Properties

Label 2-280-280.179-c0-0-1
Degree $2$
Conductor $280$
Sign $-0.0633 + 0.997i$
Analytic cond. $0.139738$
Root an. cond. $0.373815$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 7-s + 0.999·8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 − 0.866i)11-s − 13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (0.5 + 0.866i)19-s + 0.999·20-s − 0.999·22-s + (0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 7-s + 0.999·8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 − 0.866i)11-s − 13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (0.5 + 0.866i)19-s + 0.999·20-s − 0.999·22-s + (0.5 + 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.0633 + 0.997i$
Analytic conductor: \(0.139738\)
Root analytic conductor: \(0.373815\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :0),\ -0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5695104506\)
\(L(\frac12)\) \(\approx\) \(0.5695104506\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 - T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.80071508954180853926845378276, −11.20615371565957059761013023842, −9.860082458716909779214524538077, −8.973991931372176462136622755714, −8.292953158098930586523783564286, −7.38314043813070187558221167778, −5.53551928462064910010219100853, −4.38038338526091381188467776575, −3.23446085663219780964219393327, −1.29490937665423030484310327694, 2.30247443887687932688881450063, 4.45199213217626417055744200907, 5.26808382279490391258148059126, 6.79736537383575133286069677998, 7.45877146856009408561287802246, 8.244697824609083528068906561575, 9.368086750882871791009580674703, 10.48062927226413804045761639682, 11.13932298076329041515698727402, 12.13830326851865339363536546666

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