Properties

Label 1-89-89.69-r0-0-0
Degree $1$
Conductor $89$
Sign $0.821 + 0.570i$
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  + (−0.959 − 0.281i)2-s + (−0.540 − 0.841i)3-s + (0.841 + 0.540i)4-s + (0.654 + 0.755i)5-s + (0.281 + 0.959i)6-s + (−0.755 + 0.654i)7-s + (−0.654 − 0.755i)8-s + (−0.415 + 0.909i)9-s + (−0.415 − 0.909i)10-s + (−0.654 + 0.755i)11-s i·12-s + (0.540 + 0.841i)13-s + (0.909 − 0.415i)14-s + (0.281 − 0.959i)15-s + (0.415 + 0.909i)16-s + (0.959 − 0.281i)17-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (−0.540 − 0.841i)3-s + (0.841 + 0.540i)4-s + (0.654 + 0.755i)5-s + (0.281 + 0.959i)6-s + (−0.755 + 0.654i)7-s + (−0.654 − 0.755i)8-s + (−0.415 + 0.909i)9-s + (−0.415 − 0.909i)10-s + (−0.654 + 0.755i)11-s i·12-s + (0.540 + 0.841i)13-s + (0.909 − 0.415i)14-s + (0.281 − 0.959i)15-s + (0.415 + 0.909i)16-s + (0.959 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.821 + 0.570i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ 0.821 + 0.570i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4950154381 + 0.1551951723i\)
\(L(\frac12)\) \(\approx\) \(0.4950154381 + 0.1551951723i\)
\(L(1)\) \(\approx\) \(0.5992546385 + 0.01634883912i\)
\(L(1)\) \(\approx\) \(0.5992546385 + 0.01634883912i\)

Euler product

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

−29.78617355714973116388765200738, −29.08725086084046674185073204993, −28.241315742889274861384840092695, −27.362750273719649200113324832409, −26.2117736170017675719766168291, −25.569075148733380799046251511627, −24.127862565906414871170466838603, −23.189948045016227533599914043509, −21.69039724051710694401845587374, −20.65472244650618671539934811066, −19.85483173468467913927159487369, −18.22060954211923798936383956645, −17.31586122623710956364215703836, −16.206223699829315362647194394580, −15.932574971438514498949172236302, −14.1281015842017372059490167382, −12.61773609089414291031135207473, −11.00825182040328371483934897327, −10.09726385108738910612843355018, −9.291492136803960982676577673681, −7.9289705520401758674731059362, −6.13056335969247341631918601468, −5.36216306222168502458198472189, −3.29045229056601040967992385923, −0.826925468539966862104423331719, 1.77677006470218510785723770909, 2.9264577800547961865205592927, 5.80073145394823440523726805245, 6.73080154517392605686858432158, 7.83187287208858716907074302420, 9.47377560936466196660984867192, 10.41864329388023072689322049991, 11.74229577661865972118572103909, 12.62373316469645112876139659374, 14.01533855190995028012254082839, 15.77023042048105783246239722730, 16.792274863686596790890305299831, 18.25979933517605133234690117885, 18.33909314657266827649201928795, 19.47313047743710201629216863293, 20.954455759598103590402393072922, 22.10642409884919586338356025122, 23.13890143938449393762650352557, 24.632791808534736060154293988522, 25.621652327518771122527802933312, 26.11947855241695188855446488697, 27.76599792900863761644156040833, 28.85616895445999797512796856932, 29.13312576782505744721762473205, 30.39146382299904019454523418566

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