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Properties

Label 1-532-532.55-r0-0-0
Degree $1$
Conductor $532$
Sign $-0.999 + 0.0158i$
Analytic cond. $2.47059$
Root an. cond. $2.47059$
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  + (0.173 + 0.984i)3-s + (−0.766 + 0.642i)5-s + (−0.939 + 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.173 + 0.984i)13-s + (−0.766 − 0.642i)15-s + (0.939 + 0.342i)17-s + (−0.766 − 0.642i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (−0.939 + 0.342i)29-s + (−0.5 + 0.866i)31-s + (−0.766 + 0.642i)33-s + 37-s − 39-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)3-s + (−0.766 + 0.642i)5-s + (−0.939 + 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.173 + 0.984i)13-s + (−0.766 − 0.642i)15-s + (0.939 + 0.342i)17-s + (−0.766 − 0.642i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (−0.939 + 0.342i)29-s + (−0.5 + 0.866i)31-s + (−0.766 + 0.642i)33-s + 37-s − 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $-0.999 + 0.0158i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ -0.999 + 0.0158i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006797683423 + 0.8601421683i\)
\(L(\frac12)\) \(\approx\) \(0.006797683423 + 0.8601421683i\)
\(L(1)\) \(\approx\) \(0.6940306807 + 0.5337373716i\)
\(L(1)\) \(\approx\) \(0.6940306807 + 0.5337373716i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.173 - 0.984i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−23.21351091503251575954457339161, −22.4518405258423131931306946463, −21.26968126871528350319305745156, −20.1335370001892252164614941934, −19.83513166824399618026913630499, −18.83942073787715915428171884644, −18.20499606219220964797369201873, −17.00436309718069798927520783394, −16.49644910429231724577798144667, −15.29067413406297598600673014025, −14.48148184976354986913945376079, −13.3994505023847550039291083869, −12.817588545983003308464964012354, −11.76918128598471515180548608168, −11.3869543308972132556390800019, −9.84419454143005422017214637877, −8.775817495789725456056920962, −7.94214608758755293968740198637, −7.439950956357909079263862199019, −6.067844660942736234977530174920, −5.34179424454317547327675975336, −3.83530657925233808571176153910, −3.011831835062786945869706969128, −1.522308114670970352994776438789, −0.451924003004351268342329260909, 1.9343505295181912411018098488, 3.2220044689855420116240295821, 4.02862772608302506465583841702, 4.79130192829831369143014770180, 6.13347454699051695009813370817, 7.18307582590583774584088557292, 8.11647421314348149008636081595, 9.19084023551457097009731990571, 9.98886331249713865510462613882, 10.848518778199732390719599155877, 11.715197965577405074443120749739, 12.46529276208171701811013342797, 14.08504112885807962006469381779, 14.612955009793282659993790908093, 15.250798326192444935905421078, 16.29969289212034164198485926455, 16.80773940388985010096062157498, 18.03774966857998305647378722048, 18.972973767886532093941072578879, 19.821760663854199511251566512530, 20.433372254977499385758953660938, 21.54113245721669249371137612681, 22.120982467218387272732796362322, 22.99997353947608291420888830377, 23.59589991821470773367122562069

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