Properties

Label 2-570-285.272-c1-0-6
Degree $2$
Conductor $570$
Sign $-0.500 - 0.865i$
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  + (0.0871 − 0.996i)2-s + (0.451 + 1.67i)3-s + (−0.984 − 0.173i)4-s + (1.03 − 1.98i)5-s + (1.70 − 0.304i)6-s + (−4.18 + 1.12i)7-s + (−0.258 + 0.965i)8-s + (−2.59 + 1.51i)9-s + (−1.88 − 1.20i)10-s + (−4.96 + 2.86i)11-s + (−0.154 − 1.72i)12-s + (0.780 − 1.67i)13-s + (0.751 + 4.26i)14-s + (3.78 + 0.835i)15-s + (0.939 + 0.342i)16-s + (−0.558 + 6.38i)17-s + ⋯
L(s)  = 1  + (0.0616 − 0.704i)2-s + (0.260 + 0.965i)3-s + (−0.492 − 0.0868i)4-s + (0.463 − 0.886i)5-s + (0.696 − 0.124i)6-s + (−1.58 + 0.423i)7-s + (−0.0915 + 0.341i)8-s + (−0.863 + 0.503i)9-s + (−0.595 − 0.380i)10-s + (−1.49 + 0.865i)11-s + (−0.0446 − 0.498i)12-s + (0.216 − 0.464i)13-s + (0.200 + 1.13i)14-s + (0.976 + 0.215i)15-s + (0.234 + 0.0855i)16-s + (−0.135 + 1.54i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.500 - 0.865i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.500 - 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.275448 + 0.477184i\)
\(L(\frac12)\) \(\approx\) \(0.275448 + 0.477184i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0871 + 0.996i)T \)
3 \( 1 + (-0.451 - 1.67i)T \)
5 \( 1 + (-1.03 + 1.98i)T \)
19 \( 1 + (-1.95 - 3.89i)T \)
good7 \( 1 + (4.18 - 1.12i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.96 - 2.86i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.780 + 1.67i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (0.558 - 6.38i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (2.19 - 1.53i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (-2.66 - 2.23i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-2.65 + 4.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.23 + 8.23i)T + 37iT^{2} \)
41 \( 1 + (0.162 - 0.446i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-5.50 - 3.85i)T + (14.7 + 40.4i)T^{2} \)
47 \( 1 + (6.16 - 0.539i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (-1.20 + 0.845i)T + (18.1 - 49.8i)T^{2} \)
59 \( 1 + (3.96 - 3.32i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.779 - 4.42i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-0.345 - 3.95i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (-5.70 + 1.00i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (5.76 - 2.68i)T + (46.9 - 55.9i)T^{2} \)
79 \( 1 + (-3.07 + 8.45i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (13.4 - 3.59i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-2.01 + 0.733i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-3.57 - 0.313i)T + (95.5 + 16.8i)T^{2} \)
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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

−10.54714649922707828683928390542, −10.15610023407015578439010809510, −9.565324727593577855883767646376, −8.699273219990137379394841364843, −7.87018206143362134508150439003, −5.97701835152204240962461156208, −5.44772389252863612855231733829, −4.26381206678638853566667087883, −3.27165935965593228604264063780, −2.17583047755294329291754693074, 0.27318057062168699293863593237, 2.74675049101581036044236588283, 3.23378436127595075336558514549, 5.20303162982264787670628318352, 6.25987835349302238935437683148, 6.80512422683544189569657941067, 7.43616855765734240298963781065, 8.533062859773978028410141216132, 9.500103697638070998889318322098, 10.27306361562673676328376633679

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