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Properties

Label 1-89-89.28-r1-0-0
Degree $1$
Conductor $89$
Sign $0.911 - 0.411i$
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  + (−0.959 − 0.281i)2-s + (−0.212 + 0.977i)3-s + (0.841 + 0.540i)4-s + (0.755 − 0.654i)5-s + (0.479 − 0.877i)6-s + (0.997 + 0.0713i)7-s + (−0.654 − 0.755i)8-s + (−0.909 − 0.415i)9-s + (−0.909 + 0.415i)10-s + (0.654 − 0.755i)11-s + (−0.707 + 0.707i)12-s + (−0.977 − 0.212i)13-s + (−0.936 − 0.349i)14-s + (0.479 + 0.877i)15-s + (0.415 + 0.909i)16-s + (−0.281 − 0.959i)17-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (−0.212 + 0.977i)3-s + (0.841 + 0.540i)4-s + (0.755 − 0.654i)5-s + (0.479 − 0.877i)6-s + (0.997 + 0.0713i)7-s + (−0.654 − 0.755i)8-s + (−0.909 − 0.415i)9-s + (−0.909 + 0.415i)10-s + (0.654 − 0.755i)11-s + (−0.707 + 0.707i)12-s + (−0.977 − 0.212i)13-s + (−0.936 − 0.349i)14-s + (0.479 + 0.877i)15-s + (0.415 + 0.909i)16-s + (−0.281 − 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ 0.911 - 0.411i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.201367450 - 0.2589546009i\)
\(L(\frac12)\) \(\approx\) \(1.201367450 - 0.2589546009i\)
\(L(1)\) \(\approx\) \(0.8649796237 - 0.04206705749i\)
\(L(1)\) \(\approx\) \(0.8649796237 - 0.04206705749i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (-0.959 - 0.281i)T \)
3 \( 1 + (-0.212 + 0.977i)T \)
5 \( 1 + (0.755 - 0.654i)T \)
7 \( 1 + (0.997 + 0.0713i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (-0.977 - 0.212i)T \)
17 \( 1 + (-0.281 - 0.959i)T \)
19 \( 1 + (0.936 - 0.349i)T \)
23 \( 1 + (0.349 + 0.936i)T \)
29 \( 1 + (0.0713 - 0.997i)T \)
31 \( 1 + (0.349 - 0.936i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.977 - 0.212i)T \)
43 \( 1 + (0.0713 + 0.997i)T \)
47 \( 1 + (-0.540 + 0.841i)T \)
53 \( 1 + (0.540 + 0.841i)T \)
59 \( 1 + (0.212 + 0.977i)T \)
61 \( 1 + (-0.800 - 0.599i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.755 - 0.654i)T \)
73 \( 1 + (-0.415 - 0.909i)T \)
79 \( 1 + (0.909 - 0.415i)T \)
83 \( 1 + (-0.479 + 0.877i)T \)
97 \( 1 + (-0.654 - 0.755i)T \)
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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

−30.19376148307958166916548724409, −29.18026692887440928358675420154, −28.36552645897068162703755374359, −27.10030808476488001485160299105, −26.09088595862583153563864652019, −24.919710253578401108322582501471, −24.48029525962430029757328820522, −23.18167068846686080522998267252, −21.821871802833340133206492141734, −20.339813483850745394888936544778, −19.32979491418074767835100658814, −18.14413023957308429477707393105, −17.633435042458801747369403244518, −16.78484142736907243580569548300, −14.73171622038616386427962068397, −14.28505128769356726835984161206, −12.41611974706032615400330336113, −11.2713213658337787359252675929, −10.16804135097775367120922573754, −8.74759037294301315834189644010, −7.416902030527137404826765161107, −6.66961610596499015510774545774, −5.31970480453072962182874402865, −2.35783504203768932544779590336, −1.39498532730516997691634433302, 0.89979089810158511031411917582, 2.71015848476210799892755930080, 4.61815591549117529872395308431, 5.9030493496686325056489147028, 7.82041545036125390269234642892, 9.16390837202192692012439548008, 9.70126411529674544487785124436, 11.1791373709872158986685496589, 11.888919116143199503523748570950, 13.79765411992024811406574114660, 15.18018931083309481207465092880, 16.415529773463466785635402772512, 17.23605947719368919240322548129, 17.95607331460038727279438417353, 19.67042879650907888230273542000, 20.63889400482252870602434299413, 21.386981112251812869393307868657, 22.25744056137666303669987329341, 24.350624157896617173949961726058, 24.939590093958396994938831522680, 26.36319571846278759948223968752, 27.23448117891687051400875250199, 27.87099067111954279053477400983, 28.995371832275926583350770485894, 29.71451921608896378905548071013

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