Properties

Label 2-280-1.1-c5-0-15
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  − 21.3·3-s − 25·5-s + 49·7-s + 211.·9-s − 381.·11-s + 372.·13-s + 533.·15-s − 1.09e3·17-s + 2.85e3·19-s − 1.04e3·21-s + 2.94e3·23-s + 625·25-s + 666.·27-s + 2.41e3·29-s − 463.·31-s + 8.13e3·33-s − 1.22e3·35-s + 1.80e3·37-s − 7.93e3·39-s − 9.01e3·41-s − 33.7·43-s − 5.29e3·45-s − 1.51e4·47-s + 2.40e3·49-s + 2.34e4·51-s − 3.13e4·53-s + 9.53e3·55-s + ⋯
L(s)  = 1  − 1.36·3-s − 0.447·5-s + 0.377·7-s + 0.871·9-s − 0.950·11-s + 0.610·13-s + 0.611·15-s − 0.922·17-s + 1.81·19-s − 0.517·21-s + 1.16·23-s + 0.200·25-s + 0.176·27-s + 0.532·29-s − 0.0866·31-s + 1.30·33-s − 0.169·35-s + 0.216·37-s − 0.835·39-s − 0.837·41-s − 0.00278·43-s − 0.389·45-s − 0.999·47-s + 0.142·49-s + 1.26·51-s − 1.53·53-s + 0.425·55-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: Trivial
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 \( 1 + 21.3T + 243T^{2} \)
11 \( 1 + 381.T + 1.61e5T^{2} \)
13 \( 1 - 372.T + 3.71e5T^{2} \)
17 \( 1 + 1.09e3T + 1.41e6T^{2} \)
19 \( 1 - 2.85e3T + 2.47e6T^{2} \)
23 \( 1 - 2.94e3T + 6.43e6T^{2} \)
29 \( 1 - 2.41e3T + 2.05e7T^{2} \)
31 \( 1 + 463.T + 2.86e7T^{2} \)
37 \( 1 - 1.80e3T + 6.93e7T^{2} \)
41 \( 1 + 9.01e3T + 1.15e8T^{2} \)
43 \( 1 + 33.7T + 1.47e8T^{2} \)
47 \( 1 + 1.51e4T + 2.29e8T^{2} \)
53 \( 1 + 3.13e4T + 4.18e8T^{2} \)
59 \( 1 + 3.02e4T + 7.14e8T^{2} \)
61 \( 1 + 1.00e4T + 8.44e8T^{2} \)
67 \( 1 - 5.28e4T + 1.35e9T^{2} \)
71 \( 1 + 1.16e4T + 1.80e9T^{2} \)
73 \( 1 + 4.76e3T + 2.07e9T^{2} \)
79 \( 1 + 2.82e4T + 3.07e9T^{2} \)
83 \( 1 - 3.64e4T + 3.93e9T^{2} \)
89 \( 1 + 1.32e5T + 5.58e9T^{2} \)
97 \( 1 + 9.80e4T + 8.58e9T^{2} \)
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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

−11.00969926878326314633067705147, −9.815722166978925485791940218576, −8.522591873219258337790262324755, −7.45667061099067271546080410523, −6.47228051255136927213197238756, −5.33314083957624770944359208082, −4.70986619520904121934010860743, −3.09685932900926104405670822259, −1.19399614153027410820455406148, 0, 1.19399614153027410820455406148, 3.09685932900926104405670822259, 4.70986619520904121934010860743, 5.33314083957624770944359208082, 6.47228051255136927213197238756, 7.45667061099067271546080410523, 8.522591873219258337790262324755, 9.815722166978925485791940218576, 11.00969926878326314633067705147

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