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Properties

Label 1-89-89.52-r1-0-0
Degree $1$
Conductor $89$
Sign $-0.774 - 0.632i$
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  + 2-s + (−0.707 − 0.707i)3-s + 4-s − i·5-s + (−0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + 8-s + i·9-s − i·10-s − 11-s + (−0.707 − 0.707i)12-s + (−0.707 + 0.707i)13-s + (−0.707 − 0.707i)14-s + (−0.707 + 0.707i)15-s + 16-s − i·17-s + ⋯
L(s)  = 1  + 2-s + (−0.707 − 0.707i)3-s + 4-s − i·5-s + (−0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + 8-s + i·9-s − i·10-s − 11-s + (−0.707 − 0.707i)12-s + (−0.707 + 0.707i)13-s + (−0.707 − 0.707i)14-s + (−0.707 + 0.707i)15-s + 16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.774 - 0.632i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ -0.774 - 0.632i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5955972051 - 1.671499302i\)
\(L(\frac12)\) \(\approx\) \(0.5955972051 - 1.671499302i\)
\(L(1)\) \(\approx\) \(1.126762730 - 0.7259528613i\)
\(L(1)\) \(\approx\) \(1.126762730 - 0.7259528613i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 - iT \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 - iT \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 + T \)
71 \( 1 - iT \)
73 \( 1 - T \)
79 \( 1 + iT \)
83 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 + T \)
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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

−30.84198354500194734208997274627, −29.42656211202813178127388081788, −28.99116600991235764202702596877, −27.65249441926829655503521206252, −26.26934227638579995582732974631, −25.50225169111828283720621197513, −23.940070889659489007198535697564, −22.97597506831734584524673846775, −22.11579648013299626554688751696, −21.65476531398380170049455872454, −20.28700890638209475272824894272, −18.890911023683231954696613727250, −17.59164815734038043404599782193, −16.07358681685085131063220659475, −15.41079439945731032294173270899, −14.49403645743828877911236005808, −12.87690106608174210081842062167, −11.92402760394258602670415796100, −10.6691361446367162011331220194, −9.92689923884622319713541887803, −7.57338916025596646220236198920, −6.109975870944288042290530076233, −5.438848356626886783148091333935, −3.73152964852854731689707778440, −2.650535968879778384964262607766, 0.61392906640253287262342926104, 2.44749765130809041281572245056, 4.4657639521085657640196417002, 5.3972658490532336430220092647, 6.77198234390086034013717762006, 7.753982709308875302145281708936, 9.88637123681954282412508679219, 11.32486975490471563128176354229, 12.38293251777175411228718083223, 13.16588905840749640983513787785, 14.00888330562627989287629773656, 16.12119108530953431456553573281, 16.35954918986728922797641714030, 17.78198327021965804670571877060, 19.41334311297190501771228525804, 20.24542559502058536222084048138, 21.4946315902631723773759320414, 22.66223058515666035055420267464, 23.54587931138767101959353796897, 24.239459335275582616108830310, 25.1526218148312949257058362425, 26.55450243827309138767237055109, 28.443554183327047718244425543326, 28.93342120320627345486479566702, 29.73151999780178785327585340449

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