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Properties

Label 1-89-89.78-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: $\chi_{89} (78, \cdot )$
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

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

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