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Properties

Label 2-560-35.4-c1-0-21
Degree $2$
Conductor $560$
Sign $-0.570 - 0.821i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)3-s + (−0.5 − 2.17i)5-s + (−2.63 − 0.209i)7-s + (1.13 − 1.97i)11-s + 6.09i·13-s + (−1.13 + 3.70i)15-s + (−4.13 − 2.38i)17-s + (2.13 + 3.70i)19-s + (3.77 + 2.59i)21-s + (0.774 − 0.447i)23-s + (−4.50 + 2.17i)25-s + 5.19i·27-s + 3.27·29-s + (−2.13 + 3.70i)31-s + (−3.41 + 1.97i)33-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (−0.223 − 0.974i)5-s + (−0.996 − 0.0791i)7-s + (0.342 − 0.594i)11-s + 1.68i·13-s + (−0.293 + 0.955i)15-s + (−1.00 − 0.579i)17-s + (0.490 + 0.849i)19-s + (0.823 + 0.566i)21-s + (0.161 − 0.0932i)23-s + (−0.900 + 0.435i)25-s + 0.999i·27-s + 0.608·29-s + (−0.383 + 0.664i)31-s + (−0.594 + 0.342i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.570 - 0.821i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.570 - 0.821i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 2.17i)T \)
7 \( 1 + (2.63 + 0.209i)T \)
good3 \( 1 + (1.5 + 0.866i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.13 + 1.97i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.09iT - 13T^{2} \)
17 \( 1 + (4.13 + 2.38i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.13 - 3.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.774 + 0.447i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.27T + 29T^{2} \)
31 \( 1 + (2.13 - 3.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.86 - 2.80i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 6.50iT - 43T^{2} \)
47 \( 1 + (1.86 - 1.07i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.41 - 3.70i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.13 - 3.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.774 + 1.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.0 + 6.95i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + (-1.86 - 1.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.137 - 0.238i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.67iT - 83T^{2} \)
89 \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−10.19785621913591580640769052034, −9.070578909963455377608552617442, −8.753488204982995987849224033655, −7.13962263999134422144245725827, −6.56917539220598213069087204922, −5.67405762067632457681254579242, −4.57810750253178214018098575230, −3.45150190099898166968462534770, −1.50197930220396894248056194266, 0, 2.62295819199053262429532311105, 3.64648417391299356844107661896, 4.90671734724070603619734710480, 5.92723819618210706846980712980, 6.67401369557679282993554121574, 7.58204941632644513794665112784, 8.800848552572405956095492641662, 10.06390288497128599537674657216, 10.34211893284068332077600806682

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