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Properties

Label 2-560-7.4-c1-0-11
Degree $2$
Conductor $560$
Sign $-0.386 + 0.922i$
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 + 2.59i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 2.59i)7-s + (−3 − 5.19i)9-s + (−1 + 1.73i)11-s − 6·13-s − 3·15-s + (−1 + 1.73i)17-s + (7.5 + 2.59i)21-s + (−4.5 − 7.79i)23-s + (−0.499 + 0.866i)25-s + 9·27-s + 3·29-s + (1 − 1.73i)31-s + (−3 − 5.19i)33-s + ⋯
L(s)  = 1  + (−0.866 + 1.49i)3-s + (0.223 + 0.387i)5-s + (−0.188 − 0.981i)7-s + (−1 − 1.73i)9-s + (−0.301 + 0.522i)11-s − 1.66·13-s − 0.774·15-s + (−0.242 + 0.420i)17-s + (1.63 + 0.566i)21-s + (−0.938 − 1.62i)23-s + (−0.0999 + 0.173i)25-s + 1.73·27-s + 0.557·29-s + (0.179 − 0.311i)31-s + (−0.522 − 0.904i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.386 + 0.922i$
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: \(1\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.386 + 0.922i)\)

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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 2.59i)T \)
good3 \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - T + 83T^{2} \)
89 \( 1 + (6.5 + 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−10.32405943704433599832285826342, −10.03491122470901708091516793093, −9.135668869186383838277426400648, −7.69300552879846633339242529603, −6.71468216381318829784125468996, −5.73764375565985845850705092204, −4.58956822920757238304778436227, −4.17734235668272847511171272854, −2.67102895762323323698297291756, 0, 1.73440781798954448186915495075, 2.78382669842538044614324828656, 4.98578943868135646803071325141, 5.59833468637058787611191790817, 6.45656769547870055658996455581, 7.40348490250964578414996655636, 8.141731613205317063466886155332, 9.229950720918807290995336527717, 10.19928638796643000909459275249

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