Properties

Label 2-560-35.33-c1-0-0
Degree $2$
Conductor $560$
Sign $-0.956 - 0.290i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
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  + (0.0749 + 0.0200i)3-s + (−0.965 − 2.01i)5-s + (−1.45 + 2.20i)7-s + (−2.59 − 1.49i)9-s + (0.390 + 0.677i)11-s + (−3.28 + 3.28i)13-s + (−0.0318 − 0.170i)15-s + (1.40 − 5.24i)17-s + (−2.91 + 5.05i)19-s + (−0.153 + 0.136i)21-s + (−8.39 + 2.24i)23-s + (−3.13 + 3.89i)25-s + (−0.328 − 0.328i)27-s + 0.303i·29-s + (5.04 − 2.91i)31-s + ⋯
L(s)  = 1  + (0.0432 + 0.0115i)3-s + (−0.431 − 0.902i)5-s + (−0.551 + 0.834i)7-s + (−0.864 − 0.498i)9-s + (0.117 + 0.204i)11-s + (−0.911 + 0.911i)13-s + (−0.00821 − 0.0440i)15-s + (0.341 − 1.27i)17-s + (−0.669 + 1.15i)19-s + (−0.0335 + 0.0297i)21-s + (−1.74 + 0.468i)23-s + (−0.627 + 0.778i)25-s + (−0.0632 − 0.0632i)27-s + 0.0564i·29-s + (0.905 − 0.522i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.956 - 0.290i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.956 - 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0109973 + 0.0741281i\)
\(L(\frac12)\) \(\approx\) \(0.0109973 + 0.0741281i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.965 + 2.01i)T \)
7 \( 1 + (1.45 - 2.20i)T \)
good3 \( 1 + (-0.0749 - 0.0200i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.390 - 0.677i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.28 - 3.28i)T - 13iT^{2} \)
17 \( 1 + (-1.40 + 5.24i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.91 - 5.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (8.39 - 2.24i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.303iT - 29T^{2} \)
31 \( 1 + (-5.04 + 2.91i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.90 - 7.12i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.39iT - 41T^{2} \)
43 \( 1 + (7.33 + 7.33i)T + 43iT^{2} \)
47 \( 1 + (2.28 - 0.611i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.61 - 6.01i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.56 + 4.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.91 + 5.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.47 - 0.395i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.83T + 71T^{2} \)
73 \( 1 + (-12.9 - 3.47i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.66 + 5.00i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.94 - 2.94i)T - 83iT^{2} \)
89 \( 1 + (-6.43 + 11.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.0900 + 0.0900i)T + 97iT^{2} \)
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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

−11.63265839192984841982044694973, −9.907847559650074563656759115492, −9.453613111377833874775078849543, −8.562452023474973210023341013294, −7.80073998535950237654994574535, −6.51261039845000639912346165934, −5.64205682957186094020455952976, −4.62605839415994319383121831491, −3.46257996888424790067718274837, −2.10015871988799270827869860323, 0.03918141505437894785945799908, 2.48510639568113524618992086442, 3.43594072806759261964886135558, 4.54465054781439717851424710092, 5.97392885002955924553215787430, 6.68978748561562322327195469390, 7.82552637123431877319758535015, 8.272074390534098918737226619511, 9.764335614272224887390426157065, 10.47225085271626278630614749650

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