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Properties

Label 2-560-7.4-c1-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 2.59i)7-s + (1 + 1.73i)9-s + (−1 + 1.73i)11-s + 4·13-s − 0.999·15-s + (3 + 5.19i)19-s + (2.5 + 0.866i)21-s + (1.5 + 2.59i)23-s + (−0.499 + 0.866i)25-s − 5·27-s − 3·29-s + (−0.999 − 1.73i)33-s + (2 − 1.73i)35-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.223 + 0.387i)5-s + (−0.188 − 0.981i)7-s + (0.333 + 0.577i)9-s + (−0.301 + 0.522i)11-s + 1.10·13-s − 0.258·15-s + (0.688 + 1.19i)19-s + (0.545 + 0.188i)21-s + (0.312 + 0.541i)23-s + (−0.0999 + 0.173i)25-s − 0.962·27-s − 0.557·29-s + (−0.174 − 0.301i)33-s + (0.338 − 0.292i)35-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: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13657 + 0.756028i\)
\(L(\frac12)\) \(\approx\) \(1.13657 + 0.756028i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6 - 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 - 9T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.5 + 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-4 + 6.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (8.5 + 14.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.80723463763147944051272985690, −10.11252841241293553378422087815, −9.542037616438129192655458189346, −8.087079864340594389865829294989, −7.40026909795747506512597950255, −6.35585420377380600562678516861, −5.33030504777728677052733581525, −4.25043224926279200672368264577, −3.33099314948671508783093821856, −1.55836194380666520755259668909, 0.909456746516219948911550141527, 2.48723732736756050385556471902, 3.81197437556956402118733780452, 5.27036316245696335354382558959, 5.99838216688869170657882192375, 6.82420590118817994838208572581, 7.980099964603997934496482500941, 9.023995278603867748844269835958, 9.394540302613449384346049926533, 10.81545769711213423634493337406

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