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Properties

Label 1-89-89.73-r0-0-0
Degree $1$
Conductor $89$
Sign $0.828 + 0.559i$
Analytic cond. $0.413314$
Root an. cond. $0.413314$
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  + (−0.654 − 0.755i)2-s + (0.142 + 0.989i)3-s + (−0.142 + 0.989i)4-s + (0.841 − 0.540i)5-s + (0.654 − 0.755i)6-s + (−0.841 + 0.540i)7-s + (0.841 − 0.540i)8-s + (−0.959 + 0.281i)9-s + (−0.959 − 0.281i)10-s + (0.841 + 0.540i)11-s − 12-s + (0.142 + 0.989i)13-s + (0.959 + 0.281i)14-s + (0.654 + 0.755i)15-s + (−0.959 − 0.281i)16-s + (−0.654 + 0.755i)17-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (0.142 + 0.989i)3-s + (−0.142 + 0.989i)4-s + (0.841 − 0.540i)5-s + (0.654 − 0.755i)6-s + (−0.841 + 0.540i)7-s + (0.841 − 0.540i)8-s + (−0.959 + 0.281i)9-s + (−0.959 − 0.281i)10-s + (0.841 + 0.540i)11-s − 12-s + (0.142 + 0.989i)13-s + (0.959 + 0.281i)14-s + (0.654 + 0.755i)15-s + (−0.959 − 0.281i)16-s + (−0.654 + 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.828 + 0.559i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ 0.828 + 0.559i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7408453436 + 0.2267493548i\)
\(L(\frac12)\) \(\approx\) \(0.7408453436 + 0.2267493548i\)
\(L(1)\) \(\approx\) \(0.8270794115 + 0.08541632863i\)
\(L(1)\) \(\approx\) \(0.8270794115 + 0.08541632863i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (-0.654 - 0.755i)T \)
3 \( 1 + (0.142 + 0.989i)T \)
5 \( 1 + (0.841 - 0.540i)T \)
7 \( 1 + (-0.841 + 0.540i)T \)
11 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (0.142 + 0.989i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (-0.841 + 0.540i)T \)
31 \( 1 + (0.959 + 0.281i)T \)
37 \( 1 - T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (-0.142 + 0.989i)T \)
53 \( 1 + (-0.142 - 0.989i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (-0.415 - 0.909i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (-0.959 - 0.281i)T \)
79 \( 1 + (-0.959 - 0.281i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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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

−29.984530499272821714744813665888, −29.455550695950466359028858530090, −28.486315257456961854561724254890, −26.8904923046628300374363676995, −26.112970406920744203120624357, −24.97126160210001190436026043860, −24.67314486498279430555470757209, −22.99190959239972291016091192593, −22.5041057224778091315438246381, −20.34526121238844560993175021966, −19.370710179717695766599292437, −18.4307439503235823570710764216, −17.523417887200316055121680716267, −16.644945663869056884080064462731, −15.10504042916218263886366399157, −13.82439888410084617617856060903, −13.32266088750191259216040673548, −11.35000727479008378574660366013, −9.97533263749821370271363791262, −8.93114298447239161745994988728, −7.42327763207277984661889081609, −6.58133907819092867659464134479, −5.65415202414930474332181310113, −2.983653497166481643039095347095, −1.13286135266071606378140240934, 1.95384089292660696020859150327, 3.461027224960302856570702415624, 4.84041715654836638882570186821, 6.595726183195206574839996975, 8.8581990949102382775601953482, 9.23542388604286427817961006572, 10.21928061826296350733363255627, 11.588273454309686780988597500998, 12.78035832282441077348818977062, 14.04394418838784413465756059347, 15.69165124074613288521849342142, 16.73879323168550094788055111989, 17.490048718219708097525144013420, 19.0221458067877892971524620868, 20.034550280018908524933520317367, 20.97158520851524152536069227445, 21.909928061441614454883600629230, 22.5280698177456451243666207193, 24.735239698484256999625751517898, 25.76602733467920225766672076507, 26.394334193468699577196993586340, 27.72491969752919101715851572617, 28.59202300548651743312138810887, 29.02178870846217481933745871919, 30.62553005693433293689384199778

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