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Properties

Label 1-89-89.34-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

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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: $\chi_{89} (34, \cdot )$
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 \( 1 + iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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