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Properties

Label 1-28-28.3-r0-0-0
Degree $1$
Conductor $28$
Sign $0.605 - 0.795i$
Analytic cond. $0.130031$
Root an. cond. $0.130031$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(0.130031\)
Root analytic conductor: \(0.130031\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 28,\ (0:\ ),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6024857757 - 0.2986686198i\)
\(L(\frac12)\) \(\approx\) \(0.6024857757 - 0.2986686198i\)
\(L(1)\) \(\approx\) \(0.8272499164 - 0.2685074801i\)
\(L(1)\) \(\approx\) \(0.8272499164 - 0.2685074801i\)

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

−37.90490365720337567833703476823, −36.78694257822323976854449062769, −34.78617442325068261563198949791, −34.06310863748782348826866600632, −32.87306776536224552253934478215, −31.75307586653671109167324601716, −29.924728159971386673695517524619, −29.03229979414819473139628004895, −27.39004370038808187134093669404, −26.59554445936726606150763972526, −25.16076152890179732894662818403, −23.35211512927444184464274455383, −22.06331783005413527696043861485, −21.381734511895633760502523152499, −19.51938201887740370352982841590, −17.87467441006700468299801803899, −16.708192258780626926637165157904, −15.179005396573225630603026027661, −13.96407991421429788322836586322, −11.74999424202570888718609457837, −10.51423245698153921537833739077, −9.23360735363388848724695713493, −6.794214378265738185173637197862, −5.219666932069924128995417747349, −3.168742768140166422476904919, 1.79104060299590196928243808855, 4.90145218572609997721407147630, 6.48949348354827902421124253680, 8.20025188815811285412341405267, 10.035048155259889179243112063613, 12.08780839164623111600817757729, 12.88497151683190396572405260016, 14.53767518302005912557469765011, 16.74918897964851249538494892171, 17.45111309920348937368431191973, 19.05214065294191287469801220805, 20.39283486937043947183748069495, 22.02477477575038040580831517844, 23.41929007401572973809577896965, 24.61480591540406694707848574615, 25.492121781285813810652627990171, 27.59488233323263092361840342523, 28.68629501827149143803670306400, 29.67465296411982472065265146399, 30.97537086303880087875832717093, 32.4409782555275931724620137368, 33.72917690559028311256085310793, 35.03367287706022271367390804396, 36.17490121615785099863749800058, 36.934593998166979214811617900949

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