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Properties

Label 1-28-28.11-r1-0-0
Degree $1$
Conductor $28$
Sign $0.0633 + 0.997i$
Analytic cond. $3.00901$
Root an. cond. $3.00901$
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)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + 13-s − 15-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s + 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s + (−0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + 13-s − 15-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s + 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s + (−0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(3.00901\)
Root analytic conductor: \(3.00901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 28,\ (1:\ ),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.061967590 + 0.9967111576i\)
\(L(\frac12)\) \(\approx\) \(1.061967590 + 0.9967111576i\)
\(L(1)\) \(\approx\) \(1.063864181 + 0.5273864705i\)
\(L(1)\) \(\approx\) \(1.063864181 + 0.5273864705i\)

Euler product

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

−36.86512116938266328637593646975, −35.52027361131090869226211449370, −35.05794896882697380710761176655, −32.98148065530048461482165625489, −31.75724361571701305472450114762, −30.798286354285592283427176207998, −29.47784821121226517950153682328, −28.21598345826257101871560811332, −26.730937700287962518124878042004, −25.20558753545679701253972511291, −24.23089777903988797481216845629, −23.17898693515451491262567120763, −21.12688779466396732603496086942, −19.86647150518276420233711384665, −18.86218370939112193562322334688, −17.25063118957620484497240638305, −15.72570369010992303948758033392, −13.9701381300463337575512171827, −12.785205470326025408407458484945, −11.44929871200261681719221922040, −8.97652629868420324973907632654, −7.95856753893636974200445830183, −6.06990601579822666252741541118, −3.67950794765112727902863483477, −1.20418351389115677729564000182, 2.94219705432514001926433052879, 4.52645948358687546331705549285, 6.915016985178494680949819668177, 8.713323158777394788543792088764, 10.26436547748135958326452301573, 11.55328266174774739909001050851, 13.80075105541795513753832990302, 15.056863961563134658233346206581, 16.04534733587020059955048832507, 17.9478437550643601666056447088, 19.50748006991314000785839278723, 20.63647515783442067335218092056, 22.15369201955020997520079824129, 23.08234097223619139240865730245, 25.1122673083688761595808017840, 26.262991264143702648634788303295, 27.232290245779806271458510157928, 28.44894235023077189071065115139, 30.45265048336728424601549629085, 31.1260890119860589331427840442, 32.704472805597450589035005577067, 33.61549110402238297441508381975, 34.96326685022600165474384514866, 36.43211096391425170139995700551, 38.012056880405991369357370134

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