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Properties

Label 1-33e2-1089.1073-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.293 - 0.955i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (−0.290 − 0.956i)2-s + (−0.830 + 0.556i)4-s + (−0.988 − 0.151i)5-s + (−0.879 + 0.475i)7-s + (0.774 + 0.633i)8-s + (0.142 + 0.989i)10-s + (−0.999 − 0.0380i)13-s + (0.710 + 0.703i)14-s + (0.380 − 0.924i)16-s + (0.198 − 0.980i)17-s + (−0.993 − 0.113i)19-s + (0.905 − 0.424i)20-s + (−0.723 + 0.690i)23-s + (0.953 + 0.299i)25-s + (0.254 + 0.967i)26-s + ⋯
L(s)  = 1  + (−0.290 − 0.956i)2-s + (−0.830 + 0.556i)4-s + (−0.988 − 0.151i)5-s + (−0.879 + 0.475i)7-s + (0.774 + 0.633i)8-s + (0.142 + 0.989i)10-s + (−0.999 − 0.0380i)13-s + (0.710 + 0.703i)14-s + (0.380 − 0.924i)16-s + (0.198 − 0.980i)17-s + (−0.993 − 0.113i)19-s + (0.905 − 0.424i)20-s + (−0.723 + 0.690i)23-s + (0.953 + 0.299i)25-s + (0.254 + 0.967i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.293 - 0.955i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1073, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.293 - 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3745224643 - 0.2767560205i\)
\(L(\frac12)\) \(\approx\) \(0.3745224643 - 0.2767560205i\)
\(L(1)\) \(\approx\) \(0.5018338462 - 0.2030971387i\)
\(L(1)\) \(\approx\) \(0.5018338462 - 0.2030971387i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.290 - 0.956i)T \)
5 \( 1 + (-0.988 - 0.151i)T \)
7 \( 1 + (-0.879 + 0.475i)T \)
13 \( 1 + (-0.999 - 0.0380i)T \)
17 \( 1 + (0.198 - 0.980i)T \)
19 \( 1 + (-0.993 - 0.113i)T \)
23 \( 1 + (-0.723 + 0.690i)T \)
29 \( 1 + (-0.217 + 0.976i)T \)
31 \( 1 + (-0.969 + 0.244i)T \)
37 \( 1 + (-0.985 - 0.170i)T \)
41 \( 1 + (0.483 + 0.875i)T \)
43 \( 1 + (0.888 + 0.458i)T \)
47 \( 1 + (-0.123 - 0.992i)T \)
53 \( 1 + (-0.610 + 0.791i)T \)
59 \( 1 + (0.999 - 0.0190i)T \)
61 \( 1 + (0.683 + 0.730i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (0.870 - 0.491i)T \)
73 \( 1 + (0.564 - 0.825i)T \)
79 \( 1 + (-0.548 - 0.836i)T \)
83 \( 1 + (-0.179 - 0.983i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (0.988 - 0.151i)T \)
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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

−22.01772792732072933990933424927, −20.65497109988497967057343461435, −19.60586683810490407738061254958, −19.26026821751954364340399933939, −18.654074954206539715358642120503, −17.35438887363132303712820541549, −16.95768310495738103272066854745, −16.06753743810736788704748823945, −15.50241921866771031939976026542, −14.657144249392670712405634875771, −14.087968885654009025424134671, −12.73296019631708848270395464197, −12.54006773628414865402270874041, −11.06424761887744555113384382582, −10.28437595570699247757531278856, −9.56909269240762535455059198536, −8.49666458511347678470841121204, −7.84980853061441785373213402098, −7.01317671817874851548991689330, −6.40443213981674802834228030246, −5.36865272673871046017453869459, −4.12568886820646314927515797612, −3.81787291002983981147119551592, −2.25417722216993193803148912552, −0.54222561089960581705589382768, 0.42993402178337748912669368191, 1.961813557009686572993514702272, 2.95045926413284219039791965471, 3.64436177775895791219944426821, 4.618254070511563203481856233594, 5.48408856524152491447867729151, 6.96021808298211134381093481, 7.6427027421194587024970010763, 8.67023059721735416764397706498, 9.31547529509712272905748783575, 10.109354884638940759654781621391, 11.05216416634636529160627920480, 11.84402548019012640347361565012, 12.474818784188632393970208517080, 12.98508509278482338157982545246, 14.15214641276789718460046311747, 14.99407487188647166318907343038, 16.03562657492693221588711592335, 16.55330575555773644543074466738, 17.565627580704103473227422978629, 18.463952991615399949107282947148, 19.11870454701422927637957792346, 19.75664335633440089495333833016, 20.18634320838287152952444905971, 21.28689340356971485485249230318

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