Properties

Degree 1
Conductor 89
Sign $0.911 - 0.411i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.959 − 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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $0.911 - 0.411i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (28, \cdot )$
Sato-Tate  :  $\mu(88)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 89,\ (1:\ ),\ 0.911 - 0.411i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.201367450 - 0.2589546009i$
$L(\frac12,\chi)$  $\approx$  $1.201367450 - 0.2589546009i$
$L(\chi,1)$  $\approx$  0.8649796237 - 0.04206705749i
$L(1,\chi)$  $\approx$  0.8649796237 - 0.04206705749i

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

−30.19376148307958166916548724409, −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