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Properties

Label 2-560-112.19-c1-0-4
Degree $2$
Conductor $560$
Sign $-0.993 + 0.114i$
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  + (−1.13 − 0.838i)2-s + (−0.589 + 2.19i)3-s + (0.593 + 1.90i)4-s + (−0.965 + 0.258i)5-s + (2.51 − 2.01i)6-s + (−1.24 + 2.33i)7-s + (0.924 − 2.67i)8-s + (−1.88 − 1.09i)9-s + (1.31 + 0.515i)10-s + (−1.17 + 4.38i)11-s + (−4.54 + 0.180i)12-s + (1.38 − 1.38i)13-s + (3.37 − 1.60i)14-s − 2.27i·15-s + (−3.29 + 2.26i)16-s + (−5.68 + 3.28i)17-s + ⋯
L(s)  = 1  + (−0.805 − 0.592i)2-s + (−0.340 + 1.26i)3-s + (0.296 + 0.954i)4-s + (−0.431 + 0.115i)5-s + (1.02 − 0.820i)6-s + (−0.472 + 0.881i)7-s + (0.327 − 0.945i)8-s + (−0.629 − 0.363i)9-s + (0.416 + 0.162i)10-s + (−0.354 + 1.32i)11-s + (−1.31 + 0.0521i)12-s + (0.384 − 0.384i)13-s + (0.902 − 0.429i)14-s − 0.587i·15-s + (−0.823 + 0.567i)16-s + (−1.37 + 0.796i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.993 + 0.114i$
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: \((2,\ 560,\ (\ :1/2),\ -0.993 + 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0222066 - 0.384980i\)
\(L(\frac12)\) \(\approx\) \(0.0222066 - 0.384980i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 + 0.838i)T \)
5 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (1.24 - 2.33i)T \)
good3 \( 1 + (0.589 - 2.19i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.17 - 4.38i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.38 + 1.38i)T - 13iT^{2} \)
17 \( 1 + (5.68 - 3.28i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-7.37 + 1.97i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.0300 + 0.0519i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.86 + 2.86i)T + 29iT^{2} \)
31 \( 1 + (1.91 + 3.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.13 + 7.97i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.20T + 41T^{2} \)
43 \( 1 + (4.82 + 4.82i)T + 43iT^{2} \)
47 \( 1 + (3.96 - 6.86i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.87 + 0.502i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.74 - 1.27i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.77 - 6.63i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (6.63 + 1.77i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.38T + 71T^{2} \)
73 \( 1 + (-1.59 - 2.75i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.58 - 5.53i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.01 - 3.01i)T + 83iT^{2} \)
89 \( 1 + (7.43 - 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.0iT - 97T^{2} \)
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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

−11.05243463701155149858177386731, −10.28711038110122419717679733913, −9.502454119210069766153772447320, −9.042922369396072196914274312163, −7.87117406040281795991739770754, −6.90292976355291632771245106702, −5.52960724910644928133152165122, −4.40558815053178914088707620384, −3.49838417947692390189395496999, −2.24059298980690700629560025356, 0.31014468659382725721946831345, 1.42572801776547827806115957323, 3.26513335212591432187473818083, 5.01405996582913685425169379905, 6.16566575531436337727130827107, 6.86597256470992456529885895614, 7.48790109992406490142113737883, 8.312429350860820528342636671167, 9.218713045355741761935976558830, 10.29254472288177621927295636677

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