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Properties

Label 1-89-89.61-r1-0-0
Degree $1$
Conductor $89$
Sign $-0.995 - 0.0964i$
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  + (−0.959 − 0.281i)2-s + (0.212 − 0.977i)3-s + (0.841 + 0.540i)4-s + (0.755 − 0.654i)5-s + (−0.479 + 0.877i)6-s + (−0.997 − 0.0713i)7-s + (−0.654 − 0.755i)8-s + (−0.909 − 0.415i)9-s + (−0.909 + 0.415i)10-s + (0.654 − 0.755i)11-s + (0.707 − 0.707i)12-s + (0.977 + 0.212i)13-s + (0.936 + 0.349i)14-s + (−0.479 − 0.877i)15-s + (0.415 + 0.909i)16-s + (−0.281 − 0.959i)17-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.212 − 0.977i)3-s + (0.841 + 0.540i)4-s + (0.755 − 0.654i)5-s + (−0.479 + 0.877i)6-s + (−0.997 − 0.0713i)7-s + (−0.654 − 0.755i)8-s + (−0.909 − 0.415i)9-s + (−0.909 + 0.415i)10-s + (0.654 − 0.755i)11-s + (0.707 − 0.707i)12-s + (0.977 + 0.212i)13-s + (0.936 + 0.349i)14-s + (−0.479 − 0.877i)15-s + (0.415 + 0.909i)16-s + (−0.281 − 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.995 - 0.0964i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ -0.995 - 0.0964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04299066990 - 0.8896735233i\)
\(L(\frac12)\) \(\approx\) \(0.04299066990 - 0.8896735233i\)
\(L(1)\) \(\approx\) \(0.5575706591 - 0.4889533628i\)
\(L(1)\) \(\approx\) \(0.5575706591 - 0.4889533628i\)

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 \( 1 + (0.212 - 0.977i)T \)
5 \( 1 + (0.755 - 0.654i)T \)
7 \( 1 + (-0.997 - 0.0713i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (0.977 + 0.212i)T \)
17 \( 1 + (-0.281 - 0.959i)T \)
19 \( 1 + (-0.936 + 0.349i)T \)
23 \( 1 + (-0.349 - 0.936i)T \)
29 \( 1 + (-0.0713 + 0.997i)T \)
31 \( 1 + (-0.349 + 0.936i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.977 + 0.212i)T \)
43 \( 1 + (-0.0713 - 0.997i)T \)
47 \( 1 + (-0.540 + 0.841i)T \)
53 \( 1 + (0.540 + 0.841i)T \)
59 \( 1 + (-0.212 - 0.977i)T \)
61 \( 1 + (0.800 + 0.599i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.755 - 0.654i)T \)
73 \( 1 + (-0.415 - 0.909i)T \)
79 \( 1 + (0.909 - 0.415i)T \)
83 \( 1 + (0.479 - 0.877i)T \)
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

−30.568721918516302823243024526967, −29.47358391391531621690480297613, −28.31624919373676077522691797155, −27.70915767906134895379546485047, −26.18577264613895375247157802544, −25.90672035129868779346688002717, −25.10027632759384645988611900517, −23.23906217877669567768553367174, −22.16340235902820080423094258572, −21.08194811853680573255366716933, −19.9070449257842397414908333496, −18.99633819435152742783308830125, −17.593583548115999835262051080116, −16.81903347834051468729983751431, −15.5038723951414498927316552197, −14.882541218014318468422126275534, −13.38440508512945478446987616999, −11.37593886713339043928129454118, −10.22623246870429330781136924499, −9.6221775204916388315117366616, −8.503812937567151960349867504907, −6.692965646762301075406762155055, −5.81759392924178378848759233772, −3.64830610779337942009141578761, −2.11670333275894857803552493672, 0.52410234002922577606924384485, 1.8333988431192170139972632496, 3.32627013918824791770752003969, 6.0843868158680012002681742719, 6.82878462674098813504790966330, 8.6036101577958684372876988366, 9.083363436097127268838523069715, 10.62328095565946251477012942500, 12.085866252011025699849613512419, 12.987242726288059994310330129433, 14.05569751386739659627061834863, 16.13417829727039367532391808317, 16.852214395094390132923125672, 18.062442898126668095864148324614, 18.9074477085644078550014217045, 19.90378445888640098387063821377, 20.79691274852846054275404652286, 22.14987470396009801829577268318, 23.72286351542078371959357716036, 24.937234355560907935436228893621, 25.41138447435355996494979153167, 26.41713483572057189151535232369, 27.89167073414562055049041545197, 28.96295742797241771993573440923, 29.423702263622234237270059328658

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