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Properties

Label 1-33e2-1089.169-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.999 + 0.0161i$
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  + (−0.999 − 0.0190i)2-s + (0.999 + 0.0380i)4-s + (0.161 + 0.986i)5-s + (−0.991 − 0.132i)7-s + (−0.998 − 0.0570i)8-s + (−0.142 − 0.989i)10-s + (0.345 − 0.938i)13-s + (0.988 + 0.151i)14-s + (0.997 + 0.0760i)16-s + (0.993 − 0.113i)17-s + (−0.870 + 0.491i)19-s + (0.123 + 0.992i)20-s + (0.723 − 0.690i)23-s + (−0.948 + 0.318i)25-s + (−0.362 + 0.931i)26-s + ⋯
L(s)  = 1  + (−0.999 − 0.0190i)2-s + (0.999 + 0.0380i)4-s + (0.161 + 0.986i)5-s + (−0.991 − 0.132i)7-s + (−0.998 − 0.0570i)8-s + (−0.142 − 0.989i)10-s + (0.345 − 0.938i)13-s + (0.988 + 0.151i)14-s + (0.997 + 0.0760i)16-s + (0.993 − 0.113i)17-s + (−0.870 + 0.491i)19-s + (0.123 + 0.992i)20-s + (0.723 − 0.690i)23-s + (−0.948 + 0.318i)25-s + (−0.362 + 0.931i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.0161i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.999 + 0.0161i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8156946799 + 0.006588963468i\)
\(L(\frac12)\) \(\approx\) \(0.8156946799 + 0.006588963468i\)
\(L(1)\) \(\approx\) \(0.6752959469 + 0.04517694661i\)
\(L(1)\) \(\approx\) \(0.6752959469 + 0.04517694661i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.999 - 0.0190i)T \)
5 \( 1 + (0.161 + 0.986i)T \)
7 \( 1 + (-0.991 - 0.132i)T \)
13 \( 1 + (0.345 - 0.938i)T \)
17 \( 1 + (0.993 - 0.113i)T \)
19 \( 1 + (-0.870 + 0.491i)T \)
23 \( 1 + (0.723 - 0.690i)T \)
29 \( 1 + (0.749 - 0.662i)T \)
31 \( 1 + (-0.0665 + 0.997i)T \)
37 \( 1 + (0.696 + 0.717i)T \)
41 \( 1 + (-0.905 - 0.424i)T \)
43 \( 1 + (-0.888 - 0.458i)T \)
47 \( 1 + (-0.683 - 0.730i)T \)
53 \( 1 + (-0.564 - 0.825i)T \)
59 \( 1 + (0.820 + 0.572i)T \)
61 \( 1 + (0.483 - 0.875i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (-0.736 - 0.676i)T \)
73 \( 1 + (0.941 - 0.336i)T \)
79 \( 1 + (0.964 - 0.263i)T \)
83 \( 1 + (0.879 - 0.475i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.161 - 0.986i)T \)
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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

−21.30405292072327475859216808974, −20.54980467352690848858115365394, −19.6543462082954343312417185632, −19.197901849949025191693840966651, −18.44084148934178566192670338457, −17.407102555096881784333136463262, −16.71746121154176277558706500772, −16.290381317734135805768731336376, −15.52214570654275301519106641955, −14.5698942284795413929197602698, −13.33984341168460090900620561290, −12.69318335147704857258847118467, −11.87437859641482944579228342346, −11.066184285022365393534978702880, −9.932895783767419436310108293927, −9.42990060153657501526022547433, −8.74278585526071210706824609908, −7.9471046809348446002922521624, −6.84756863792429593184455775589, −6.18406267410488840675395281296, −5.23276269928434358442672987252, −3.96897882607996169149444310692, −2.90970375903128995703858788449, −1.792589958712420769600424652849, −0.82551697213888996713905528016, 0.659898963830442043018193778244, 2.05033179497133498590019982136, 3.08629803082390237868503151224, 3.51652826480725480112917752107, 5.34654693469523608091946692016, 6.45113562314434700853540546124, 6.68877739589453766904254176530, 7.83612245057873184519259371722, 8.512424521109773335493496908460, 9.689191410511206788378716470208, 10.261791724820688870078765367926, 10.707230365484158197978085221135, 11.81187207896835631068381562293, 12.61405150039099512462255056597, 13.54625733008808014255816927429, 14.70475391662585669136169687776, 15.23379557816503448543054386300, 16.137434125047721236740480656929, 16.83932631114525132706015061474, 17.63394090535529027083288303182, 18.4923198809944871274232872360, 18.97063624258870291345467251381, 19.62605260408282831440051832781, 20.52902329844636306149077006405, 21.29756061496385438047250339323

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