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  :  $\chi_{89} (49, \cdot )$
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

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

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