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Properties

Label 2-560-7.2-c1-0-10
Degree $2$
Conductor $560$
Sign $-0.266 + 0.963i$
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 − 1.73i)3-s + (0.5 − 0.866i)5-s + (2 − 1.73i)7-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)11-s + 5·13-s − 1.99·15-s + (−3 − 5.19i)17-s + (−0.5 + 0.866i)19-s + (−5 − 1.73i)21-s + (1.5 − 2.59i)23-s + (−0.499 − 0.866i)25-s − 4.00·27-s − 6·29-s + (−2 − 3.46i)31-s + ⋯
L(s)  = 1  + (−0.577 − 0.999i)3-s + (0.223 − 0.387i)5-s + (0.755 − 0.654i)7-s + (−0.166 + 0.288i)9-s + (0.452 + 0.783i)11-s + 1.38·13-s − 0.516·15-s + (−0.727 − 1.26i)17-s + (−0.114 + 0.198i)19-s + (−1.09 − 0.377i)21-s + (0.312 − 0.541i)23-s + (−0.0999 − 0.173i)25-s − 0.769·27-s − 1.11·29-s + (−0.359 − 0.622i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.825728 - 1.08540i\)
\(L(\frac12)\) \(\approx\) \(0.825728 - 1.08540i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2 + 1.73i)T \)
good3 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−10.79227449960127041815417537956, −9.560861411194934036492115524667, −8.686707550985788046102918238370, −7.60586686002355157655090275629, −6.91868861120015509505675193345, −6.07507628220102194036638442666, −4.94008285553480310087553101806, −3.92642610587442570958112972783, −1.95672506842265544406176821053, −0.927073704962082531506322323716, 1.78896972707807543948051188093, 3.53197388284067456217804941364, 4.36823258912413996399271939857, 5.72139668689148666345464718967, 5.96606522622766865774084291812, 7.46262926777427564232229994049, 8.761698362322245369220241515060, 9.082209274511416262401810216190, 10.49370229258119774458345302366, 11.02591315657370832896903779873

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