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Properties

Label 1-1872-1872.1211-r1-0-0
Degree $1$
Conductor $1872$
Sign $0.970 - 0.241i$
Analytic cond. $201.174$
Root an. cond. $201.174$
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  + (0.5 + 0.866i)5-s + i·7-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + 23-s + (−0.5 + 0.866i)25-s + (0.866 − 0.5i)29-s + (0.866 − 0.5i)31-s + (−0.866 + 0.5i)35-s + (0.5 − 0.866i)37-s − i·41-s − i·43-s + (−0.866 − 0.5i)47-s − 49-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + i·7-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + 23-s + (−0.5 + 0.866i)25-s + (0.866 − 0.5i)29-s + (0.866 − 0.5i)31-s + (−0.866 + 0.5i)35-s + (0.5 − 0.866i)37-s − i·41-s − i·43-s + (−0.866 − 0.5i)47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.970 - 0.241i$
Analytic conductor: \(201.174\)
Root analytic conductor: \(201.174\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (1211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1872,\ (1:\ ),\ 0.970 - 0.241i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.330497723 - 0.2856880148i\)
\(L(\frac12)\) \(\approx\) \(2.330497723 - 0.2856880148i\)
\(L(1)\) \(\approx\) \(1.204231992 + 0.1893028826i\)
\(L(1)\) \(\approx\) \(1.204231992 + 0.1893028826i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + iT \)
11 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - iT \)
43 \( 1 - iT \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 - iT \)
67 \( 1 + T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 - iT \)
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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

−19.82350500552764999610254913382, −19.48641781562457773189517426661, −18.48432449163821470277971567371, −17.4960947434082381573144192294, −16.94329009753430569427346125214, −16.56796925955981507008088086197, −15.668361752281231753683802971188, −14.639769250921214547997334314832, −13.94242082083858942014989734097, −13.30481530100670430123022014269, −12.68542223696697770102148281820, −11.8032163475368387643454823601, −10.85733609607952791055614938108, −10.25827015496310056563109329605, −9.42142480451210699835368475630, −8.46885451502038491920634988142, −8.14197480715818317785034355783, −6.77775262533757292544592222745, −6.287714356312929098294178778807, −5.28868223054680382932024055981, −4.4191851193626055191706214094, −3.763824170256310228922344987911, −2.661818237922980630419373270066, −1.29928980328550317689348686134, −0.99308875508269452661388611353, 0.461826765683096052538330813426, 1.962136611946939016986170095903, 2.45893983948561245380172540680, 3.30201205779842435940614562711, 4.571394451817404508016071589086, 5.20872396009536819214095455857, 6.388953525358784788139622193295, 6.68618876205579985459773960727, 7.65508892853469008216210268281, 8.73893681205243074192224259950, 9.434591433417541421628604319555, 9.98141228254069147986466127837, 11.120352828834510939470588525181, 11.52819067968446397909658243257, 12.50007317036870269214652572054, 13.22899832142180064847738049629, 14.07592105657441335881069337516, 14.85502383522669084869115013292, 15.31560908848163240694387412102, 16.057000310693732745968339192340, 17.29723864409200216938568016245, 17.684031673312673293691361574319, 18.36512763917157668007642334924, 19.157717738008517062988802072316, 19.69664937813319085448128119389

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