Properties

Label 2-570-285.269-c1-0-15
Degree $2$
Conductor $570$
Sign $0.892 - 0.451i$
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  + (0.642 − 0.766i)2-s + (−0.417 + 1.68i)3-s + (−0.173 − 0.984i)4-s + (−0.0444 + 2.23i)5-s + (1.01 + 1.40i)6-s + (4.02 − 2.32i)7-s + (−0.866 − 0.500i)8-s + (−2.65 − 1.40i)9-s + (1.68 + 1.47i)10-s + (0.0906 + 0.0523i)11-s + (1.72 + 0.119i)12-s + (1.33 + 0.485i)13-s + (0.807 − 4.57i)14-s + (−3.73 − 1.00i)15-s + (−0.939 + 0.342i)16-s + (3.79 + 3.18i)17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (−0.240 + 0.970i)3-s + (−0.0868 − 0.492i)4-s + (−0.0198 + 0.999i)5-s + (0.416 + 0.571i)6-s + (1.52 − 0.878i)7-s + (−0.306 − 0.176i)8-s + (−0.883 − 0.467i)9-s + (0.532 + 0.465i)10-s + (0.0273 + 0.0157i)11-s + (0.498 + 0.0343i)12-s + (0.369 + 0.134i)13-s + (0.215 − 1.22i)14-s + (−0.965 − 0.260i)15-s + (−0.234 + 0.0855i)16-s + (0.921 + 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.892 - 0.451i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.892 - 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88242 + 0.449169i\)
\(L(\frac12)\) \(\approx\) \(1.88242 + 0.449169i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
3 \( 1 + (0.417 - 1.68i)T \)
5 \( 1 + (0.0444 - 2.23i)T \)
19 \( 1 + (-3.84 - 2.05i)T \)
good7 \( 1 + (-4.02 + 2.32i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0906 - 0.0523i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.33 - 0.485i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-3.79 - 3.18i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.445 - 2.52i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (5.99 - 5.02i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.96 - 1.71i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.83T + 37T^{2} \)
41 \( 1 + (-0.912 + 0.332i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (3.52 + 0.620i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-4.06 + 3.40i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-3.46 + 0.611i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-1.98 - 1.66i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.42 + 13.7i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (5.34 - 4.48i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-2.73 + 15.5i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (2.57 + 7.07i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (1.78 + 4.90i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-2.49 - 4.31i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (16.3 + 5.93i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (7.91 + 6.63i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.83708005819827014741077628124, −10.32344154323682876827715323668, −9.396688758186458487519431432349, −8.119996574765789492032197434931, −7.25492965312267238836746208227, −5.87700882028086022882515895056, −5.10078832695074617487390646902, −3.95533859582102913445453883433, −3.36141709476220777747868118654, −1.59974051229365732742830018372, 1.19903395264807724689887359540, 2.55358312065326295477105604848, 4.39215513436478299897033154389, 5.53230317103299730074867569614, 5.61439616529474783916019327588, 7.25219592391468105322237046223, 7.940513908477996633901976241974, 8.553695302959221259222850802313, 9.442152292754022751220259821966, 11.24099549167143884523385037407

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