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Properties

Label 2-560-112.19-c1-0-9
Degree $2$
Conductor $560$
Sign $0.157 - 0.987i$
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  + (−1.39 − 0.208i)2-s + (0.308 − 1.14i)3-s + (1.91 + 0.584i)4-s + (0.965 − 0.258i)5-s + (−0.671 + 1.54i)6-s + (−2.40 + 1.10i)7-s + (−2.55 − 1.21i)8-s + (1.37 + 0.791i)9-s + (−1.40 + 0.160i)10-s + (−1.19 + 4.45i)11-s + (1.26 − 2.01i)12-s + (−3.04 + 3.04i)13-s + (3.59 − 1.04i)14-s − 1.19i·15-s + (3.31 + 2.23i)16-s + (−1.57 + 0.909i)17-s + ⋯
L(s)  = 1  + (−0.989 − 0.147i)2-s + (0.177 − 0.663i)3-s + (0.956 + 0.292i)4-s + (0.431 − 0.115i)5-s + (−0.273 + 0.630i)6-s + (−0.908 + 0.417i)7-s + (−0.902 − 0.430i)8-s + (0.457 + 0.263i)9-s + (−0.444 + 0.0506i)10-s + (−0.359 + 1.34i)11-s + (0.364 − 0.582i)12-s + (−0.844 + 0.844i)13-s + (0.960 − 0.279i)14-s − 0.307i·15-s + (0.829 + 0.559i)16-s + (−0.382 + 0.220i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.157 - 0.987i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.157 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.478821 + 0.408635i\)
\(L(\frac12)\) \(\approx\) \(0.478821 + 0.408635i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.208i)T \)
5 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (2.40 - 1.10i)T \)
good3 \( 1 + (-0.308 + 1.14i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.19 - 4.45i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.04 - 3.04i)T - 13iT^{2} \)
17 \( 1 + (1.57 - 0.909i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.17 - 1.11i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.32 - 5.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.52 - 3.52i)T + 29iT^{2} \)
31 \( 1 + (3.34 + 5.78i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.41 + 9.01i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.385T + 41T^{2} \)
43 \( 1 + (-2.99 - 2.99i)T + 43iT^{2} \)
47 \( 1 + (4.73 - 8.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.29 - 1.15i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.66 - 0.446i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.95 - 7.28i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-15.6 - 4.18i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.16T + 71T^{2} \)
73 \( 1 + (-2.50 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.961 - 0.555i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.433 + 0.433i)T + 83iT^{2} \)
89 \( 1 + (5.26 - 9.11i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.7iT - 97T^{2} \)
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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

−10.70680981812175998527559997775, −9.749905562919677689206745564812, −9.491323763776020047464488699859, −8.333218237134895224446979664059, −7.27839115424248272712407668583, −6.87317019224261630068331177709, −5.77908000165223822878629647038, −4.20254095122818952874104053681, −2.41820220544152614413007034262, −1.85602686087216906841388288069, 0.44904588467130368764151811392, 2.54620045898171333591480897324, 3.52612862989670730882815537849, 5.07799658137644368322298498359, 6.32309414141384119640576549220, 6.86688251007319943767361326011, 8.184300096406298832849917522138, 8.860160348183578432578877429389, 9.883323017448961258767808587678, 10.29082814200321141622270345344

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