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Properties

Label 1-89-89.8-r0-0-0
Degree $1$
Conductor $89$
Sign $0.980 - 0.195i$
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  + (−0.142 − 0.989i)2-s + (−0.959 − 0.281i)3-s + (−0.959 + 0.281i)4-s + (0.415 + 0.909i)5-s + (−0.142 + 0.989i)6-s + (0.415 + 0.909i)7-s + (0.415 + 0.909i)8-s + (0.841 + 0.540i)9-s + (0.841 − 0.540i)10-s + (0.415 − 0.909i)11-s + 12-s + (−0.959 − 0.281i)13-s + (0.841 − 0.540i)14-s + (−0.142 − 0.989i)15-s + (0.841 − 0.540i)16-s + (−0.142 + 0.989i)17-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s + (−0.959 − 0.281i)3-s + (−0.959 + 0.281i)4-s + (0.415 + 0.909i)5-s + (−0.142 + 0.989i)6-s + (0.415 + 0.909i)7-s + (0.415 + 0.909i)8-s + (0.841 + 0.540i)9-s + (0.841 − 0.540i)10-s + (0.415 − 0.909i)11-s + 12-s + (−0.959 − 0.281i)13-s + (0.841 − 0.540i)14-s + (−0.142 − 0.989i)15-s + (0.841 − 0.540i)16-s + (−0.142 + 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ 0.980 - 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6791497674 - 0.06692960138i\)
\(L(\frac12)\) \(\approx\) \(0.6791497674 - 0.06692960138i\)
\(L(1)\) \(\approx\) \(0.7370647878 - 0.1631927868i\)
\(L(1)\) \(\approx\) \(0.7370647878 - 0.1631927868i\)

Euler product

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

−30.63006775939404166348250896897, −29.15328640437406204892868548217, −28.33867218404258449210199754444, −27.25758240622550697798001340779, −26.59630409523879891589263032867, −24.95980799569405546981598813413, −24.30882625961822250974488303834, −23.24529403268961740751166712476, −22.421735747523857543735402180418, −21.128286636500580039585050698134, −19.87205124891631869537909330968, −18.06186199364868406798615994881, −17.271118977221128415480802034438, −16.71297591747085439220188290789, −15.59369488953458570745907716444, −14.233952104331098475713360777034, −13.04366448555428459539872217563, −11.78880481811048534527166903904, −10.05240272681308335979205807981, −9.32516631238324600740130218195, −7.54599314716454736613194901944, −6.55112663407281487575407268837, −4.85092331871173585457982901544, −4.63459045388468678780012786261, −0.99942414189777924611223691702, 1.6560026624364633336959113780, 3.12458831777370659578238631493, 5.048884785606512708442533771965, 6.15859282659899275674822797071, 7.8766955799275735100365710250, 9.50302511243374269474314745871, 10.67724839427371382955122603901, 11.520760142072593793447010643268, 12.45486425282055737562657188984, 13.77241511728383476794314526137, 15.0243433602302244632760454477, 16.88986851615408421209715412494, 17.79839320792667676019168481877, 18.64735984630307273324014578191, 19.42920505180046106470336944516, 21.31861157795657811029877491870, 21.9386722176054408537923088651, 22.59320914093869702433562257133, 23.96900413812397740185154217473, 25.18733476570176343561310547345, 26.81996516736978215104220511057, 27.43405500954556939703755692898, 28.69481834531185334368777982763, 29.35683247396687602091114319777, 30.23701296243032623983969385220

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