Properties

Label 2-280-1.1-c5-0-8
Degree $2$
Conductor $280$
Sign $1$
Analytic cond. $44.9074$
Root an. cond. $6.70130$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11.5·3-s + 25·5-s + 49·7-s − 110.·9-s + 311.·11-s + 474.·13-s − 288.·15-s − 1.01e3·17-s − 225.·19-s − 564.·21-s − 1.77e3·23-s + 625·25-s + 4.07e3·27-s + 2.81e3·29-s − 4.25e3·31-s − 3.58e3·33-s + 1.22e3·35-s + 5.26e3·37-s − 5.46e3·39-s + 4.86e3·41-s − 1.69e4·43-s − 2.75e3·45-s + 800.·47-s + 2.40e3·49-s + 1.16e4·51-s + 3.69e4·53-s + 7.77e3·55-s + ⋯
L(s)  = 1  − 0.739·3-s + 0.447·5-s + 0.377·7-s − 0.453·9-s + 0.775·11-s + 0.778·13-s − 0.330·15-s − 0.850·17-s − 0.143·19-s − 0.279·21-s − 0.698·23-s + 0.200·25-s + 1.07·27-s + 0.620·29-s − 0.795·31-s − 0.573·33-s + 0.169·35-s + 0.632·37-s − 0.575·39-s + 0.451·41-s − 1.39·43-s − 0.202·45-s + 0.0528·47-s + 0.142·49-s + 0.628·51-s + 1.80·53-s + 0.346·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(44.9074\)
Root analytic conductor: \(6.70130\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.672043525\)
\(L(\frac12)\) \(\approx\) \(1.672043525\)
\(L(\frac{7}{2})\) 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 \( 1 - 25T \)
7 \( 1 - 49T \)
good3 \( 1 + 11.5T + 243T^{2} \)
11 \( 1 - 311.T + 1.61e5T^{2} \)
13 \( 1 - 474.T + 3.71e5T^{2} \)
17 \( 1 + 1.01e3T + 1.41e6T^{2} \)
19 \( 1 + 225.T + 2.47e6T^{2} \)
23 \( 1 + 1.77e3T + 6.43e6T^{2} \)
29 \( 1 - 2.81e3T + 2.05e7T^{2} \)
31 \( 1 + 4.25e3T + 2.86e7T^{2} \)
37 \( 1 - 5.26e3T + 6.93e7T^{2} \)
41 \( 1 - 4.86e3T + 1.15e8T^{2} \)
43 \( 1 + 1.69e4T + 1.47e8T^{2} \)
47 \( 1 - 800.T + 2.29e8T^{2} \)
53 \( 1 - 3.69e4T + 4.18e8T^{2} \)
59 \( 1 + 4.24e3T + 7.14e8T^{2} \)
61 \( 1 - 4.08e4T + 8.44e8T^{2} \)
67 \( 1 - 1.47e4T + 1.35e9T^{2} \)
71 \( 1 - 6.39e4T + 1.80e9T^{2} \)
73 \( 1 - 6.80e4T + 2.07e9T^{2} \)
79 \( 1 - 2.61e4T + 3.07e9T^{2} \)
83 \( 1 - 3.16e4T + 3.93e9T^{2} \)
89 \( 1 + 7.84e4T + 5.58e9T^{2} \)
97 \( 1 - 1.23e5T + 8.58e9T^{2} \)
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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

−11.17673782297527145950851742936, −10.22923318044835764736595746376, −9.053673878477203604306308877121, −8.272699668758403834319146773506, −6.76247075379932351377638451064, −6.05840469128741337504519881562, −5.05431980456017445520473720436, −3.81042601850168948400750303476, −2.15540302193688309202603114133, −0.77675293585368835529628785413, 0.77675293585368835529628785413, 2.15540302193688309202603114133, 3.81042601850168948400750303476, 5.05431980456017445520473720436, 6.05840469128741337504519881562, 6.76247075379932351377638451064, 8.272699668758403834319146773506, 9.053673878477203604306308877121, 10.22923318044835764736595746376, 11.17673782297527145950851742936

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