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Properties

Label 2-570-285.284-c1-0-9
Degree $2$
Conductor $570$
Sign $-0.437 - 0.899i$
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  + i·2-s − 1.73·3-s − 4-s + (1.41 + 1.73i)5-s − 1.73i·6-s − 2.44i·7-s − i·8-s + 2.99·9-s + (−1.73 + 1.41i)10-s + 3.46i·11-s + 1.73·12-s + 2.44·14-s + (−2.44 − 2.99i)15-s + 16-s + 7.07·17-s + 2.99i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.00·3-s − 0.5·4-s + (0.632 + 0.774i)5-s − 0.707i·6-s − 0.925i·7-s − 0.353i·8-s + 0.999·9-s + (−0.547 + 0.447i)10-s + 1.04i·11-s + 0.500·12-s + 0.654·14-s + (−0.632 − 0.774i)15-s + 0.250·16-s + 1.71·17-s + 0.707i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.437 - 0.899i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.437 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.551881 + 0.882660i\)
\(L(\frac12)\) \(\approx\) \(0.551881 + 0.882660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + 1.73T \)
5 \( 1 + (-1.41 - 1.73i)T \)
19 \( 1 + (1 - 4.24i)T \)
good7 \( 1 + 2.44iT - 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 7.07T + 17T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 - 4.89T + 29T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + 6.92T + 37T^{2} \)
41 \( 1 + 2.44T + 41T^{2} \)
43 \( 1 - 9.79iT - 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 - 7.34T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 13.8T + 67T^{2} \)
71 \( 1 - 4.89T + 71T^{2} \)
73 \( 1 + 9.79iT - 73T^{2} \)
79 \( 1 + 4.24iT - 79T^{2} \)
83 \( 1 - 1.41T + 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 - 3.46T + 97T^{2} \)
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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

−10.66350297498140249129071606554, −10.14849001498968541830933042682, −9.687396633338696557269077839344, −7.942159345499573000859569132989, −7.27150567047686326077226063968, −6.47857595125367046978245423293, −5.71335034550734694579490509986, −4.69745480235158199326640262675, −3.57635907990383575505143292215, −1.49828849099031985661421758002, 0.74000035077477428108076209636, 2.15579943594176016556091806640, 3.70485922896856506949062630135, 5.13423532581442251527061602772, 5.54039623623618548772322100810, 6.46928967225605498466442355877, 8.073365578028049073591848044961, 8.862225910745974298528228639802, 9.825904069008390315292530319539, 10.40328735877923161625275781212

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