Properties

Label 2-560-35.34-c4-0-79
Degree $2$
Conductor $560$
Sign $-0.542 + 0.839i$
Analytic cond. $57.8871$
Root an. cond. $7.60836$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·3-s + (−5 − 24.4i)5-s + (35 + 34.2i)7-s − 56·9-s − 89·11-s + 5·13-s + (−25 − 122. i)15-s + 485·17-s + 220. i·19-s + (175 + 171. i)21-s − 700. i·23-s + (−575 + 244. i)25-s − 685·27-s + 191·29-s − 1.05e3i·31-s + ⋯
L(s)  = 1  + 0.555·3-s + (−0.200 − 0.979i)5-s + (0.714 + 0.699i)7-s − 0.691·9-s − 0.735·11-s + 0.0295·13-s + (−0.111 − 0.544i)15-s + 1.67·17-s + 0.610i·19-s + (0.396 + 0.388i)21-s − 1.32i·23-s + (−0.920 + 0.391i)25-s − 0.939·27-s + 0.227·29-s − 1.09i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.542 + 0.839i$
Analytic conductor: \(57.8871\)
Root analytic conductor: \(7.60836\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :2),\ -0.542 + 0.839i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.518885521\)
\(L(\frac12)\) \(\approx\) \(1.518885521\)
\(L(3)\) 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 \( 1 + (5 + 24.4i)T \)
7 \( 1 + (-35 - 34.2i)T \)
good3 \( 1 - 5T + 81T^{2} \)
11 \( 1 + 89T + 1.46e4T^{2} \)
13 \( 1 - 5T + 2.85e4T^{2} \)
17 \( 1 - 485T + 8.35e4T^{2} \)
19 \( 1 - 220. iT - 1.30e5T^{2} \)
23 \( 1 + 700. iT - 2.79e5T^{2} \)
29 \( 1 - 191T + 7.07e5T^{2} \)
31 \( 1 + 1.05e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.63e3iT - 1.87e6T^{2} \)
41 \( 1 + 2.91e3iT - 2.82e6T^{2} \)
43 \( 1 - 377. iT - 3.41e6T^{2} \)
47 \( 1 + 2.19e3T + 4.87e6T^{2} \)
53 \( 1 - 1.58e3iT - 7.89e6T^{2} \)
59 \( 1 + 3.62e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.93e3iT - 1.38e7T^{2} \)
67 \( 1 - 2.04e3iT - 2.01e7T^{2} \)
71 \( 1 + 4.45e3T + 2.54e7T^{2} \)
73 \( 1 + 8.65e3T + 2.83e7T^{2} \)
79 \( 1 + 5.56e3T + 3.89e7T^{2} \)
83 \( 1 - 1.99e3T + 4.74e7T^{2} \)
89 \( 1 - 808. iT - 6.27e7T^{2} \)
97 \( 1 + 9.23e3T + 8.85e7T^{2} \)
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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

−9.681463842586660857416526784702, −8.787761909895477426881795191793, −8.167390419072266469307605054443, −7.63745353849207534770860862889, −5.80694317173141389687053077443, −5.34541316038813786733643071299, −4.15823128441811982825984381604, −2.89266650976249348140608549521, −1.79877274001089526237048989461, −0.34749033290520041302759223999, 1.37694216485085279819232481642, 2.87536811959914912302884501385, 3.43496154670577408214325277645, 4.85956034237170837785253310564, 5.87938254775186359666683035391, 7.15557679342411792904436353800, 7.79448981805437990122000260119, 8.445719265420990285627196221615, 9.790080537596316144482601479161, 10.39295705880686500987646311617

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