Properties

Label 2-570-57.8-c1-0-21
Degree $2$
Conductor $570$
Sign $-0.934 - 0.357i$
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  + (0.5 − 0.866i)2-s + (−1.71 + 0.249i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + (−0.641 + 1.60i)6-s − 2.43·7-s − 0.999·8-s + (2.87 − 0.854i)9-s + (0.866 − 0.499i)10-s − 2.32i·11-s + (1.07 + 1.35i)12-s + (−0.190 + 0.109i)13-s + (−1.21 + 2.10i)14-s + (−1.60 − 0.641i)15-s + (−0.5 + 0.866i)16-s + (−4.43 − 2.56i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.989 + 0.143i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + (−0.261 + 0.656i)6-s − 0.919·7-s − 0.353·8-s + (0.958 − 0.284i)9-s + (0.273 − 0.158i)10-s − 0.700i·11-s + (0.309 + 0.392i)12-s + (−0.0528 + 0.0304i)13-s + (−0.324 + 0.562i)14-s + (−0.415 − 0.165i)15-s + (−0.125 + 0.216i)16-s + (−1.07 − 0.621i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.934 - 0.357i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.934 - 0.357i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0424669 + 0.229952i\)
\(L(\frac12)\) \(\approx\) \(0.0424669 + 0.229952i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (1.71 - 0.249i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (3.94 - 1.85i)T \)
good7 \( 1 + 2.43T + 7T^{2} \)
11 \( 1 + 2.32iT - 11T^{2} \)
13 \( 1 + (0.190 - 0.109i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.43 + 2.56i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (7.51 - 4.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.98 + 6.90i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.02iT - 31T^{2} \)
37 \( 1 + 1.28iT - 37T^{2} \)
41 \( 1 + (-1.19 + 2.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.705 - 1.22i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.74 + 2.74i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.04 + 3.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.478 - 0.828i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.12 - 7.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.63 - 3.83i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.05 + 8.75i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.78 - 3.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (13.0 + 7.53i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.36iT - 83T^{2} \)
89 \( 1 + (2.03 + 3.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.10 - 3.52i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.28086639817346903453824025031, −9.783606302461577053772948881063, −8.801561338270236984320566340728, −7.27393531539261265795093640412, −6.16757906141168536556598243545, −5.82433592161954865052562011172, −4.47462132329702461859266347119, −3.52695365225462732839757802451, −2.05756760580941830255252280337, −0.12425217173797307356865321647, 2.14664737515904840323821429871, 4.01108611377836972063943315896, 4.79882660864245717306974739917, 6.01127345076411020540120581819, 6.45321769020333062497925948208, 7.31157476232164617224482373146, 8.531030694112494784548951022376, 9.553574248262156935013254015297, 10.33999649205731736253980151624, 11.23214770162338902179180931922

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