Properties

Label 2-570-285.269-c1-0-1
Degree $2$
Conductor $570$
Sign $0.110 + 0.993i$
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 + (−0.213 + 1.71i)3-s + (−0.173 − 0.984i)4-s + (−2.00 + 0.981i)5-s + (−1.17 − 1.26i)6-s + (−3.44 + 1.98i)7-s + (0.866 + 0.500i)8-s + (−2.90 − 0.734i)9-s + (0.539 − 2.16i)10-s + (3.19 + 1.84i)11-s + (1.72 − 0.0881i)12-s + (−2.53 − 0.924i)13-s + (0.689 − 3.91i)14-s + (−1.25 − 3.66i)15-s + (−0.939 + 0.342i)16-s + (4.70 + 3.94i)17-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (−0.123 + 0.992i)3-s + (−0.0868 − 0.492i)4-s + (−0.898 + 0.438i)5-s + (−0.481 − 0.517i)6-s + (−1.30 + 0.750i)7-s + (0.306 + 0.176i)8-s + (−0.969 − 0.244i)9-s + (0.170 − 0.686i)10-s + (0.962 + 0.555i)11-s + (0.499 − 0.0254i)12-s + (−0.704 − 0.256i)13-s + (0.184 − 1.04i)14-s + (−0.324 − 0.945i)15-s + (−0.234 + 0.0855i)16-s + (1.14 + 0.956i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.110 + 0.993i$
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: \((2,\ 570,\ (\ :1/2),\ 0.110 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0859980 - 0.0769821i\)
\(L(\frac12)\) \(\approx\) \(0.0859980 - 0.0769821i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.213 - 1.71i)T \)
5 \( 1 + (2.00 - 0.981i)T \)
19 \( 1 + (0.557 + 4.32i)T \)
good7 \( 1 + (3.44 - 1.98i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.19 - 1.84i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.53 + 0.924i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-4.70 - 3.94i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (1.43 + 8.11i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (2.49 - 2.09i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (5.40 - 3.12i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.48T + 37T^{2} \)
41 \( 1 + (-5.54 + 2.01i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (9.57 + 1.68i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (8.28 - 6.95i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (7.81 - 1.37i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-3.14 - 2.63i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.625 + 3.54i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-4.03 + 3.38i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.156 + 0.885i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-0.0635 - 0.174i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (2.72 + 7.49i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-2.29 - 3.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.92 + 2.51i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (6.07 + 5.09i)T + (16.8 + 95.5i)T^{2} \)
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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

−11.17881458709066785463157027182, −10.31744033856101151443983877939, −9.595656735909841648853301060251, −8.937305791095378772079682290359, −7.983386607454875424818013478961, −6.77575257259749900046886233016, −6.16834620445576782875842703542, −4.93089466022953160053092402091, −3.83795348380703656017347177676, −2.82750324302540542412092457850, 0.082596148950308821971488046490, 1.34377657709656654139328180887, 3.19618254627992438200831286375, 3.85669316116341579265083623802, 5.54727427299348683393464849499, 6.73812569085887958436687067515, 7.49547649232380143910232670668, 8.110756332345558342668390226122, 9.411893904709817761652220511757, 9.809155318592166343468065567504

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