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Properties

Label 2-280-5.4-c1-0-6
Degree $2$
Conductor $280$
Sign $-0.193 + 0.981i$
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  − 1.76i·3-s + (0.432 − 2.19i)5-s + i·7-s − 0.103·9-s − 0.626·11-s − 5.49i·13-s + (−3.86 − 0.761i)15-s + 0.896i·17-s − 6.38·19-s + 1.76·21-s + 3.72i·23-s + (−4.62 − 1.89i)25-s − 5.10i·27-s + 7.87·29-s + 7.52·31-s + ⋯
L(s)  = 1  − 1.01i·3-s + (0.193 − 0.981i)5-s + 0.377i·7-s − 0.0343·9-s − 0.188·11-s − 1.52i·13-s + (−0.997 − 0.196i)15-s + 0.217i·17-s − 1.46·19-s + 0.384·21-s + 0.777i·23-s + (−0.925 − 0.379i)25-s − 0.982i·27-s + 1.46·29-s + 1.35·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.193 + 0.981i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.193 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.808528 - 0.983403i\)
\(L(\frac12)\) \(\approx\) \(0.808528 - 0.983403i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.432 + 2.19i)T \)
7 \( 1 - iT \)
good3 \( 1 + 1.76iT - 3T^{2} \)
11 \( 1 + 0.626T + 11T^{2} \)
13 \( 1 + 5.49iT - 13T^{2} \)
17 \( 1 - 0.896iT - 17T^{2} \)
19 \( 1 + 6.38T + 19T^{2} \)
23 \( 1 - 3.72iT - 23T^{2} \)
29 \( 1 - 7.87T + 29T^{2} \)
31 \( 1 - 7.52T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 7.72T + 41T^{2} \)
43 \( 1 + 1.72iT - 43T^{2} \)
47 \( 1 - 5.87iT - 47T^{2} \)
53 \( 1 - 6.77iT - 53T^{2} \)
59 \( 1 - 0.593T + 59T^{2} \)
61 \( 1 - 7.13T + 61T^{2} \)
67 \( 1 + 5.79iT - 67T^{2} \)
71 \( 1 - 5.52T + 71T^{2} \)
73 \( 1 - 3.72iT - 73T^{2} \)
79 \( 1 - 5.67T + 79T^{2} \)
83 \( 1 - 17.4iT - 83T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 + 10.1iT - 97T^{2} \)
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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

−12.00689102009852133158099906776, −10.63146638746084162857167213338, −9.679234758791504517425711675737, −8.325527074602906438747407241624, −8.013764963333671511435130694462, −6.57245077497587463430219597539, −5.66766889283579745312532935898, −4.46192528904856440559900897225, −2.57075760395248597019951817067, −1.05356713822094753555425434045, 2.39981295589077538341173400792, 3.92790858080743725235610648313, 4.66450928892987930586168625508, 6.32782239262657766005977900302, 7.01503221393120805024727746374, 8.454360297186717670259032976698, 9.527837538195298445800113066285, 10.32745331223125913080300113591, 10.88657931952175062855583144307, 11.87955474170534826565177723309

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