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Properties

Label 2-570-57.8-c1-0-3
Degree $2$
Conductor $570$
Sign $0.130 - 0.991i$
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 + (−0.641 − 1.60i)3-s + (−0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + (1.71 + 0.249i)6-s − 2.43·7-s + 0.999·8-s + (−2.17 + 2.06i)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 + (−0.249 + 1.71i)15-s + (−0.5 + 0.866i)16-s + (4.43 + 2.56i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.370 − 0.928i)3-s + (−0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + (0.699 + 0.101i)6-s − 0.919·7-s + 0.353·8-s + (−0.726 + 0.687i)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 + (−0.0643 + 0.442i)15-s + (−0.125 + 0.216i)16-s + (1.07 + 0.621i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.130 - 0.991i$
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: \((2,\ 570,\ (\ :1/2),\ 0.130 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446766 + 0.391939i\)
\(L(\frac12)\) \(\approx\) \(0.446766 + 0.391939i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.641 + 1.60i)T \)
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

−10.77443671487598471218412147932, −10.10839580142776072145007383161, −8.889250383432334361325382184707, −8.232848399844274303656320292742, −7.15690834054160488107709837897, −6.67707542915433517273095249621, −5.68695484594292092241408862191, −4.61868035349356100619673894077, −3.01747248677259195398038829821, −1.27698254268935400251640784032, 0.44642407337435493765059575338, 2.91755923589596152493776133113, 3.54427486224073310505014466786, 4.72456680175477584824742822852, 5.86387915002662477796522378283, 6.90620689643540111705230084356, 8.138418135762439399326205075274, 9.097509848193832229249704027518, 9.772574586997851722613477135993, 10.48602759683842289897734820456

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