Properties

Degree 1
Conductor 89
Sign $0.743 - 0.669i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.142 − 0.989i)2-s + (0.479 + 0.877i)3-s + (−0.959 + 0.281i)4-s + (0.909 − 0.415i)5-s + (0.800 − 0.599i)6-s + (0.349 − 0.936i)7-s + (0.415 + 0.909i)8-s + (−0.540 + 0.841i)9-s + (−0.540 − 0.841i)10-s + (−0.415 + 0.909i)11-s + (−0.707 − 0.707i)12-s + (0.877 − 0.479i)13-s + (−0.977 − 0.212i)14-s + (0.800 + 0.599i)15-s + (0.841 − 0.540i)16-s + (0.989 + 0.142i)17-s + ⋯
L(s,χ)  = 1  + (−0.142 − 0.989i)2-s + (0.479 + 0.877i)3-s + (−0.959 + 0.281i)4-s + (0.909 − 0.415i)5-s + (0.800 − 0.599i)6-s + (0.349 − 0.936i)7-s + (0.415 + 0.909i)8-s + (−0.540 + 0.841i)9-s + (−0.540 − 0.841i)10-s + (−0.415 + 0.909i)11-s + (−0.707 − 0.707i)12-s + (0.877 − 0.479i)13-s + (−0.977 − 0.212i)14-s + (0.800 + 0.599i)15-s + (0.841 − 0.540i)16-s + (0.989 + 0.142i)17-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $0.743 - 0.669i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (60, \cdot )$
Sato-Tate  :  $\mu(88)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 89,\ (1:\ ),\ 0.743 - 0.669i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.917130953 - 0.7359954346i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.917130953 - 0.7359954346i\)
\(L(\chi,1)\)  \(\approx\)  \(1.295936163 - 0.3768634466i\)
\(L(1,\chi)\)  \(\approx\)  \(1.295936163 - 0.3768634466i\)

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

−30.617676221274975482255715395714, −29.3251480866350589867801624110, −28.32449274544669823894342945596, −26.85714003622456731428462292780, −25.77664673804168639163264412687, −25.223705421073165519691357473256, −24.290222354798723388088823465794, −23.35305657593734884435488280811, −21.96901712948705413376142837462, −20.94992075121350240181772359351, −18.97304011557003591357704090046, −18.47270600708047497674810120084, −17.65748288450779069089121380924, −16.24216101862248706998526301117, −14.9154444582714920148383021006, −13.92612921900149011816391424323, −13.30205533344789626753404462177, −11.661214310339753826995545752259, −9.740937894896860746666384189410, −8.676308508035221244787308293936, −7.66957196744606758773469641655, −6.20012487235712350344155659665, −5.57552559000400205622004321541, −3.12816161830662613504142931817, −1.36074329362082791131536492567, 1.27397238038732689406135095447, 2.880429540744238184535205455490, 4.296474653074909375165035150305, 5.33126997264259611601808539630, 7.84888226363747921093409770033, 9.061101663314365362696015670, 10.17567030999640027031432782217, 10.67778422389887492775226557260, 12.47010918355830191495836484660, 13.68619245164497713559225630633, 14.37890466159056811260030990756, 16.16945588118465628244226811022, 17.311667228704688824121550718171, 18.24485524295940447147240704917, 19.95980381749484494070309789232, 20.58979477578021908013157806291, 21.18530433087096349323804002614, 22.390793841498443832649453793683, 23.44229316473786883654877505648, 25.30244337637121879478059987549, 26.11770446006453020236829138754, 27.1393990406695431438027110660, 28.143709899160777071655632078002, 28.90165135795852269273657441301, 30.29079234020570132414459297524

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