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Properties

Label 2-560-140.27-c2-0-14
Degree $2$
Conductor $560$
Sign $0.522 - 0.852i$
Analytic cond. $15.2588$
Root an. cond. $3.90626$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.33 − 2.33i)3-s + (−4.70 + 1.70i)5-s + (6.43 + 2.74i)7-s − 1.92i·9-s + 11.8i·11-s + (−0.694 − 0.694i)13-s + (−7.01 + 14.9i)15-s + (−16.8 + 16.8i)17-s + 10.7i·19-s + (21.4 − 8.62i)21-s + (−6.37 − 6.37i)23-s + (19.2 − 16.0i)25-s + (16.5 + 16.5i)27-s + 1.46i·29-s − 32.9·31-s + ⋯
L(s)  = 1  + (0.779 − 0.779i)3-s + (−0.940 + 0.340i)5-s + (0.919 + 0.392i)7-s − 0.214i·9-s + 1.07i·11-s + (−0.0533 − 0.0533i)13-s + (−0.467 + 0.997i)15-s + (−0.990 + 0.990i)17-s + 0.564i·19-s + (1.02 − 0.410i)21-s + (−0.277 − 0.277i)23-s + (0.768 − 0.640i)25-s + (0.612 + 0.612i)27-s + 0.0506i·29-s − 1.06·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.522 - 0.852i$
Analytic conductor: \(15.2588\)
Root analytic conductor: \(3.90626\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1),\ 0.522 - 0.852i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.802496592\)
\(L(\frac12)\) \(\approx\) \(1.802496592\)
\(L(2)\) 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 \( 1 + (4.70 - 1.70i)T \)
7 \( 1 + (-6.43 - 2.74i)T \)
good3 \( 1 + (-2.33 + 2.33i)T - 9iT^{2} \)
11 \( 1 - 11.8iT - 121T^{2} \)
13 \( 1 + (0.694 + 0.694i)T + 169iT^{2} \)
17 \( 1 + (16.8 - 16.8i)T - 289iT^{2} \)
19 \( 1 - 10.7iT - 361T^{2} \)
23 \( 1 + (6.37 + 6.37i)T + 529iT^{2} \)
29 \( 1 - 1.46iT - 841T^{2} \)
31 \( 1 + 32.9T + 961T^{2} \)
37 \( 1 + (-47.6 - 47.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 33.1iT - 1.68e3T^{2} \)
43 \( 1 + (-43.6 - 43.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (34.1 + 34.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (-56.5 + 56.5i)T - 2.80e3iT^{2} \)
59 \( 1 + 22.3iT - 3.48e3T^{2} \)
61 \( 1 + 68.2iT - 3.72e3T^{2} \)
67 \( 1 + (-67.8 + 67.8i)T - 4.48e3iT^{2} \)
71 \( 1 - 71.4iT - 5.04e3T^{2} \)
73 \( 1 + (-10.2 - 10.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 61.2T + 6.24e3T^{2} \)
83 \( 1 + (80.3 - 80.3i)T - 6.88e3iT^{2} \)
89 \( 1 - 27.7T + 7.92e3T^{2} \)
97 \( 1 + (-4.28 + 4.28i)T - 9.40e3iT^{2} \)
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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

−10.90281641584820760967840044087, −9.757373993849653435546612278424, −8.467252865594524725360943078627, −8.118171011274139983249543491613, −7.33516142415856605175803676161, −6.44840954566302515764450744113, −4.90963245379070242673795891399, −3.98327421575247791008406201494, −2.55759535144240429602346596458, −1.64399080336466348010636501121, 0.64284294709788847243918975187, 2.62775717677442342629861443846, 3.85866392177074797954034566220, 4.40449939547280196195392674192, 5.53722125613033266739094576440, 7.12357286408440778381838748600, 7.86959367235908678871799403231, 8.931532776772956353614199765723, 9.074312998715637458980271354686, 10.56025271940973366275840241377

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