Properties

Degree 1
Conductor $ 2^{3} \cdot 11 $
Sign $0.0694 - 0.997i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.809 + 0.587i)3-s + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)15-s + (−0.309 − 0.951i)17-s + (0.809 − 0.587i)19-s + 21-s − 23-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (−0.809 − 0.587i)29-s + (−0.309 + 0.951i)31-s + (−0.309 + 0.951i)35-s + ⋯
L(s,χ)  = 1  + (−0.809 + 0.587i)3-s + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)15-s + (−0.309 − 0.951i)17-s + (0.809 − 0.587i)19-s + 21-s − 23-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (−0.809 − 0.587i)29-s + (−0.309 + 0.951i)31-s + (−0.309 + 0.951i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0694 - 0.997i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0694 - 0.997i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(88\)    =    \(2^{3} \cdot 11\)
\( \varepsilon \)  =  $0.0694 - 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{88} (19, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 88,\ (0:\ ),\ 0.0694 - 0.997i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.3993263170 - 0.3724822098i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.3993263170 - 0.3724822098i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6511998565 - 0.1691030041i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6511998565 - 0.1691030041i\)

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

−30.70893128527336636085071586713, −29.73840506480344196746090531603, −28.77019452360659402392553364447, −27.95035823637056982002728213219, −26.51980993386904844611656024126, −25.63296929619169314578889525074, −24.30690786395969530056456566479, −23.33299980424584101588597259494, −22.34838352224467146167472126830, −21.72037190153563865092499952300, −19.74478703322054946154751739927, −18.74639336295990101897218168864, −18.182297020305848063206113976867, −16.68470150962378205705302441197, −15.73503815556105305394697873528, −14.33958443889979078147039412861, −13.01391950443119377870755231272, −11.89742657093131609595517938831, −10.97502698410018377562435467139, −9.65545083914649846185624880254, −7.85754244879477082874470058159, −6.61033101181637062184098660776, −5.837641127410206019420611613822, −3.8325979303635796474161533732, −2.10579865845431469113989746090, 0.64820396794601972718801604188, 3.490087663771641315866403262917, 4.74351485585035498836217597940, 5.91354156194166474795720552202, 7.43718119626603726998764887568, 9.12338134511641725793708558653, 10.105392733237627371851516793084, 11.40979936600001759350941875620, 12.51669672961695940352484854460, 13.58206509377565419165009291057, 15.533727943879889723251744949625, 16.13624430958668194312003575203, 17.10454449068520966488626544543, 18.2186459006013946919096410136, 19.97948368370428560071277490483, 20.51275472771878567823296017407, 21.984685361679824037337430526045, 22.88687695821551701342466745430, 23.76190095950076553155583017055, 24.95126818769146770010766919196, 26.36022141471776852090606573934, 27.31559330771876736912893266293, 28.28250985061461518310673051557, 29.03949374369125982664707195113, 30.10478521302104464781812904979

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