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Properties

Label 1-89-89.86-r1-0-0
Degree $1$
Conductor $89$
Sign $0.993 + 0.112i$
Analytic cond. $9.56437$
Root an. cond. $9.56437$
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  + (0.415 + 0.909i)2-s + (−0.997 − 0.0713i)3-s + (−0.654 + 0.755i)4-s + (0.281 − 0.959i)5-s + (−0.349 − 0.936i)6-s + (−0.877 + 0.479i)7-s + (−0.959 − 0.281i)8-s + (0.989 + 0.142i)9-s + (0.989 − 0.142i)10-s + (0.959 − 0.281i)11-s + (0.707 − 0.707i)12-s + (0.0713 − 0.997i)13-s + (−0.800 − 0.599i)14-s + (−0.349 + 0.936i)15-s + (−0.142 − 0.989i)16-s + (0.909 + 0.415i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (−0.997 − 0.0713i)3-s + (−0.654 + 0.755i)4-s + (0.281 − 0.959i)5-s + (−0.349 − 0.936i)6-s + (−0.877 + 0.479i)7-s + (−0.959 − 0.281i)8-s + (0.989 + 0.142i)9-s + (0.989 − 0.142i)10-s + (0.959 − 0.281i)11-s + (0.707 − 0.707i)12-s + (0.0713 − 0.997i)13-s + (−0.800 − 0.599i)14-s + (−0.349 + 0.936i)15-s + (−0.142 − 0.989i)16-s + (0.909 + 0.415i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.993 + 0.112i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ 0.993 + 0.112i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.287186569 + 0.07257925097i\)
\(L(\frac12)\) \(\approx\) \(1.287186569 + 0.07257925097i\)
\(L(1)\) \(\approx\) \(0.9365363292 + 0.2293253503i\)
\(L(1)\) \(\approx\) \(0.9365363292 + 0.2293253503i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (0.415 + 0.909i)T \)
3 \( 1 + (-0.997 - 0.0713i)T \)
5 \( 1 + (0.281 - 0.959i)T \)
7 \( 1 + (-0.877 + 0.479i)T \)
11 \( 1 + (0.959 - 0.281i)T \)
13 \( 1 + (0.0713 - 0.997i)T \)
17 \( 1 + (0.909 + 0.415i)T \)
19 \( 1 + (0.800 - 0.599i)T \)
23 \( 1 + (0.599 + 0.800i)T \)
29 \( 1 + (0.479 + 0.877i)T \)
31 \( 1 + (0.599 - 0.800i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.0713 - 0.997i)T \)
43 \( 1 + (0.479 - 0.877i)T \)
47 \( 1 + (-0.755 - 0.654i)T \)
53 \( 1 + (0.755 - 0.654i)T \)
59 \( 1 + (0.997 - 0.0713i)T \)
61 \( 1 + (-0.212 + 0.977i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.281 - 0.959i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (-0.989 + 0.142i)T \)
83 \( 1 + (0.349 + 0.936i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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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

−29.999503095292096521165161505240, −29.28352854289622991273797268780, −28.52554575300039069966303854943, −27.253653862774732657183743158889, −26.435246230920120893215867831717, −24.76351126323300444644853241758, −23.15384265568270033621885688173, −22.82794095294987833740528248099, −21.91972377330894647315953501404, −20.88999484490119801057444594593, −19.31937250708993407219933363475, −18.638323187422628463474654509948, −17.4245444647473014125796877058, −16.21212440748705001950533452957, −14.6048098954963644325778190542, −13.6406376688043782254866043986, −12.235009392823113509221450309560, −11.43960750430184847765820064148, −10.1932264053406239852090069540, −9.5946844210075829398429260283, −6.890286911971764278760756082719, −6.07365765339053998064335529395, −4.4357383026491742357024202563, −3.17593449359751002685011321563, −1.23398275626777150447937631923, 0.72643145848883250358934971689, 3.62646727490773945977489206329, 5.25512919958767577895493271078, 5.859113264117447127070424139145, 7.15057781433577855459518425421, 8.75504709757822555083262515956, 9.86598333856154096727890194708, 11.89553447544808374622264401500, 12.65155650186180526781366529048, 13.598772397839825576098168620772, 15.34950611329169739672101387267, 16.2432244402608751521754186504, 17.04797333860093780057522454854, 17.9210032596688480210315865353, 19.38460675715778195269063133493, 21.12068424861546887097844404355, 22.13177928428780951744189352169, 22.85123622945486361242720936028, 24.018758303894207253459663251202, 24.84771560013435940707947162365, 25.68192575218153677398114164295, 27.34866046639650693555069712045, 28.01180804743773571956448456906, 29.24224119017397071361671398791, 30.220168002777672591256437565576

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