Properties

Label 2-570-285.59-c1-0-0
Degree $2$
Conductor $570$
Sign $-0.972 - 0.231i$
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  + (−0.984 − 0.173i)2-s + (−1.64 − 0.541i)3-s + (0.939 + 0.342i)4-s + (2.21 + 0.302i)5-s + (1.52 + 0.819i)6-s + (−2.76 + 1.59i)7-s + (−0.866 − 0.5i)8-s + (2.41 + 1.78i)9-s + (−2.12 − 0.682i)10-s + (−1.63 − 0.942i)11-s + (−1.36 − 1.07i)12-s + (−1.97 + 1.65i)13-s + (3.00 − 1.09i)14-s + (−3.48 − 1.69i)15-s + (0.766 + 0.642i)16-s + (0.781 − 4.43i)17-s + ⋯
L(s)  = 1  + (−0.696 − 0.122i)2-s + (−0.949 − 0.312i)3-s + (0.469 + 0.171i)4-s + (0.990 + 0.135i)5-s + (0.623 + 0.334i)6-s + (−1.04 + 0.603i)7-s + (−0.306 − 0.176i)8-s + (0.804 + 0.594i)9-s + (−0.673 − 0.215i)10-s + (−0.492 − 0.284i)11-s + (−0.392 − 0.309i)12-s + (−0.546 + 0.458i)13-s + (0.801 − 0.291i)14-s + (−0.898 − 0.438i)15-s + (0.191 + 0.160i)16-s + (0.189 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.972 - 0.231i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.972 - 0.231i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00412967 + 0.0351312i\)
\(L(\frac12)\) \(\approx\) \(0.00412967 + 0.0351312i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 + 0.173i)T \)
3 \( 1 + (1.64 + 0.541i)T \)
5 \( 1 + (-2.21 - 0.302i)T \)
19 \( 1 + (3.91 + 1.91i)T \)
good7 \( 1 + (2.76 - 1.59i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.63 + 0.942i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.97 - 1.65i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.781 + 4.43i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-0.269 - 0.0979i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.186 - 1.05i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (6.53 - 3.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.87T + 37T^{2} \)
41 \( 1 + (6.45 + 5.41i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.955 + 2.62i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.01 - 5.78i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (3.28 - 9.01i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.980 + 5.55i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (11.2 + 4.09i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.624 - 3.54i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (9.79 - 3.56i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (3.19 - 3.81i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (0.568 - 0.678i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.89 - 6.74i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-12.5 + 10.5i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (0.803 - 4.55i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.88160240932039681603223755334, −10.32787053479015271689406588274, −9.444774357293750044136840541767, −8.831556523921480456398247819268, −7.28077853799409165636514971306, −6.70053006634275693461347740888, −5.82039467710917030838622298930, −4.98699885849456338179066958418, −3.00031998677232854533573153009, −1.88654790083867303248093482774, 0.02632907952067336040804385710, 1.80599763886958433365447735488, 3.51167688223929831041038843826, 4.95334165272851290181612376144, 5.95861788503081220775080932392, 6.55253750001386643968282922656, 7.50799345598089029239836312103, 8.799546278303148353171063199505, 9.788888715572373130359841472747, 10.30503219028520630298636297809

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