Properties

Label 2-570-285.224-c1-0-8
Degree $2$
Conductor $570$
Sign $0.999 + 0.0215i$
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  + (0.342 − 0.939i)2-s + (−1.52 − 0.821i)3-s + (−0.766 − 0.642i)4-s + (0.0113 + 2.23i)5-s + (−1.29 + 1.15i)6-s + (−1.80 − 1.04i)7-s + (−0.866 + 0.500i)8-s + (1.64 + 2.50i)9-s + (2.10 + 0.754i)10-s + (0.547 − 0.315i)11-s + (0.639 + 1.60i)12-s + (−0.454 + 2.57i)13-s + (−1.59 + 1.33i)14-s + (1.82 − 3.41i)15-s + (0.173 + 0.984i)16-s + (4.93 + 1.79i)17-s + ⋯
L(s)  = 1  + (0.241 − 0.664i)2-s + (−0.880 − 0.474i)3-s + (−0.383 − 0.321i)4-s + (0.00506 + 0.999i)5-s + (−0.528 + 0.470i)6-s + (−0.681 − 0.393i)7-s + (−0.306 + 0.176i)8-s + (0.549 + 0.835i)9-s + (0.665 + 0.238i)10-s + (0.165 − 0.0952i)11-s + (0.184 + 0.464i)12-s + (−0.126 + 0.715i)13-s + (−0.425 + 0.357i)14-s + (0.470 − 0.882i)15-s + (0.0434 + 0.246i)16-s + (1.19 + 0.435i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.999 + 0.0215i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.999 + 0.0215i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04048 - 0.0112228i\)
\(L(\frac12)\) \(\approx\) \(1.04048 - 0.0112228i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 + 0.939i)T \)
3 \( 1 + (1.52 + 0.821i)T \)
5 \( 1 + (-0.0113 - 2.23i)T \)
19 \( 1 + (-4.23 + 1.03i)T \)
good7 \( 1 + (1.80 + 1.04i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.547 + 0.315i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.454 - 2.57i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-4.93 - 1.79i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-4.64 - 3.90i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (3.31 - 1.20i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-1.55 - 0.898i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 + (-1.55 - 8.80i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-2.63 - 3.13i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-5.84 + 2.12i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (1.62 - 1.93i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (8.57 + 3.11i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-5.16 - 4.33i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.73 + 0.995i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (6.82 - 5.72i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-10.6 + 1.87i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-4.54 + 0.802i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (8.24 - 14.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.584 + 3.31i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (13.6 + 4.96i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.98150362453905982526729699489, −10.01018715335810851271298928835, −9.474818230298466029163746362488, −7.76932218710104539267633541763, −7.00685640111756393738920466474, −6.17651519698946604171320210905, −5.24640079960938925299086872651, −3.88211219964715662122920860737, −2.86986957773828528804566785029, −1.27965678442297832856423761041, 0.72528443529167692027995468296, 3.24471604926748207106764178170, 4.41255800669888279380896444966, 5.44025035183759477644086158914, 5.78456102850877149544973251346, 7.02780670561072132929390704751, 7.981693109872772823091156308467, 9.225702222735859020456437397562, 9.596045901623621827013710751102, 10.66897774401238321908244081712

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