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Properties

Label 2-560-28.3-c1-0-12
Degree $2$
Conductor $560$
Sign $0.0691 + 0.997i$
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  + (1.32 − 2.30i)3-s + (0.866 − 0.5i)5-s + (1.45 − 2.21i)7-s + (−2.03 − 3.52i)9-s + (4.36 + 2.51i)11-s + 3.01i·13-s − 2.65i·15-s + (−1.94 − 1.12i)17-s + (−1.03 − 1.79i)19-s + (−3.16 − 6.28i)21-s + (0.724 − 0.418i)23-s + (0.499 − 0.866i)25-s − 2.84·27-s − 8.81·29-s + (−5.36 + 9.29i)31-s + ⋯
L(s)  = 1  + (0.767 − 1.32i)3-s + (0.387 − 0.223i)5-s + (0.548 − 0.835i)7-s + (−0.678 − 1.17i)9-s + (1.31 + 0.759i)11-s + 0.835i·13-s − 0.686i·15-s + (−0.471 − 0.272i)17-s + (−0.237 − 0.410i)19-s + (−0.689 − 1.37i)21-s + (0.151 − 0.0872i)23-s + (0.0999 − 0.173i)25-s − 0.546·27-s − 1.63·29-s + (−0.963 + 1.66i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0691 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0691 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.0691 + 0.997i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.0691 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55813 - 1.45388i\)
\(L(\frac12)\) \(\approx\) \(1.55813 - 1.45388i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-1.45 + 2.21i)T \)
good3 \( 1 + (-1.32 + 2.30i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-4.36 - 2.51i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.01iT - 13T^{2} \)
17 \( 1 + (1.94 + 1.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.03 + 1.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.724 + 0.418i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.81T + 29T^{2} \)
31 \( 1 + (5.36 - 9.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.14 + 1.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.10iT - 41T^{2} \)
43 \( 1 + 7.14iT - 43T^{2} \)
47 \( 1 + (-5.30 - 9.17i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.263 + 0.456i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.57 + 4.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.9 - 6.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.17 + 1.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.22iT - 71T^{2} \)
73 \( 1 + (-7.28 - 4.20i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.91 + 3.41i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.66T + 83T^{2} \)
89 \( 1 + (-6.78 + 3.91i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.8iT - 97T^{2} \)
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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

−10.63816121674103963959156376392, −9.185748641826037735625312515397, −8.974678459943837441132732306067, −7.64241334550277545574808526538, −7.06835070585966863996978546673, −6.41700421046222648625321148036, −4.81556416748181160803249878434, −3.72771894225646032622634434287, −2.07292382202667991703623946362, −1.36020295002089131280266502116, 2.06782189987973239498531073278, 3.35439503196192572744404618632, 4.13042811498379256267851970970, 5.39433318469734798870510913497, 6.12679810394966373420796003180, 7.68465496791226667266734017039, 8.730567200301933558247702283080, 9.120238001324018581453533233552, 9.937889730539126638498899144074, 10.93272383084364045832203304386

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