.658 - 0.752i$
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: \((2,\ 570,\ (\ :1/2),\ -0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.533144 + 1.17505i\)
\(L(\frac12)\) \(\approx\) \(0.533144 + 1.17505i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.573 - 1.63i)T \)
5 \( 1 + (-1.58 - 1.57i)T \)
19 \( 1 + (4.35 + 0.273i)T \)
good7 \( 1 + (-0.599 + 2.23i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-4.47 - 2.58i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.74 - 3.32i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (1.73 - 3.71i)T + (-10.9 - 13.0i)T^{2} \)
23 \( 1 + (-2.89 - 0.253i)T + (22.6 + 3.99i)T^{2} \)
29 \( 1 + (3.37 + 1.22i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (2.06 + 3.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.36 + 4.36i)T + 37iT^{2} \)
41 \( 1 + (5.60 + 0.988i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-10.8 + 0.946i)T + (42.3 - 7.46i)T^{2} \)
47 \( 1 + (-0.499 + 0.232i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (7.02 + 0.614i)T + (52.1 + 9.20i)T^{2} \)
59 \( 1 + (3.30 - 1.20i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (4.90 - 4.11i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-5.13 - 11.0i)T + (-43.0 + 51.3i)T^{2} \)
71 \( 1 + (4.62 - 5.51i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 + (-5.55 - 7.92i)T + (-24.9 + 68.5i)T^{2} \)
79 \( 1 + (-0.848 - 0.149i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-4.46 + 16.6i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.20 + 6.84i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (7.44 + 3.47i)T + (62.3 + 74.3i)T^{2} \)
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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

−10.86936813281518909512766102124, −10.19305221057671066559475491467, −9.209188987375599674591507188179, −8.778271469155276718639169366835, −7.20488202165316157893822116083, −6.47476285714401683920268834327, −5.83758868325701332191544668420, −4.30768176051356990251010759683, −3.85551234968007649772535552085, −1.68340034590375802935801466990, 0.949374817855599356250359323120, 1.95980795869149597016881755213, 3.30673027965139193907139409872, 4.95090487301828490519355372598, 5.90715401955880814794279673190, 6.59903080624507786296076232890, 8.103178253359138895847499565998, 8.815106653152361809215223803822, 9.203503963861638944621768926776, 10.76131830748244858503473241525

Graph of the $Z$-function along the critical line