Properties

Label 2-280-1.1-c5-0-21
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  + 30.3·3-s + 25·5-s + 49·7-s + 677.·9-s + 604.·11-s + 383.·13-s + 758.·15-s − 791.·17-s − 2.07e3·19-s + 1.48e3·21-s − 2.52e3·23-s + 625·25-s + 1.31e4·27-s + 372.·29-s − 2.48e3·31-s + 1.83e4·33-s + 1.22e3·35-s − 6.55e3·37-s + 1.16e4·39-s − 1.61e4·41-s + 1.54e4·43-s + 1.69e4·45-s + 2.22e4·47-s + 2.40e3·49-s − 2.40e4·51-s − 2.62e4·53-s + 1.51e4·55-s + ⋯
L(s)  = 1  + 1.94·3-s + 0.447·5-s + 0.377·7-s + 2.78·9-s + 1.50·11-s + 0.630·13-s + 0.870·15-s − 0.664·17-s − 1.32·19-s + 0.735·21-s − 0.994·23-s + 0.200·25-s + 3.47·27-s + 0.0821·29-s − 0.464·31-s + 2.93·33-s + 0.169·35-s − 0.786·37-s + 1.22·39-s − 1.50·41-s + 1.27·43-s + 1.24·45-s + 1.47·47-s + 0.142·49-s − 1.29·51-s − 1.28·53-s + 0.673·55-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

\(L(3)\) \(\approx\) \(5.322876041\)
\(L(\frac12)\) \(\approx\) \(5.322876041\)
\(L(\frac{7}{2})\) 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 \( 1 - 30.3T + 243T^{2} \)
11 \( 1 - 604.T + 1.61e5T^{2} \)
13 \( 1 - 383.T + 3.71e5T^{2} \)
17 \( 1 + 791.T + 1.41e6T^{2} \)
19 \( 1 + 2.07e3T + 2.47e6T^{2} \)
23 \( 1 + 2.52e3T + 6.43e6T^{2} \)
29 \( 1 - 372.T + 2.05e7T^{2} \)
31 \( 1 + 2.48e3T + 2.86e7T^{2} \)
37 \( 1 + 6.55e3T + 6.93e7T^{2} \)
41 \( 1 + 1.61e4T + 1.15e8T^{2} \)
43 \( 1 - 1.54e4T + 1.47e8T^{2} \)
47 \( 1 - 2.22e4T + 2.29e8T^{2} \)
53 \( 1 + 2.62e4T + 4.18e8T^{2} \)
59 \( 1 - 3.98e4T + 7.14e8T^{2} \)
61 \( 1 - 5.89e3T + 8.44e8T^{2} \)
67 \( 1 + 3.20e4T + 1.35e9T^{2} \)
71 \( 1 + 5.79e4T + 1.80e9T^{2} \)
73 \( 1 - 3.94e4T + 2.07e9T^{2} \)
79 \( 1 + 8.98e3T + 3.07e9T^{2} \)
83 \( 1 + 5.25e4T + 3.93e9T^{2} \)
89 \( 1 - 5.20e4T + 5.58e9T^{2} \)
97 \( 1 - 1.49e4T + 8.58e9T^{2} \)
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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

−10.76258148432783031401690249236, −9.774863763247863647977465474549, −8.811141341444328614954385961373, −8.543856633598979092115045294914, −7.25277371116993730539449923949, −6.31271445615688204044001848628, −4.35418940734519868825688185379, −3.66708966895661366146336163250, −2.24429078518721282654680591133, −1.47507092495482398870471603134, 1.47507092495482398870471603134, 2.24429078518721282654680591133, 3.66708966895661366146336163250, 4.35418940734519868825688185379, 6.31271445615688204044001848628, 7.25277371116993730539449923949, 8.543856633598979092115045294914, 8.811141341444328614954385961373, 9.774863763247863647977465474549, 10.76258148432783031401690249236

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