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Properties

Label 1-33e2-1089.70-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.857 + 0.514i$
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  + (−0.851 + 0.524i)2-s + (0.449 − 0.893i)4-s + (−0.999 + 0.0190i)5-s + (0.749 − 0.662i)7-s + (0.0855 + 0.996i)8-s + (0.841 − 0.540i)10-s + (−0.710 − 0.703i)13-s + (−0.290 + 0.956i)14-s + (−0.595 − 0.803i)16-s + (−0.985 − 0.170i)17-s + (0.696 − 0.717i)19-s + (−0.432 + 0.901i)20-s + (−0.995 − 0.0950i)23-s + (0.999 − 0.0380i)25-s + (0.974 + 0.226i)26-s + ⋯
L(s)  = 1  + (−0.851 + 0.524i)2-s + (0.449 − 0.893i)4-s + (−0.999 + 0.0190i)5-s + (0.749 − 0.662i)7-s + (0.0855 + 0.996i)8-s + (0.841 − 0.540i)10-s + (−0.710 − 0.703i)13-s + (−0.290 + 0.956i)14-s + (−0.595 − 0.803i)16-s + (−0.985 − 0.170i)17-s + (0.696 − 0.717i)19-s + (−0.432 + 0.901i)20-s + (−0.995 − 0.0950i)23-s + (0.999 − 0.0380i)25-s + (0.974 + 0.226i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.857 + 0.514i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.857 + 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04379889219 + 0.1580664998i\)
\(L(\frac12)\) \(\approx\) \(0.04379889219 + 0.1580664998i\)
\(L(1)\) \(\approx\) \(0.5130408882 + 0.05419407442i\)
\(L(1)\) \(\approx\) \(0.5130408882 + 0.05419407442i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.851 + 0.524i)T \)
5 \( 1 + (-0.999 + 0.0190i)T \)
7 \( 1 + (0.749 - 0.662i)T \)
13 \( 1 + (-0.710 - 0.703i)T \)
17 \( 1 + (-0.985 - 0.170i)T \)
19 \( 1 + (0.696 - 0.717i)T \)
23 \( 1 + (-0.995 - 0.0950i)T \)
29 \( 1 + (-0.532 + 0.846i)T \)
31 \( 1 + (-0.935 + 0.353i)T \)
37 \( 1 + (-0.362 + 0.931i)T \)
41 \( 1 + (-0.991 + 0.132i)T \)
43 \( 1 + (-0.327 + 0.945i)T \)
47 \( 1 + (-0.179 - 0.983i)T \)
53 \( 1 + (0.993 + 0.113i)T \)
59 \( 1 + (0.380 + 0.924i)T \)
61 \( 1 + (0.879 - 0.475i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (0.897 + 0.441i)T \)
73 \( 1 + (-0.870 - 0.491i)T \)
79 \( 1 + (0.123 + 0.992i)T \)
83 \( 1 + (-0.217 + 0.976i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (-0.999 - 0.0190i)T \)
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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

−20.80377641997424140774014096295, −20.31486684281438146640104850172, −19.46049476658910119685726694488, −18.84293020728421782304376495074, −18.1623310606833992896730249978, −17.388067900394969371783946855485, −16.49462015736637722856448638513, −15.7861155494289295162573599702, −15.05991336524663406154800143337, −14.14491133110946194290878074053, −12.89909689470398339186120961004, −12.00870132684495800529683945105, −11.65106539518712395524105114637, −10.93098261882246386329848451695, −9.88306178784508720963839737812, −9.01636422719610408260755646014, −8.30718588656372398807083458106, −7.60210814191390006207905885137, −6.862206574369134874301108054408, −5.53590330205203851535554487565, −4.32586936175686642708788356075, −3.67765669441812606474014336818, −2.37330335181363501757200844837, −1.71625756713549771800462865601, −0.1013308867946580591579948641, 1.072843447379236064411597906331, 2.28289866075085480634457144398, 3.54918136296306578848941028097, 4.74030208591515717890627248520, 5.310918330088560346671893878861, 6.82765733871529897276809162956, 7.239526045903317273581984552744, 8.09341473924482517350444223749, 8.662176915157585679596103650050, 9.77847250928018722730475476117, 10.61334903216809248012862930770, 11.30355397122084142697111963018, 11.952632172312094452842813039158, 13.213554906866242587870946977852, 14.208431103453233486628180104383, 14.968308168109075933441837394737, 15.53306038313819292487304442192, 16.39177383556942834360289445387, 17.06133412480510761406455889296, 18.02780508375830484796817044359, 18.32791189288068933155705422696, 19.64599103723389665301528351750, 20.0261297603494462735490736523, 20.43464273535355180981196250947, 21.86483557457956748929674431516

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