Properties

Label 2-570-19.4-c1-0-5
Degree $2$
Conductor $570$
Sign $-0.0540 - 0.998i$
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  + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (0.766 − 0.642i)5-s + (−0.173 + 0.984i)6-s + (−1.43 + 2.49i)7-s + (0.500 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.939 − 0.342i)10-s + (0.266 + 0.460i)11-s + (−0.5 + 0.866i)12-s + (−0.673 + 3.82i)13-s + (−2.20 + 1.85i)14-s + (0.766 + 0.642i)15-s + (0.173 + 0.984i)16-s + (−0.673 − 0.245i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.100 + 0.568i)3-s + (0.383 + 0.321i)4-s + (0.342 − 0.287i)5-s + (−0.0708 + 0.402i)6-s + (−0.544 + 0.942i)7-s + (0.176 + 0.306i)8-s + (−0.313 + 0.114i)9-s + (0.297 − 0.108i)10-s + (0.0802 + 0.138i)11-s + (−0.144 + 0.249i)12-s + (−0.186 + 1.05i)13-s + (−0.589 + 0.494i)14-s + (0.197 + 0.165i)15-s + (0.0434 + 0.246i)16-s + (−0.163 − 0.0594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0540 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0540 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.0540 - 0.998i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.0540 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47605 + 1.55818i\)
\(L(\frac12)\) \(\approx\) \(1.47605 + 1.55818i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-4.21 - 1.10i)T \)
good7 \( 1 + (1.43 - 2.49i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.266 - 0.460i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.673 - 3.82i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.673 + 0.245i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-1.20 - 1.01i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-2.91 + 1.06i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.41 + 4.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.92T + 37T^{2} \)
41 \( 1 + (0.687 + 3.89i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.748 - 0.628i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-7.23 + 2.63i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (2.07 + 1.74i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-7.21 - 2.62i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (11.1 + 9.38i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.27 + 0.829i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-6.16 + 5.17i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.982 + 5.57i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (0.254 + 1.44i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.85 - 3.21i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.68 - 15.2i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (12.4 + 4.53i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.11520527093406458746891394349, −9.891614141408115526352792058622, −9.281571309554199322059613092262, −8.471266902428253369322140786277, −7.19669489355301649428620613796, −6.19899035435690428823345651684, −5.38807292867576968396106507784, −4.47736126460670508978353838089, −3.31344501039702360923875571529, −2.14773225257693278840125029835, 1.03759598966275093788417511417, 2.72808740786159478078645061087, 3.53530980784843203903915413623, 4.92232895274743741530142983152, 5.95186798666922957488862669524, 6.88450591764365129507382916750, 7.50893299843282682956828736898, 8.732547889369331736015577722961, 9.984853803567504199606717981106, 10.48559336212040965836927447130

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