Properties

Degree 1
Conductor $ 7 \cdot 13 $
Sign $-0.923 - 0.384i$
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)2-s − 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + 8-s + 9-s − 10-s + 11-s + (0.5 − 0.866i)12-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s − 19-s + (0.5 + 0.866i)20-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + 8-s + 9-s − 10-s + 11-s + (0.5 − 0.866i)12-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s − 19-s + (0.5 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.923 - 0.384i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.923 - 0.384i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(91\)    =    \(7 \cdot 13\)
\( \varepsilon \)  =  $-0.923 - 0.384i$
motivic weight  =  \(0\)
character  :  $\chi_{91} (3, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 91,\ (1:\ ),\ -0.923 - 0.384i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1512255299 - 0.7557925848i$
$L(\frac12,\chi)$  $\approx$  $0.1512255299 - 0.7557925848i$
$L(\chi,1)$  $\approx$  0.5234534264 - 0.3997815344i
$L(1,\chi)$  $\approx$  0.5234534264 - 0.3997815344i

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

−30.26020819263716198251503718578, −29.52209131975747567035149527636, −28.20429814511270278992021366073, −27.52820089356858923524516040959, −26.38791776361962019808846498945, −25.423567147005789338239259539, −24.31494749574185643721424591237, −23.24356437938922612915550902098, −22.4155208346557511213635272859, −21.46574159277190215799967430721, −19.37749679568628657720458280901, −18.56513032450260499824848317740, −17.3579973129817124789236045662, −16.97663340195688245770833628682, −15.478722672070317512553155233749, −14.55174248976097183915867058881, −13.24162128519766141202459668757, −11.54058957708874492838034122484, −10.40752432869681735468911099028, −9.5043805629563773883594769841, −7.73125102796981878987243598106, −6.43001739875837904281724998824, −5.91017388346939354382044741038, −4.20811761443913365093335840779, −1.51850597070835552789696205004, 0.51238739618614739824727418895, 1.81612344955041348547661007999, 4.041935943480720662590243350492, 5.24208522100883392858129576107, 6.85137626998681131772796836786, 8.6072349910705262675799654125, 9.67718348576715315172449182014, 10.766078087090484339003311708227, 12.05025667728532363517472742391, 12.629485794387940303463370577290, 14.00792042605566973772546283491, 16.18625421378666952790054859295, 16.94959098970928487100567695132, 17.75339052628012184958923588972, 18.88379329678289210855001218273, 20.16321163526392838280376555100, 21.1934636881654880772292825015, 22.05934188741325581774618179592, 23.084248348064119651557482638402, 24.46404768304903191402374447628, 25.52313113053089055889227815982, 27.09024111354318011287451735399, 27.849123689443346575775874018029, 28.576915596872486255137071319682, 29.57965127051317828050222396072

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