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Properties

Label 2-2280-2280.1829-c0-0-1
Degree $2$
Conductor $2280$
Sign $0.135 - 0.990i$
Analytic cond. $1.13786$
Root an. cond. $1.06670$
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  + (−0.342 + 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 − 0.642i)4-s + (−0.642 + 0.766i)5-s + (0.5 − 0.866i)6-s + (0.866 − 0.500i)8-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)10-s + (0.642 + 0.766i)12-s + (0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (−0.642 + 0.233i)17-s + (−0.642 + 0.766i)18-s + (0.173 − 0.984i)19-s + (0.984 − 0.173i)20-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 − 0.642i)4-s + (−0.642 + 0.766i)5-s + (0.5 − 0.866i)6-s + (0.866 − 0.500i)8-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)10-s + (0.642 + 0.766i)12-s + (0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (−0.642 + 0.233i)17-s + (−0.642 + 0.766i)18-s + (0.173 − 0.984i)19-s + (0.984 − 0.173i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.135 - 0.990i$
Analytic conductor: \(1.13786\)
Root analytic conductor: \(1.06670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :0),\ 0.135 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5506172141\)
\(L(\frac12)\) \(\approx\) \(0.5506172141\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 - 0.939i)T \)
3 \( 1 + (0.984 + 0.173i)T \)
5 \( 1 + (0.642 - 0.766i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
good7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.642 - 0.233i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (-1.32 + 1.11i)T + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-1.20 - 0.439i)T + (0.766 + 0.642i)T^{2} \)
53 \( 1 + (-1.20 - 1.43i)T + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (-1.11 - 1.32i)T + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.300 + 0.173i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.766 + 0.642i)T^{2} \)
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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

−9.211041453728485636260851089336, −8.519586356809444881415952162990, −7.52015090991106059242046997429, −6.99362298947547794026673109942, −6.49943002221257292372008388277, −5.64539094180390104241866205200, −4.70017586423578490824065918782, −4.12695518296395992249198361501, −2.62536450429777691141114643240, −0.880530865632672690758849659232, 0.73281189529996120359568314187, 1.85073107225216538967555931561, 3.43448618128553185737001462906, 4.08397210589824934295566127570, 4.99498858845023268585657509379, 5.50187008080294982738374100879, 6.85500991071664662358984420489, 7.59153453664560612408907972004, 8.437626555978202224225479488538, 9.204943023469737599548138571853

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