↑

Properties

Label 1-89-89.53-r0-0-0
Degree $1$
Conductor $89$
Sign $0.769 + 0.639i$
Analytic cond. $0.413314$
Root an. cond. $0.413314$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.769 + 0.639i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ 0.769 + 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.411435152 + 0.5098995591i\)
\(L(\frac12)\) \(\approx\) \(1.411435152 + 0.5098995591i\)
\(L(1)\) \(\approx\) \(1.445713982 + 0.4195262532i\)
\(L(1)\) \(\approx\) \(1.445713982 + 0.4195262532i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (0.415 + 0.909i)T \)
3 \( 1 + (0.755 - 0.654i)T \)
5 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (0.281 - 0.959i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (-0.755 + 0.654i)T \)
17 \( 1 + (-0.415 + 0.909i)T \)
19 \( 1 + (0.989 + 0.142i)T \)
23 \( 1 + (-0.989 - 0.142i)T \)
29 \( 1 + (-0.281 + 0.959i)T \)
31 \( 1 + (-0.989 + 0.142i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.755 - 0.654i)T \)
43 \( 1 + (-0.281 - 0.959i)T \)
47 \( 1 + (0.654 - 0.755i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (0.755 + 0.654i)T \)
61 \( 1 + (0.540 + 0.841i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (0.909 + 0.415i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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