Properties

Degree 1
Conductor 89
Sign $-0.616 - 0.787i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $-0.616 - 0.787i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (85, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 89,\ (0:\ ),\ -0.616 - 0.787i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5369875969 - 1.101981729i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5369875969 - 1.101981729i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9058141913 - 0.8859339252i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9058141913 - 0.8859339252i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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