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Properties

Label 1-89-89.22-r0-0-0
Degree $1$
Conductor $89$
Sign $-0.616 + 0.787i$
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  + (0.415 + 0.909i)2-s + (0.654 + 0.755i)3-s + (−0.654 + 0.755i)4-s + (−0.959 − 0.281i)5-s + (−0.415 + 0.909i)6-s + (0.959 + 0.281i)7-s + (−0.959 − 0.281i)8-s + (−0.142 + 0.989i)9-s + (−0.142 − 0.989i)10-s + (−0.959 + 0.281i)11-s − 12-s + (0.654 + 0.755i)13-s + (0.142 + 0.989i)14-s + (−0.415 − 0.909i)15-s + (−0.142 − 0.989i)16-s + (0.415 − 0.909i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (0.654 + 0.755i)3-s + (−0.654 + 0.755i)4-s + (−0.959 − 0.281i)5-s + (−0.415 + 0.909i)6-s + (0.959 + 0.281i)7-s + (−0.959 − 0.281i)8-s + (−0.142 + 0.989i)9-s + (−0.142 − 0.989i)10-s + (−0.959 + 0.281i)11-s − 12-s + (0.654 + 0.755i)13-s + (0.142 + 0.989i)14-s + (−0.415 − 0.909i)15-s + (−0.142 − 0.989i)16-s + (0.415 − 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.616 + 0.787i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ -0.616 + 0.787i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5369875969 + 1.101981729i\)
\(L(\frac12)\) \(\approx\) \(0.5369875969 + 1.101981729i\)
\(L(1)\) \(\approx\) \(0.9058141913 + 0.8859339252i\)
\(L(1)\) \(\approx\) \(0.9058141913 + 0.8859339252i\)

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 \( 1 + (0.654 + 0.755i)T \)
5 \( 1 + (-0.959 - 0.281i)T \)
7 \( 1 + (0.959 + 0.281i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (0.415 - 0.909i)T \)
19 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (0.959 + 0.281i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 - T \)
41 \( 1 + (0.654 - 0.755i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 + (-0.654 + 0.755i)T \)
53 \( 1 + (-0.654 - 0.755i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (-0.142 - 0.989i)T \)
83 \( 1 + (-0.415 + 0.909i)T \)
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

−30.25687170246624951668766394018, −29.53383191254500195236842588804, −28.08562683579549436511888143275, −27.17063528262302712593731626760, −26.12826680411286444141972666669, −24.5152561982935658568123710727, −23.55929838614161742182442711954, −23.052481128886112131128696295086, −21.221649408192401676938760244897, −20.5277080714896213064533244950, −19.46542719807725894899586898634, −18.62696798460073112836687590717, −17.72799786519799573094640938887, −15.481836845808224281845818441970, −14.58103518972804750447065329113, −13.517848889017461322606145157268, −12.45043432021420966998813015403, −11.3805751662185134308136764205, −10.32203884594553077424877354584, −8.39683966554668317678812677587, −7.76159571167713401224908890141, −5.773172418993071018417521063849, −4.02753409615733175543647485659, −2.94500721261204060193857328848, −1.299838144630741120154368535529, 2.95265619996264712126601673266, 4.46605405070049902943894640178, 5.06866769474062874763196049429, 7.229946601378604080703810054269, 8.2749031853274717139361230323, 9.023816821694283864743054715740, 10.91593536439972926720334916626, 12.24884810224222780285529128211, 13.75332450342663419996881410787, 14.659064830805048349160556444859, 15.764897147904104432422957285592, 16.170458183252492813452298511192, 17.79641232816734966159511122578, 19.02826701236937521216659662847, 20.66432586223943312002175905519, 21.17127380303216051114715687297, 22.55923035018738029925506555453, 23.65361280187381852127966096755, 24.493126603068903648510053501902, 25.69129697438417890663946669049, 26.63322064529138197827280695570, 27.40661030676454772170288212837, 28.35838185336298382479387598401, 30.81300234306621085885872272401, 30.90447710268608166251636027309

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