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Properties

Label 1-91-91.3-r1-0-0
Degree $1$
Conductor $91$
Sign $-0.923 - 0.384i$
Analytic cond. $9.77930$
Root an. cond. $9.77930$
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)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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.923 - 0.384i$
Analytic conductor: \(9.77930\)
Root analytic conductor: \(9.77930\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (1:\ ),\ -0.923 - 0.384i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1512255299 - 0.7557925848i\)
\(L(\frac12)\) \(\approx\) \(0.1512255299 - 0.7557925848i\)
\(L(1)\) \(\approx\) \(0.5234534264 - 0.3997815344i\)
\(L(1)\) \(\approx\) \(0.5234534264 - 0.3997815344i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 - T \)
5 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 + (0.5 + 0.866i)T \)
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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

−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