Properties

Label 2-280-1.1-c5-0-22
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  + 6.24·3-s − 25·5-s + 49·7-s − 203.·9-s + 89.4·11-s + 459.·13-s − 156.·15-s + 701.·17-s − 396.·19-s + 306.·21-s − 4.31e3·23-s + 625·25-s − 2.79e3·27-s − 1.43e3·29-s + 3.66e3·31-s + 558.·33-s − 1.22e3·35-s − 3.16e3·37-s + 2.86e3·39-s − 6.35e3·41-s − 1.55e4·43-s + 5.09e3·45-s − 3.99e3·47-s + 2.40e3·49-s + 4.38e3·51-s − 2.44e4·53-s − 2.23e3·55-s + ⋯
L(s)  = 1  + 0.400·3-s − 0.447·5-s + 0.377·7-s − 0.839·9-s + 0.222·11-s + 0.753·13-s − 0.179·15-s + 0.588·17-s − 0.251·19-s + 0.151·21-s − 1.70·23-s + 0.200·25-s − 0.737·27-s − 0.317·29-s + 0.685·31-s + 0.0893·33-s − 0.169·35-s − 0.380·37-s + 0.302·39-s − 0.590·41-s − 1.27·43-s + 0.375·45-s − 0.263·47-s + 0.142·49-s + 0.235·51-s − 1.19·53-s − 0.0996·55-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 \( 1 - 6.24T + 243T^{2} \)
11 \( 1 - 89.4T + 1.61e5T^{2} \)
13 \( 1 - 459.T + 3.71e5T^{2} \)
17 \( 1 - 701.T + 1.41e6T^{2} \)
19 \( 1 + 396.T + 2.47e6T^{2} \)
23 \( 1 + 4.31e3T + 6.43e6T^{2} \)
29 \( 1 + 1.43e3T + 2.05e7T^{2} \)
31 \( 1 - 3.66e3T + 2.86e7T^{2} \)
37 \( 1 + 3.16e3T + 6.93e7T^{2} \)
41 \( 1 + 6.35e3T + 1.15e8T^{2} \)
43 \( 1 + 1.55e4T + 1.47e8T^{2} \)
47 \( 1 + 3.99e3T + 2.29e8T^{2} \)
53 \( 1 + 2.44e4T + 4.18e8T^{2} \)
59 \( 1 + 2.39e3T + 7.14e8T^{2} \)
61 \( 1 + 2.10e4T + 8.44e8T^{2} \)
67 \( 1 - 853.T + 1.35e9T^{2} \)
71 \( 1 + 1.37e3T + 1.80e9T^{2} \)
73 \( 1 - 1.22e4T + 2.07e9T^{2} \)
79 \( 1 + 5.10e4T + 3.07e9T^{2} \)
83 \( 1 - 1.76e4T + 3.93e9T^{2} \)
89 \( 1 + 1.83e4T + 5.58e9T^{2} \)
97 \( 1 + 1.03e5T + 8.58e9T^{2} \)
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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

−10.60994099073235148263569278421, −9.504114987706198315668413648122, −8.359791248840447739601158471513, −7.981151730705663041948916114223, −6.53872771581583984915250523780, −5.49595764317791957771514262947, −4.12265324513195603555761943554, −3.11231258648179354421980482746, −1.65508158090314240164689945335, 0, 1.65508158090314240164689945335, 3.11231258648179354421980482746, 4.12265324513195603555761943554, 5.49595764317791957771514262947, 6.53872771581583984915250523780, 7.981151730705663041948916114223, 8.359791248840447739601158471513, 9.504114987706198315668413648122, 10.60994099073235148263569278421

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