Properties

Label 2-280-1.1-c5-0-24
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.08·3-s + 25·5-s + 49·7-s − 206.·9-s + 407.·11-s − 1.07e3·13-s + 152.·15-s − 1.77e3·17-s + 2.15e3·19-s + 298.·21-s − 4.54e3·23-s + 625·25-s − 2.73e3·27-s + 6.11e3·29-s − 2.85e3·31-s + 2.47e3·33-s + 1.22e3·35-s − 7.03e3·37-s − 6.54e3·39-s − 1.12e4·41-s + 1.06e4·43-s − 5.15e3·45-s − 7.63e3·47-s + 2.40e3·49-s − 1.08e4·51-s + 1.92e4·53-s + 1.01e4·55-s + ⋯
L(s)  = 1  + 0.390·3-s + 0.447·5-s + 0.377·7-s − 0.847·9-s + 1.01·11-s − 1.76·13-s + 0.174·15-s − 1.49·17-s + 1.37·19-s + 0.147·21-s − 1.78·23-s + 0.200·25-s − 0.720·27-s + 1.35·29-s − 0.533·31-s + 0.396·33-s + 0.169·35-s − 0.845·37-s − 0.688·39-s − 1.04·41-s + 0.881·43-s − 0.379·45-s − 0.504·47-s + 0.142·49-s − 0.581·51-s + 0.941·53-s + 0.454·55-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: \(1\)
Selberg data: \((2,\ 280,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - 6.08T + 243T^{2} \)
11 \( 1 - 407.T + 1.61e5T^{2} \)
13 \( 1 + 1.07e3T + 3.71e5T^{2} \)
17 \( 1 + 1.77e3T + 1.41e6T^{2} \)
19 \( 1 - 2.15e3T + 2.47e6T^{2} \)
23 \( 1 + 4.54e3T + 6.43e6T^{2} \)
29 \( 1 - 6.11e3T + 2.05e7T^{2} \)
31 \( 1 + 2.85e3T + 2.86e7T^{2} \)
37 \( 1 + 7.03e3T + 6.93e7T^{2} \)
41 \( 1 + 1.12e4T + 1.15e8T^{2} \)
43 \( 1 - 1.06e4T + 1.47e8T^{2} \)
47 \( 1 + 7.63e3T + 2.29e8T^{2} \)
53 \( 1 - 1.92e4T + 4.18e8T^{2} \)
59 \( 1 + 9.18e3T + 7.14e8T^{2} \)
61 \( 1 + 3.37e4T + 8.44e8T^{2} \)
67 \( 1 + 2.97e4T + 1.35e9T^{2} \)
71 \( 1 + 2.98e4T + 1.80e9T^{2} \)
73 \( 1 + 4.07e4T + 2.07e9T^{2} \)
79 \( 1 + 3.91e4T + 3.07e9T^{2} \)
83 \( 1 - 3.44e4T + 3.93e9T^{2} \)
89 \( 1 + 7.99e3T + 5.58e9T^{2} \)
97 \( 1 + 5.85e4T + 8.58e9T^{2} \)
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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

−10.41220645521196378254657802765, −9.476578602678035659966953188547, −8.750596375267260206693022413949, −7.64679699090966369404837863051, −6.60115478532026277543450807366, −5.43008287654672575419986607457, −4.32986042385450295655262629403, −2.84381915186886150968011683042, −1.81229171657368070915484761308, 0, 1.81229171657368070915484761308, 2.84381915186886150968011683042, 4.32986042385450295655262629403, 5.43008287654672575419986607457, 6.60115478532026277543450807366, 7.64679699090966369404837863051, 8.750596375267260206693022413949, 9.476578602678035659966953188547, 10.41220645521196378254657802765

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