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Properties

Label 2-560-140.19-c1-0-3
Degree $2$
Conductor $560$
Sign $0.147 - 0.989i$
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  + (0.686 − 0.396i)3-s + (−2.18 − 0.469i)5-s + (−2 + 1.73i)7-s + (−1.18 + 2.05i)9-s + (1.5 − 0.866i)11-s + 4.37·13-s + (−1.68 + 0.543i)15-s + (3.68 + 6.38i)17-s + (−2.87 + 4.97i)19-s + (−0.686 + 1.98i)21-s + (−2.87 + 4.97i)23-s + (4.55 + 2.05i)25-s + 4.25i·27-s − 2.74·29-s + (−5.05 − 8.76i)31-s + ⋯
L(s)  = 1  + (0.396 − 0.228i)3-s + (−0.977 − 0.210i)5-s + (−0.755 + 0.654i)7-s + (−0.395 + 0.684i)9-s + (0.452 − 0.261i)11-s + 1.21·13-s + (−0.435 + 0.140i)15-s + (0.894 + 1.54i)17-s + (−0.658 + 1.14i)19-s + (−0.149 + 0.432i)21-s + (−0.598 + 1.03i)23-s + (0.911 + 0.410i)25-s + 0.819i·27-s − 0.509·29-s + (−0.908 − 1.57i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.147 - 0.989i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.147 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.798595 + 0.688122i\)
\(L(\frac12)\) \(\approx\) \(0.798595 + 0.688122i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 + 0.469i)T \)
7 \( 1 + (2 - 1.73i)T \)
good3 \( 1 + (-0.686 + 0.396i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.37T + 13T^{2} \)
17 \( 1 + (-3.68 - 6.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.87 - 4.97i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.87 - 4.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.74T + 29T^{2} \)
31 \( 1 + (5.05 + 8.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.5 - 4.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.939iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (10.2 + 5.91i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.61 - 3.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.31 + 4.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.941 + 0.543i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.05 - 7.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (2.31 + 4.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.05 - 4.65i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.51iT - 83T^{2} \)
89 \( 1 + (3.68 + 2.12i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−11.14305940711227884110194331698, −10.05658319743549644200572814664, −8.993985235353077459618958930024, −8.170068144650776211690488194979, −7.79323322668466159306883663149, −6.23701936120246801571141608537, −5.68332952615910741673439160117, −3.94080515921684334826231077560, −3.39158512841271673555800644796, −1.73051593240918835650625456656, 0.59283861136912058279327041443, 2.98470283398387308455912301391, 3.66592400380222981038638555382, 4.61953192056722791616825053958, 6.24821033901441935129601121017, 6.94193572688654139317425615693, 7.894355003947555601354090713640, 8.943630382505353522205966092927, 9.492778762588781713057121789079, 10.68668042826532857131182129235

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