Properties

Degree 1
Conductor 89
Sign $-0.995 - 0.0964i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $-0.995 - 0.0964i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (61, \cdot )$
Sato-Tate  :  $\mu(88)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 89,\ (1:\ ),\ -0.995 - 0.0964i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.04299066990 - 0.8896735233i$
$L(\frac12,\chi)$  $\approx$  $0.04299066990 - 0.8896735233i$
$L(\chi,1)$  $\approx$  0.5575706591 - 0.4889533628i
$L(1,\chi)$  $\approx$  0.5575706591 - 0.4889533628i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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