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Properties

Label 1-224-224.27-r0-0-0
Degree $1$
Conductor $224$
Sign $0.555 + 0.831i$
Analytic cond. $1.04025$
Root an. cond. $1.04025$
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  + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)5-s − i·9-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s + 15-s + 17-s + (0.707 − 0.707i)19-s + i·23-s + i·25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s + 31-s − 33-s + (0.707 + 0.707i)37-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)5-s − i·9-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s + 15-s + 17-s + (0.707 − 0.707i)19-s + i·23-s + i·25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s + 31-s − 33-s + (0.707 + 0.707i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.555 + 0.831i$
Analytic conductor: \(1.04025\)
Root analytic conductor: \(1.04025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 224,\ (0:\ ),\ 0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7041456197 + 0.3763736750i\)
\(L(\frac12)\) \(\approx\) \(0.7041456197 + 0.3763736750i\)
\(L(1)\) \(\approx\) \(0.7714055603 + 0.1752713426i\)
\(L(1)\) \(\approx\) \(0.7714055603 + 0.1752713426i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + iT \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (0.707 - 0.707i)T \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 - iT \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 - T \)
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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

−26.50219192833942226409911145889, −25.08129808316934753868074335315, −24.46467264713009920602524395490, −23.43055173901658357710228260949, −22.56924441160975745626127409787, −22.13374236347352256584190130332, −20.6150890266785909456986798433, −19.31432200221626655635355342497, −18.92746620204390747422840423261, −17.8884540179846142153017257940, −16.87413435734086149860473066504, −16.03311379769907920229508251144, −14.71486964392655910573279799789, −13.926050593516021841048649078714, −12.51489246678784335251515615646, −11.86449758606426675890430052674, −10.94288360585096408252238287322, −9.94709898089309447863058336306, −8.1810537265539234573700034268, −7.465792559537697753929223227086, −6.386634657177452339126750745, −5.41060242718449337447479859238, −3.86736150936024432793533329182, −2.57603458010095088720656535270, −0.78537340950087120718937157029, 1.22760792963433838039895572634, 3.39345166518824661224970858399, 4.506217389279392412442867126446, 5.20810539255516691076234871613, 6.6659496829359099577923325169, 7.78397827994218375366280185135, 9.30055695772888926750721959921, 9.77492005946730277289528905007, 11.416391492180130359241381282598, 11.84153729245017915650825285670, 12.84943829578295235630834618375, 14.44026538242305198771851465444, 15.304856209399394120011497354548, 16.2933661297669095334041998973, 16.94252899145532130455660876383, 17.851022222345333118095634912574, 19.27014424221249755313232098758, 20.10630363497358275544415953452, 21.05174048390574372163184524387, 21.98260957495937713833919971567, 22.90284531407045992258106529447, 23.70628537413281900039785336586, 24.534083245270655381119201385910, 25.79917401321993203636115334236, 26.85941261109527943086457391902

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