Properties

Degree 1
Conductor 89
Sign $-0.848 + 0.529i$
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 + (−0.977 + 0.212i)3-s + (0.841 − 0.540i)4-s + (−0.755 − 0.654i)5-s + (0.877 − 0.479i)6-s + (−0.0713 − 0.997i)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 + (−0.212 − 0.977i)13-s + (0.349 + 0.936i)14-s + (0.877 + 0.479i)15-s + (0.415 − 0.909i)16-s + (0.281 − 0.959i)17-s + ⋯
L(s,χ)  = 1  + (−0.959 + 0.281i)2-s + (−0.977 + 0.212i)3-s + (0.841 − 0.540i)4-s + (−0.755 − 0.654i)5-s + (0.877 − 0.479i)6-s + (−0.0713 − 0.997i)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 + (−0.212 − 0.977i)13-s + (0.349 + 0.936i)14-s + (0.877 + 0.479i)15-s + (0.415 − 0.909i)16-s + (0.281 − 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.848 + 0.529i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.848 + 0.529i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $-0.848 + 0.529i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (56, \cdot )$
Sato-Tate  :  $\mu(88)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 89,\ (1:\ ),\ -0.848 + 0.529i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.008406988493 + 0.02933968357i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.008406988493 + 0.02933968357i\)
\(L(\chi,1)\)  \(\approx\)  \(0.3886846172 + 0.02571157685i\)
\(L(1,\chi)\)  \(\approx\)  \(0.3886846172 + 0.02571157685i\)

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

−29.78086529480620478729103129835, −28.46073155460101361579813154366, −27.93809413871664694736519012944, −26.911579527900479579688640417554, −25.88962922335204359973859803988, −24.47775976525096943399208554471, −23.67284912936445500345584094325, −21.88417813592971424475471436297, −21.77072408326699900197465653598, −19.6374755147270618748529984671, −18.92146466335947916552515768803, −18.16098146694335030640772526790, −16.88658520150889450567342115374, −16.01005457359672803906464481594, −14.86727741872399096305408448846, −12.69857467786038488273152762804, −11.54373187675026232730648804092, −11.20834152798496649318154472445, −9.65478290884369131057841666735, −8.3011747202760283153130972627, −6.94880819326144988692865146784, −6.000195300900770178086311453332, −3.811245118619601529786373831906, −1.98803201699936633884042758855, −0.024691728035828855224792067908, 1.19958432457352569707188104155, 4.01418104515317284102358564842, 5.4422622998523961762136471113, 6.975521372125305263938846025158, 7.85458085989118686776617859508, 9.53242529246640178261645003739, 10.474995283896960305696689056006, 11.64516051678016991324760946461, 12.61863763403292459157032834580, 14.71229783914827748691828679319, 15.94223762766433572577520013199, 16.70713665502911835698700125916, 17.46946065406744639761793992916, 18.6556549397371757252946873349, 20.120167670907669584069239524905, 20.5474376149442398419471747181, 22.557235574286015826846270175626, 23.397779801354477811184531582923, 24.31821907015321298110722144161, 25.46169197449475742655599898649, 27.06368967863747439109046424533, 27.395417093865847830897587809950, 28.30568332423153626897852297386, 29.434075644386826909040220595957, 30.21310442970779155683428319063

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