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Properties

Label 2-280-56.27-c1-0-24
Degree $2$
Conductor $280$
Sign $0.594 + 0.803i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 0.0765i)2-s − 2.21i·3-s + (1.98 − 0.216i)4-s + 5-s + (−0.169 − 3.13i)6-s + (1.20 + 2.35i)7-s + (2.79 − 0.457i)8-s − 1.92·9-s + (1.41 − 0.0765i)10-s − 3.88·11-s + (−0.479 − 4.41i)12-s − 5.67·13-s + (1.88 + 3.23i)14-s − 2.21i·15-s + (3.90 − 0.859i)16-s + 5.63i·17-s + ⋯
L(s)  = 1  + (0.998 − 0.0541i)2-s − 1.28i·3-s + (0.994 − 0.108i)4-s + 0.447·5-s + (−0.0693 − 1.27i)6-s + (0.457 + 0.889i)7-s + (0.986 − 0.161i)8-s − 0.641·9-s + (0.446 − 0.0242i)10-s − 1.17·11-s + (−0.138 − 1.27i)12-s − 1.57·13-s + (0.504 + 0.863i)14-s − 0.572i·15-s + (0.976 − 0.214i)16-s + 1.36i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.594 + 0.803i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.594 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12208 - 1.06955i\)
\(L(\frac12)\) \(\approx\) \(2.12208 - 1.06955i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0765i)T \)
5 \( 1 - T \)
7 \( 1 + (-1.20 - 2.35i)T \)
good3 \( 1 + 2.21iT - 3T^{2} \)
11 \( 1 + 3.88T + 11T^{2} \)
13 \( 1 + 5.67T + 13T^{2} \)
17 \( 1 - 5.63iT - 17T^{2} \)
19 \( 1 + 1.31iT - 19T^{2} \)
23 \( 1 + 7.37iT - 23T^{2} \)
29 \( 1 - 9.07iT - 29T^{2} \)
31 \( 1 + 2.23T + 31T^{2} \)
37 \( 1 + 6.98iT - 37T^{2} \)
41 \( 1 - 7.47iT - 41T^{2} \)
43 \( 1 + 1.46T + 43T^{2} \)
47 \( 1 - 0.567T + 47T^{2} \)
53 \( 1 + 0.100iT - 53T^{2} \)
59 \( 1 - 2.93iT - 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 - 5.54T + 67T^{2} \)
71 \( 1 + 2.42iT - 71T^{2} \)
73 \( 1 + 6.08iT - 73T^{2} \)
79 \( 1 - 2.83iT - 79T^{2} \)
83 \( 1 + 2.52iT - 83T^{2} \)
89 \( 1 + 10.9iT - 89T^{2} \)
97 \( 1 + 9.93iT - 97T^{2} \)
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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

−12.30630825021736779579149156800, −11.04055124832184339465729467047, −10.13235382135660641724714500712, −8.493750873393390951845646909898, −7.57053726493223921267110337755, −6.67619074279668021922556507538, −5.65092994045240056348518486826, −4.78889198523495903299502945777, −2.67434687563554975762415297729, −1.94429898334409791617361001881, 2.49499067021003374299123461651, 3.83009246659029200246039803076, 4.96686673966736616266000417578, 5.31788223786157776101850468542, 7.09622336994217823571604007503, 7.83908950471242374627021749504, 9.744234682001624285425815499607, 10.05442543096129718343942039587, 11.07544895988698208386266164148, 11.88210931123598165614829184059

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