↑

Properties

Label 1-89-89.55-r0-0-0
Degree $1$
Conductor $89$
Sign $0.484 - 0.874i$
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  + 2-s − i·3-s + 4-s − 5-s − i·6-s − i·7-s + 8-s − 9-s − 10-s + 11-s − i·12-s + i·13-s − i·14-s + i·15-s + 16-s − 17-s + ⋯
L(s)  = 1  + 2-s − i·3-s + 4-s − 5-s − i·6-s − i·7-s + 8-s − 9-s − 10-s + 11-s − i·12-s + i·13-s − i·14-s + i·15-s + 16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.484 - 0.874i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ 0.484 - 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.324926898 - 0.7804835063i\)
\(L(\frac12)\) \(\approx\) \(1.324926898 - 0.7804835063i\)
\(L(1)\) \(\approx\) \(1.458930928 - 0.5361010417i\)
\(L(1)\) \(\approx\) \(1.458930928 - 0.5361010417i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 - iT \)
19 \( 1 - iT \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 - iT \)
43 \( 1 + iT \)
47 \( 1 - iT \)
53 \( 1 + iT \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + iT \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + T \)
97 \( 1 - iT \)
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

−31.027361342133376036253146188342, −29.957957018950188288737441916178, −28.354655671660951866265094936274, −27.797248870510678532264831105912, −26.5185998523721547251219847123, −25.23619957009223231068896614787, −24.30531408254983922627821949603, −22.89209869300491176415927137603, −22.289022433166629384820523559825, −21.392116987238389511617637619071, −20.08218993908565204338371687636, −19.48536347401554739706442737574, −17.4365568199838830922652930190, −15.98921200670468247754884149546, −15.36378186383131340059075611938, −14.68331326375918010396058374346, −13.03908073151316568514997447617, −11.6474568207043950457341890063, −11.181348007350197762536313777093, −9.40911372682890798580170218377, −8.02359385676240097161956022844, −6.29337903453591675290385538101, −4.99738269249039486316128482560, −3.92246409257427397426013719061, −2.75482227101391584300687549320, 1.57404352839967406894521973360, 3.46877851821792659348806544178, 4.554613943933911784643158574603, 6.57237050077876382733769496013, 7.09435151499964120408591342888, 8.47585384910133472899248660900, 10.81821546780621806039247161320, 11.78106077376468305004864166793, 12.62584807429615109525581777958, 13.938594498176910523354785030711, 14.58218709958702500990474974055, 16.21793304053621051478655555198, 17.0817526168963492892882222521, 18.86950959229036022390400049364, 19.80393065542177522264588424815, 20.46274568972432433332176462551, 22.25078226942616270084308110695, 23.09335847704627941105054773056, 23.92034326078899276213011734460, 24.58108101195344141033398302890, 25.903670485465215074571079820126, 27.133569396121674944716721059154, 28.7281983788327584663659934410, 29.54887026476552853211386055622, 30.68456949914262977715167037849

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