Properties

Degree 1
Conductor $ 2^{2} \cdot 7 $
Sign $0.605 - 0.795i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(28\)    =    \(2^{2} \cdot 7\)
\( \varepsilon \)  =  $0.605 - 0.795i$
motivic weight  =  \(0\)
character  :  $\chi_{28} (3, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 28,\ (0:\ ),\ 0.605 - 0.795i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6024857757 - 0.2986686198i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6024857757 - 0.2986686198i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8272499164 - 0.2685074801i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8272499164 - 0.2685074801i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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