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Properties

Label 2-2289-2289.158-c0-0-0
Degree $2$
Conductor $2289$
Sign $0.778 + 0.627i$
Analytic cond. $1.14235$
Root an. cond. $1.06881$
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  + (−0.835 − 0.549i)3-s + (−0.939 − 0.342i)4-s + (−0.939 − 0.342i)7-s + (0.396 + 0.918i)9-s + (0.597 + 0.802i)12-s + (−0.744 + 1.72i)13-s + (0.766 + 0.642i)16-s + (1.57 − 0.571i)19-s + (0.597 + 0.802i)21-s + (−0.686 − 0.727i)25-s + (0.173 − 0.984i)27-s + (0.766 + 0.642i)28-s + (1.86 + 0.218i)31-s + (−0.0581 − 0.998i)36-s + (−0.786 − 0.0919i)37-s + ⋯
L(s)  = 1  + (−0.835 − 0.549i)3-s + (−0.939 − 0.342i)4-s + (−0.939 − 0.342i)7-s + (0.396 + 0.918i)9-s + (0.597 + 0.802i)12-s + (−0.744 + 1.72i)13-s + (0.766 + 0.642i)16-s + (1.57 − 0.571i)19-s + (0.597 + 0.802i)21-s + (−0.686 − 0.727i)25-s + (0.173 − 0.984i)27-s + (0.766 + 0.642i)28-s + (1.86 + 0.218i)31-s + (−0.0581 − 0.998i)36-s + (−0.786 − 0.0919i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2289\)    =    \(3 \cdot 7 \cdot 109\)
Sign: $0.778 + 0.627i$
Analytic conductor: \(1.14235\)
Root analytic conductor: \(1.06881\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2289} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2289,\ (\ :0),\ 0.778 + 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5567266884\)
\(L(\frac12)\) \(\approx\) \(0.5567266884\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.835 + 0.549i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
109 \( 1 + (-0.173 - 0.984i)T \)
good2 \( 1 + (0.939 + 0.342i)T^{2} \)
5 \( 1 + (0.686 + 0.727i)T^{2} \)
11 \( 1 + (-0.597 - 0.802i)T^{2} \)
13 \( 1 + (0.744 - 1.72i)T + (-0.686 - 0.727i)T^{2} \)
17 \( 1 + (0.939 + 0.342i)T^{2} \)
19 \( 1 + (-1.57 + 0.571i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.396 - 0.918i)T^{2} \)
31 \( 1 + (-1.86 - 0.218i)T + (0.973 + 0.230i)T^{2} \)
37 \( 1 + (0.786 + 0.0919i)T + (0.973 + 0.230i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (1.05 + 0.882i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.893 + 0.448i)T^{2} \)
53 \( 1 + (-0.973 + 0.230i)T^{2} \)
59 \( 1 + (0.993 - 0.116i)T^{2} \)
61 \( 1 + (-1.36 + 1.44i)T + (-0.0581 - 0.998i)T^{2} \)
67 \( 1 + (-0.993 - 0.116i)T + (0.973 + 0.230i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.512 + 0.257i)T + (0.597 + 0.802i)T^{2} \)
79 \( 1 + (0.227 + 0.526i)T + (-0.686 + 0.727i)T^{2} \)
83 \( 1 + (-0.597 + 0.802i)T^{2} \)
89 \( 1 + (-0.893 - 0.448i)T^{2} \)
97 \( 1 + (-1.86 + 0.218i)T + (0.973 - 0.230i)T^{2} \)
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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

−9.293299869608253909345579848392, −8.379036247496729964633854825781, −7.35083986081024360299390399152, −6.73901813972314340034594324206, −6.08398791737533154872254007267, −5.05525976612517559362708246098, −4.55343672766466416164411750068, −3.50406628008638303046006059505, −2.07288758834603746361836107850, −0.71530486341201516784758423617, 0.77302799325488865562171610855, 3.03113729050813361823283238096, 3.48278107267100786548289316275, 4.60625198119360027993905514589, 5.44158058308378497125446491464, 5.77011405791363398472394003428, 6.95145326930984359027755973128, 7.80131532353915280553804901828, 8.572400129430174497864876071506, 9.709212594203701584413813099047

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