Properties

Degree 1
Conductor 89
Sign $0.769 + 0.639i$
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 + (0.755 − 0.654i)3-s + (−0.654 + 0.755i)4-s + (0.959 + 0.281i)5-s + (0.909 + 0.415i)6-s + (0.281 − 0.959i)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 + i·12-s + (−0.755 + 0.654i)13-s + (0.989 − 0.142i)14-s + (0.909 − 0.415i)15-s + (−0.142 − 0.989i)16-s + (−0.415 + 0.909i)17-s + ⋯
L(s,χ)  = 1  + (0.415 + 0.909i)2-s + (0.755 − 0.654i)3-s + (−0.654 + 0.755i)4-s + (0.959 + 0.281i)5-s + (0.909 + 0.415i)6-s + (0.281 − 0.959i)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 + i·12-s + (−0.755 + 0.654i)13-s + (0.989 − 0.142i)14-s + (0.909 − 0.415i)15-s + (−0.142 − 0.989i)16-s + (−0.415 + 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.769 + 0.639i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.769 + 0.639i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $0.769 + 0.639i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (53, \cdot )$
Sato-Tate  :  $\mu(44)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 89,\ (0:\ ),\ 0.769 + 0.639i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.411435152 + 0.5098995591i$
$L(\frac12,\chi)$  $\approx$  $1.411435152 + 0.5098995591i$
$L(\chi,1)$  $\approx$  1.445713982 + 0.4195262532i
$L(1,\chi)$  $\approx$  1.445713982 + 0.4195262532i

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

−30.54252130359318923041406764161, −29.291485737742921053454028728000, −28.4606461377308966670655563392, −27.438095950893663879900192535732, −26.36163785128190655153606195617, −25.03667793893898866343088837223, −24.242502622397990530386237762445, −22.29789210952991059354838638788, −21.82319666157730626880612602654, −20.77771363182433579013978736122, −20.153935271724170774225529300716, −18.68729176389436774431714513980, −17.82344809783457171576685382695, −15.94829116110219762468454208074, −14.86455842787810402918861223738, −13.795937570660653431795830935750, −12.93925842703346345767777405968, −11.48281052433769157442351206638, −10.04711212470243205035310813091, −9.4353664730701720967684326888, −8.17512736305045583992895331149, −5.564863913100771609501037523517, −4.8871400565729498921582688766, −2.96079263373527314565564988319, −2.155222429791525245369835483294, 2.11462035027123270784458606154, 3.77101590867553713427928545261, 5.41277493284578893744855313750, 6.88273508505335137463314773387, 7.58324289028274243528280086045, 8.97554323058272114945333704436, 10.25780747495640310334941781538, 12.384851760131397260995257194733, 13.46558916396753157594279190673, 14.097062254300044398766399888060, 15.01332063006341367649105728319, 16.59183333471468156952737242515, 17.722266096716298185524062891452, 18.39796069191305114026087753935, 20.05949417453839225237806225540, 21.14235805244204075932679984257, 22.22381590961635849851131454044, 23.7281302646049478707525278766, 24.18035277884915411285641007924, 25.42523052686571882754913381670, 26.226380446358526765357055572830, 26.77449353515381062274018695430, 28.85715648315330690909723797487, 29.9058606215198020446871939196, 30.75142235854965365158808019748

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