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Properties

Label 2-560-7.4-c1-0-2
Degree $2$
Conductor $560$
Sign $-0.749 - 0.661i$
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  + (−1.20 + 2.09i)3-s + (−0.5 − 0.866i)5-s + (2.62 + 0.358i)7-s + (−1.41 − 2.44i)9-s + (−2.41 + 4.18i)11-s + 2·13-s + 2.41·15-s + (−1.82 + 3.16i)17-s + (2.82 + 4.89i)19-s + (−3.91 + 5.04i)21-s + (−4.20 − 7.28i)23-s + (−0.499 + 0.866i)25-s − 0.414·27-s − 2.17·29-s + (−2.41 + 4.18i)31-s + ⋯
L(s)  = 1  + (−0.696 + 1.20i)3-s + (−0.223 − 0.387i)5-s + (0.990 + 0.135i)7-s + (−0.471 − 0.816i)9-s + (−0.727 + 1.26i)11-s + 0.554·13-s + 0.623·15-s + (−0.443 + 0.768i)17-s + (0.648 + 1.12i)19-s + (−0.854 + 1.10i)21-s + (−0.877 − 1.51i)23-s + (−0.0999 + 0.173i)25-s − 0.0797·27-s − 0.403·29-s + (−0.433 + 0.751i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.749 - 0.661i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.749 - 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.331922 + 0.877863i\)
\(L(\frac12)\) \(\approx\) \(0.331922 + 0.877863i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 - 0.358i)T \)
good3 \( 1 + (1.20 - 2.09i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.41 - 4.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (1.82 - 3.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.82 - 4.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.20 + 7.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.17T + 29T^{2} \)
31 \( 1 + (2.41 - 4.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.82 - 4.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.171T + 41T^{2} \)
43 \( 1 + 12.8T + 43T^{2} \)
47 \( 1 + (-0.171 - 0.297i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.82 - 4.89i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.32 - 4.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.44 + 5.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-3.82 + 6.63i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (-8.32 - 14.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−10.84493345289755166030405283701, −10.39669838012188770387227581917, −9.578261094447041826102826630391, −8.429481148880763210938450712041, −7.77867468965175396418522739312, −6.31983640096746890680335091522, −5.21591744697219998737670193448, −4.66578558696794469330022086297, −3.80036155467176529376893035210, −1.86449883122570750352496448683, 0.58943952282039319654075048682, 2.03853526150052686716080141763, 3.51280139105421303898109979248, 5.12877884704281348068737638064, 5.84083781942698524420119909191, 6.89858435197585086374222599199, 7.65913339828836832565532654024, 8.295018786914272658419260795863, 9.537160280393539664018497677968, 10.98563834906788495621406431740

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