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Properties

Label 1-89-89.47-r0-0-0
Degree $1$
Conductor $89$
Sign $-0.908 - 0.417i$
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  + (0.415 − 0.909i)2-s + (−0.755 − 0.654i)3-s + (−0.654 − 0.755i)4-s + (0.959 − 0.281i)5-s + (−0.909 + 0.415i)6-s + (−0.281 − 0.959i)7-s + (−0.959 + 0.281i)8-s + (0.142 + 0.989i)9-s + (0.142 − 0.989i)10-s + (−0.959 − 0.281i)11-s + i·12-s + (0.755 + 0.654i)13-s + (−0.989 − 0.142i)14-s + (−0.909 − 0.415i)15-s + (−0.142 + 0.989i)16-s + (−0.415 − 0.909i)17-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (−0.755 − 0.654i)3-s + (−0.654 − 0.755i)4-s + (0.959 − 0.281i)5-s + (−0.909 + 0.415i)6-s + (−0.281 − 0.959i)7-s + (−0.959 + 0.281i)8-s + (0.142 + 0.989i)9-s + (0.142 − 0.989i)10-s + (−0.959 − 0.281i)11-s + i·12-s + (0.755 + 0.654i)13-s + (−0.989 − 0.142i)14-s + (−0.909 − 0.415i)15-s + (−0.142 + 0.989i)16-s + (−0.415 − 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.908 - 0.417i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ -0.908 - 0.417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1971100644 - 0.9013703736i\)
\(L(\frac12)\) \(\approx\) \(0.1971100644 - 0.9013703736i\)
\(L(1)\) \(\approx\) \(0.6329260015 - 0.7684163223i\)
\(L(1)\) \(\approx\) \(0.6329260015 - 0.7684163223i\)

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 \( 1 + (-0.755 - 0.654i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (-0.281 - 0.959i)T \)
11 \( 1 + (-0.959 - 0.281i)T \)
13 \( 1 + (0.755 + 0.654i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (-0.989 + 0.142i)T \)
23 \( 1 + (0.989 - 0.142i)T \)
29 \( 1 + (0.281 + 0.959i)T \)
31 \( 1 + (0.989 + 0.142i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.755 - 0.654i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (0.654 + 0.755i)T \)
53 \( 1 + (0.654 - 0.755i)T \)
59 \( 1 + (-0.755 + 0.654i)T \)
61 \( 1 + (-0.540 + 0.841i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (-0.142 + 0.989i)T \)
79 \( 1 + (0.142 - 0.989i)T \)
83 \( 1 + (-0.909 + 0.415i)T \)
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

−31.25022648391430503968681126431, −29.97038745610098001966858182598, −28.672295554122685593891579605730, −27.903746800286966513119400347743, −26.444283723962633352845081637522, −25.7354489882773606849623962595, −24.742181552873241643604855254722, −23.34508347390585619813222731769, −22.60762819704416014780613293370, −21.48409970575138485636302289420, −21.0949475063271184244014532251, −18.6015402031261981413845712115, −17.73287635342402584691287100797, −16.89549076818252887256926638944, −15.44781161717319254715446961841, −15.143749173462028719433601619288, −13.35157443314764581440931880993, −12.53222006848648057842655874360, −10.84018941997441085867000107538, −9.62156046663412546402485075378, −8.406413725053483914052654297454, −6.41382316431981779673557417548, −5.829638582943672860583887601592, −4.68688154741790479613863219990, −2.89830116620136352664079214688, 1.017165222665814032333735633986, 2.49420740846511726148041689351, 4.499935480238040221208295703313, 5.682395375351529365938937770296, 6.8630207939931577983083540647, 8.86909504947037851059280699983, 10.4016104099468156871618561035, 11.0006222720482699158756775074, 12.553737070319072769721094234076, 13.377665080590435313317039739470, 14.000169340218498928895197023856, 16.14927424195404294907057155842, 17.35058383132890657662122556609, 18.30039832166600602797431161238, 19.27127092152912245621927824534, 20.68298591081695271031888252590, 21.42806881218519230838153716109, 22.739807634884858713713702074101, 23.46611512832990134760955663555, 24.40653758633829221747949017244, 25.85235501194697625681993856765, 27.23857557258237528255303952556, 28.58798181814164278263944672272, 29.09574408730276979982497387371, 29.793867649734320516595247305392

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