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Properties

Label 1-89-89.67-r0-0-0
Degree $1$
Conductor $89$
Sign $-0.743 - 0.669i$
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 + (−0.654 − 0.755i)3-s + (−0.654 + 0.755i)4-s + (−0.959 − 0.281i)5-s + (0.415 − 0.909i)6-s + (−0.959 − 0.281i)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 + 12-s + (−0.654 − 0.755i)13-s + (−0.142 − 0.989i)14-s + (0.415 + 0.909i)15-s + (−0.142 − 0.989i)16-s + (0.415 − 0.909i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (−0.654 − 0.755i)3-s + (−0.654 + 0.755i)4-s + (−0.959 − 0.281i)5-s + (0.415 − 0.909i)6-s + (−0.959 − 0.281i)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 + 12-s + (−0.654 − 0.755i)13-s + (−0.142 − 0.989i)14-s + (0.415 + 0.909i)15-s + (−0.142 − 0.989i)16-s + (0.415 − 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.743 - 0.669i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ -0.743 - 0.669i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01988453163 - 0.05178633920i\)
\(L(\frac12)\) \(\approx\) \(0.01988453163 - 0.05178633920i\)
\(L(1)\) \(\approx\) \(0.4949180180 + 0.09769598297i\)
\(L(1)\) \(\approx\) \(0.4949180180 + 0.09769598297i\)

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

−31.02892974434381267674029007656, −29.74480157658320592466594090688, −28.626566842079682220744958124411, −28.193954246879523545879085405176, −26.838136836896723308036714280359, −26.24287666411197171069028488102, −23.8971510347986230365806977013, −23.3486885663312339634931338434, −22.192769636705940566470111703405, −21.64458331940841313993025661153, −20.29304571916963886724585503947, −19.24866637243612208625333033736, −18.35856870051689135651265072332, −16.66642031765614802955599316028, −15.5695220360027605678873502999, −14.679916554516754503700845764620, −12.903544679114425661105974624442, −12.01939909101581993204955980267, −10.93946058131732662058978956074, −10.06699121032228427621877801159, −8.77608391800371857414685382234, −6.63139687244420846765563102064, −5.19984088665246501521922009897, −3.989487797388288974793667898602, −2.87392267501658199772831979340, 0.05322816866237985853650637423, 3.19443380494253293655656731691, 4.84326299932843789181174912222, 5.972770261604119314970932991274, 7.4533845720202277460435258724, 7.837436130740461574094227060641, 9.82642776590423822520032986419, 11.65039936549172504744798178227, 12.70329596186738067940398557326, 13.31862845463952566050109679941, 15.00558700071444013395012749178, 16.13678749495868486883312997491, 16.79488768793636839986558902955, 18.138710103770505597573467289306, 19.09688273542237146707221173780, 20.45589861144763214436175553394, 22.24084466411317583871274798784, 23.04917108890106469446113233838, 23.56955982606155185779738443810, 24.70538762479291148585887207528, 25.620480858068931522749335760811, 26.92076513860708902881760185061, 27.8965454697907026582576271367, 29.248997478229605164221169329348, 30.17607663233540894056733832164

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