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Properties

Label 1-1053-1053.1019-r0-0-0
Degree $1$
Conductor $1053$
Sign $-0.900 - 0.434i$
Analytic cond. $4.89011$
Root an. cond. $4.89011$
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  + (0.727 − 0.686i)2-s + (0.0581 − 0.998i)4-s + (−0.116 − 0.993i)5-s + (−0.448 + 0.893i)7-s + (−0.642 − 0.766i)8-s + (−0.766 − 0.642i)10-s + (0.918 − 0.396i)11-s + (0.286 + 0.957i)14-s + (−0.993 − 0.116i)16-s + (0.173 − 0.984i)17-s + (0.984 − 0.173i)19-s + (−0.998 + 0.0581i)20-s + (0.396 − 0.918i)22-s + (0.893 − 0.448i)23-s + (−0.973 + 0.230i)25-s + ⋯
L(s)  = 1  + (0.727 − 0.686i)2-s + (0.0581 − 0.998i)4-s + (−0.116 − 0.993i)5-s + (−0.448 + 0.893i)7-s + (−0.642 − 0.766i)8-s + (−0.766 − 0.642i)10-s + (0.918 − 0.396i)11-s + (0.286 + 0.957i)14-s + (−0.993 − 0.116i)16-s + (0.173 − 0.984i)17-s + (0.984 − 0.173i)19-s + (−0.998 + 0.0581i)20-s + (0.396 − 0.918i)22-s + (0.893 − 0.448i)23-s + (−0.973 + 0.230i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $-0.900 - 0.434i$
Analytic conductor: \(4.89011\)
Root analytic conductor: \(4.89011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1053,\ (0:\ ),\ -0.900 - 0.434i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4319799019 - 1.891133727i\)
\(L(\frac12)\) \(\approx\) \(0.4319799019 - 1.891133727i\)
\(L(1)\) \(\approx\) \(1.094112335 - 0.9487987528i\)
\(L(1)\) \(\approx\) \(1.094112335 - 0.9487987528i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.727 - 0.686i)T \)
5 \( 1 + (-0.116 - 0.993i)T \)
7 \( 1 + (-0.448 + 0.893i)T \)
11 \( 1 + (0.918 - 0.396i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.984 - 0.173i)T \)
23 \( 1 + (0.893 - 0.448i)T \)
29 \( 1 + (0.286 - 0.957i)T \)
31 \( 1 + (-0.549 + 0.835i)T \)
37 \( 1 + (-0.342 - 0.939i)T \)
41 \( 1 + (-0.727 - 0.686i)T \)
43 \( 1 + (-0.597 + 0.802i)T \)
47 \( 1 + (0.549 + 0.835i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.918 - 0.396i)T \)
61 \( 1 + (-0.0581 - 0.998i)T \)
67 \( 1 + (-0.957 + 0.286i)T \)
71 \( 1 + (-0.642 + 0.766i)T \)
73 \( 1 + (0.642 + 0.766i)T \)
79 \( 1 + (-0.686 - 0.727i)T \)
83 \( 1 + (0.727 - 0.686i)T \)
89 \( 1 + (-0.642 - 0.766i)T \)
97 \( 1 + (-0.116 + 0.993i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.25001982027779823426176889761, −21.34482042155717153602343795476, −20.270324631019267882878733257004, −19.69412934229826411586492024901, −18.647628616835440395798241396135, −17.7876512355595047867701967677, −16.91693653999165845860932514149, −16.51820371553995315924960069509, −15.12735020686072278827112914191, −15.05232557849157032783916061863, −13.896399498862589293257187689812, −13.54879975576886650477307091745, −12.42755013177650054339111648675, −11.684411952038979474592335283909, −10.77750263449023730541905579716, −9.90980095979584962826847592018, −8.874684953971608246048686473803, −7.69254458523652869152757061701, −7.08259922815284361274521459507, −6.51863966554143765980133746878, −5.595474625623098514506595351849, −4.40033872501894186267812856477, −3.584525777578290147464218146507, −3.062487644784988914223839766, −1.555568923955259663835138154429, 0.64915963266124498905062060557, 1.67055621773747460005069536738, 2.83739632136850266847396492050, 3.61031134603136025528302958162, 4.7240558457949577284193438401, 5.35719809650905593744060880464, 6.1697370621010229246693551074, 7.18945629089532990892879245674, 8.65584107191622019067746049291, 9.22161841709110356086138118522, 9.84614889908052677292169268590, 11.16786480128226626289199367643, 11.86164078885456512877210535334, 12.346538471342916442293723917081, 13.18425882270644783288454794812, 13.93737661172613242780217755383, 14.75129912278339934286712940136, 15.82450543772515118239819438349, 16.14287729969802925296160011721, 17.28650942958784013116011419228, 18.3444029073607962695355769205, 19.10190098046030407164258524844, 19.758626054910318347379831206763, 20.4713316922022079986854227078, 21.21133066053041501904114094057

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