Properties

Label 2-280-1.1-c5-0-28
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  + 13.9·3-s + 25·5-s + 49·7-s − 48.2·9-s − 473.·11-s + 22.8·13-s + 348.·15-s − 1.78e3·17-s − 2.95e3·19-s + 683.·21-s + 3.33e3·23-s + 625·25-s − 4.06e3·27-s − 4.71e3·29-s − 1.59e3·31-s − 6.60e3·33-s + 1.22e3·35-s + 7.78e3·37-s + 319.·39-s − 5.78e3·41-s − 4.47e3·43-s − 1.20e3·45-s + 2.95e4·47-s + 2.40e3·49-s − 2.48e4·51-s − 2.43e4·53-s − 1.18e4·55-s + ⋯
L(s)  = 1  + 0.895·3-s + 0.447·5-s + 0.377·7-s − 0.198·9-s − 1.17·11-s + 0.0375·13-s + 0.400·15-s − 1.49·17-s − 1.87·19-s + 0.338·21-s + 1.31·23-s + 0.200·25-s − 1.07·27-s − 1.04·29-s − 0.297·31-s − 1.05·33-s + 0.169·35-s + 0.935·37-s + 0.0336·39-s − 0.537·41-s − 0.369·43-s − 0.0887·45-s + 1.95·47-s + 0.142·49-s − 1.33·51-s − 1.18·53-s − 0.527·55-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 \( 1 - 13.9T + 243T^{2} \)
11 \( 1 + 473.T + 1.61e5T^{2} \)
13 \( 1 - 22.8T + 3.71e5T^{2} \)
17 \( 1 + 1.78e3T + 1.41e6T^{2} \)
19 \( 1 + 2.95e3T + 2.47e6T^{2} \)
23 \( 1 - 3.33e3T + 6.43e6T^{2} \)
29 \( 1 + 4.71e3T + 2.05e7T^{2} \)
31 \( 1 + 1.59e3T + 2.86e7T^{2} \)
37 \( 1 - 7.78e3T + 6.93e7T^{2} \)
41 \( 1 + 5.78e3T + 1.15e8T^{2} \)
43 \( 1 + 4.47e3T + 1.47e8T^{2} \)
47 \( 1 - 2.95e4T + 2.29e8T^{2} \)
53 \( 1 + 2.43e4T + 4.18e8T^{2} \)
59 \( 1 + 3.58e4T + 7.14e8T^{2} \)
61 \( 1 - 1.23e4T + 8.44e8T^{2} \)
67 \( 1 - 3.03e4T + 1.35e9T^{2} \)
71 \( 1 - 9.50e3T + 1.80e9T^{2} \)
73 \( 1 - 5.66e4T + 2.07e9T^{2} \)
79 \( 1 + 7.52e4T + 3.07e9T^{2} \)
83 \( 1 - 3.69e4T + 3.93e9T^{2} \)
89 \( 1 + 7.79e4T + 5.58e9T^{2} \)
97 \( 1 + 1.57e4T + 8.58e9T^{2} \)
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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

−10.69878128479496992391839404731, −9.320074672698247077801301368512, −8.668158274588073224379254174996, −7.83464341061390383836449758844, −6.63400957874845551085782197836, −5.40567821544252933332267483474, −4.24315584067897383890586769303, −2.74245815212572737521802825782, −2.00852005888322678381302962654, 0, 2.00852005888322678381302962654, 2.74245815212572737521802825782, 4.24315584067897383890586769303, 5.40567821544252933332267483474, 6.63400957874845551085782197836, 7.83464341061390383836449758844, 8.668158274588073224379254174996, 9.320074672698247077801301368512, 10.69878128479496992391839404731

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