Properties

Label 1-89-89.41-r1-0-0
Degree $1$
Conductor $89$
Sign $-0.496 + 0.867i$
Analytic cond. $9.56437$
Root an. cond. $9.56437$
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  + (0.415 − 0.909i)2-s + (0.0713 + 0.997i)3-s + (−0.654 − 0.755i)4-s + (−0.281 − 0.959i)5-s + (0.936 + 0.349i)6-s + (−0.479 + 0.877i)7-s + (−0.959 + 0.281i)8-s + (−0.989 + 0.142i)9-s + (−0.989 − 0.142i)10-s + (0.959 + 0.281i)11-s + (0.707 − 0.707i)12-s + (−0.997 + 0.0713i)13-s + (0.599 + 0.800i)14-s + (0.936 − 0.349i)15-s + (−0.142 + 0.989i)16-s + (−0.909 + 0.415i)17-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (0.0713 + 0.997i)3-s + (−0.654 − 0.755i)4-s + (−0.281 − 0.959i)5-s + (0.936 + 0.349i)6-s + (−0.479 + 0.877i)7-s + (−0.959 + 0.281i)8-s + (−0.989 + 0.142i)9-s + (−0.989 − 0.142i)10-s + (0.959 + 0.281i)11-s + (0.707 − 0.707i)12-s + (−0.997 + 0.0713i)13-s + (0.599 + 0.800i)14-s + (0.936 − 0.349i)15-s + (−0.142 + 0.989i)16-s + (−0.909 + 0.415i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.496 + 0.867i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ -0.496 + 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1831816354 + 0.3158812837i\)
\(L(\frac12)\) \(\approx\) \(0.1831816354 + 0.3158812837i\)
\(L(1)\) \(\approx\) \(0.8001618971 - 0.1188056122i\)
\(L(1)\) \(\approx\) \(0.8001618971 - 0.1188056122i\)

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 \( 1 + (0.0713 + 0.997i)T \)
5 \( 1 + (-0.281 - 0.959i)T \)
7 \( 1 + (-0.479 + 0.877i)T \)
11 \( 1 + (0.959 + 0.281i)T \)
13 \( 1 + (-0.997 + 0.0713i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (-0.599 + 0.800i)T \)
23 \( 1 + (-0.800 - 0.599i)T \)
29 \( 1 + (0.877 + 0.479i)T \)
31 \( 1 + (-0.800 + 0.599i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (0.997 + 0.0713i)T \)
43 \( 1 + (0.877 - 0.479i)T \)
47 \( 1 + (0.755 - 0.654i)T \)
53 \( 1 + (-0.755 - 0.654i)T \)
59 \( 1 + (-0.0713 + 0.997i)T \)
61 \( 1 + (-0.977 + 0.212i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (0.281 - 0.959i)T \)
73 \( 1 + (0.142 - 0.989i)T \)
79 \( 1 + (0.989 + 0.142i)T \)
83 \( 1 + (-0.936 - 0.349i)T \)
97 \( 1 + (-0.959 + 0.281i)T \)
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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

−30.0086312300939820020443007733, −29.4420259430157164679725550670, −27.41962971608722371689437848886, −26.41952216909926221272485414642, −25.66436587611970566399706310084, −24.486739250586873046198439473331, −23.68051942158008112360408574798, −22.650102890096695615162790388193, −22.01376580819113991818097102524, −19.911520198258235771172234233651, −19.120200412840181890275450433448, −17.73208000470149080318964840325, −17.08919340661608031032923373581, −15.581994097488076343489867100250, −14.31424840865448932137067834973, −13.72522827215439487299937448500, −12.45989439857189983719118081693, −11.23601875293781571275337555609, −9.35791011569327400469128915968, −7.774449911382348834355916315794, −6.942498298214649707233203945687, −6.223324639082622136682631763105, −4.18644191899963260588469129486, −2.77710684574886797553608201831, −0.13774037018465449805985997959, 2.16601957202303590150489216281, 3.82055266817745228016017158351, 4.73824102048913784280744714949, 5.970420914006943853082927279412, 8.70513345071131424466580829915, 9.30144020675461988933056919582, 10.504037312991240945446367720045, 12.00010444658074406812149782278, 12.522419887947121852811408047037, 14.24117818534094400927309112668, 15.198304329601384605331451240425, 16.357793701121642863287594342076, 17.62235164908592676770958180638, 19.44581432065221471365651462369, 19.905860190995646277988341302970, 21.10877064182848956778843312694, 21.97626301265196023205294202273, 22.71226995979487200554488950592, 24.16072457558463858681259696397, 25.26815452305391665725736020685, 26.89222020358345709304386496440, 27.72379578697460771154637376304, 28.44353915777448556156067160288, 29.30171270608628129708573583524, 30.92994151308611233530709406007

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