Properties

Degree 1
Conductor 89
Sign $0.644 + 0.764i$
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 + (0.540 + 0.841i)3-s + (0.841 + 0.540i)4-s + (0.654 + 0.755i)5-s + (−0.281 − 0.959i)6-s + (0.755 − 0.654i)7-s + (−0.654 − 0.755i)8-s + (−0.415 + 0.909i)9-s + (−0.415 − 0.909i)10-s + (−0.654 + 0.755i)11-s + i·12-s + (−0.540 − 0.841i)13-s + (−0.909 + 0.415i)14-s + (−0.281 + 0.959i)15-s + (0.415 + 0.909i)16-s + (0.959 − 0.281i)17-s + ⋯
L(s,χ)  = 1  + (−0.959 − 0.281i)2-s + (0.540 + 0.841i)3-s + (0.841 + 0.540i)4-s + (0.654 + 0.755i)5-s + (−0.281 − 0.959i)6-s + (0.755 − 0.654i)7-s + (−0.654 − 0.755i)8-s + (−0.415 + 0.909i)9-s + (−0.415 − 0.909i)10-s + (−0.654 + 0.755i)11-s + i·12-s + (−0.540 − 0.841i)13-s + (−0.909 + 0.415i)14-s + (−0.281 + 0.959i)15-s + (0.415 + 0.909i)16-s + (0.959 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.644 + 0.764i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.644 + 0.764i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $0.644 + 0.764i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (20, \cdot )$
Sato-Tate  :  $\mu(44)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 89,\ (0:\ ),\ 0.644 + 0.764i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7832647388 + 0.3640741605i$
$L(\frac12,\chi)$  $\approx$  $0.7832647388 + 0.3640741605i$
$L(\chi,1)$  $\approx$  0.8724062558 + 0.2307338927i
$L(1,\chi)$  $\approx$  0.8724062558 + 0.2307338927i

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

−30.09461005689675254859615222086, −29.140841225855586269316079280209, −28.41673226819460179036646953657, −27.16429411924707487166110534136, −25.989179368603373774640270595114, −25.09681968534921478262746192469, −24.3458399721870803555790764093, −23.67004908607635269338282909191, −21.209127906193883425928487937609, −20.80320321516436715598886461358, −19.20031965698901900946220408563, −18.66006931339693122553868361609, −17.4609322058762362864581308885, −16.63686697030834683750961532741, −15.046865693874670683239753245592, −14.0720054202244312007786359028, −12.63642620626927584671657130263, −11.50628231311443655904586921530, −9.83433083317842519503812875060, −8.61849149739960086130657352117, −8.08765492480206546355736025261, −6.48848293971930584134433718334, −5.3159603930604495992139735159, −2.50576069255957969189034257257, −1.404254545389082849124713107627, 2.090787036141883966917376810047, 3.28408229321167864819200028942, 5.133521344850328819266289679492, 7.16924363807255495406753711034, 8.11543924516714901748812069046, 9.646456858570780536483312739074, 10.341596284021097538779701428042, 11.1702943943639361385573111023, 13.0701508769595979945463976243, 14.60595887624910534419002817679, 15.322715378353767470270607107532, 16.899533441566101589725651582152, 17.63082057910353798037119506812, 18.84002180025377246724376616538, 20.0984155281977287977329839984, 20.91569853412031181251101969751, 21.68595523583293651862245850091, 23.10166836735379514160197731021, 24.942335524951793006563412848806, 25.70257933564481451389604072306, 26.56612522608644914496134325843, 27.36750380646756417458618873139, 28.28072242198335568144379221949, 29.75638331636278612561632197793, 30.31022566967857003349237511042

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