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Properties

Label 1-88-88.19-r0-0-0
Degree $1$
Conductor $88$
Sign $0.0694 - 0.997i$
Analytic cond. $0.408670$
Root an. cond. $0.408670$
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  + (−0.809 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.0694 - 0.997i$
Analytic conductor: \(0.408670\)
Root analytic conductor: \(0.408670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 88,\ (0:\ ),\ 0.0694 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3993263170 - 0.3724822098i\)
\(L(\frac12)\) \(\approx\) \(0.3993263170 - 0.3724822098i\)
\(L(1)\) \(\approx\) \(0.6511998565 - 0.1691030041i\)
\(L(1)\) \(\approx\) \(0.6511998565 - 0.1691030041i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−30.70893128527336636085071586713, −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