Properties

Degree 1
Conductor 389
Sign $-0.0928 + 0.995i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.868 + 0.495i)2-s + (−0.590 + 0.807i)3-s + (0.509 + 0.860i)4-s + (0.898 − 0.438i)5-s + (−0.912 + 0.408i)6-s + (0.948 + 0.318i)7-s + (0.0161 + 0.999i)8-s + (−0.302 − 0.953i)9-s + (0.997 + 0.0647i)10-s + (0.393 + 0.919i)11-s + (−0.995 − 0.0970i)12-s + (−0.641 − 0.767i)13-s + (0.665 + 0.746i)14-s + (−0.177 + 0.984i)15-s + (−0.481 + 0.876i)16-s + (0.997 + 0.0647i)17-s + ⋯
L(s,χ)  = 1  + (0.868 + 0.495i)2-s + (−0.590 + 0.807i)3-s + (0.509 + 0.860i)4-s + (0.898 − 0.438i)5-s + (−0.912 + 0.408i)6-s + (0.948 + 0.318i)7-s + (0.0161 + 0.999i)8-s + (−0.302 − 0.953i)9-s + (0.997 + 0.0647i)10-s + (0.393 + 0.919i)11-s + (−0.995 − 0.0970i)12-s + (−0.641 − 0.767i)13-s + (0.665 + 0.746i)14-s + (−0.177 + 0.984i)15-s + (−0.481 + 0.876i)16-s + (0.997 + 0.0647i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.0928 + 0.995i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.0928 + 0.995i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(389\)
\( \varepsilon \)  =  $-0.0928 + 0.995i$
motivic weight  =  \(0\)
character  :  $\chi_{389} (13, \cdot )$
Sato-Tate  :  $\mu(97)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 389,\ (0:\ ),\ -0.0928 + 0.995i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.534092431 + 1.683794966i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.534092431 + 1.683794966i\)
\(L(\chi,1)\)  \(\approx\)  \(1.487063977 + 0.9424310964i\)
\(L(1,\chi)\)  \(\approx\)  \(1.487063977 + 0.9424310964i\)

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

−24.0758190572122838210108351552, −23.40993022098319699605014353975, −22.48534212558610367871073244010, −21.568729725293097713735351212767, −21.18344197305321436744428809309, −19.909033942720736647433919312345, −18.798564522126788715990195733501, −18.481873261888547932668048575948, −17.01789706406146950337389105781, −16.697412283686095414387819202220, −14.85955216872009545150683512398, −14.14387955991164988943056807614, −13.69751945226889217037825844810, −12.535236102872838892407246935519, −11.76853995224738636765616105087, −10.863166222007223564137537209442, −10.264270809319713410006325027700, −8.74140561330583090969714763028, −7.25746249590226204467277477976, −6.51069158098825398962850612975, −5.55288106716145770792335495516, −4.805553613241236234668065172977, −3.274585368536685048996643363115, −1.99330790423773032781525344238, −1.296672240257542011232503483411, 1.739164129240813460075200102095, 3.090083966986140558644816250408, 4.580527203210081838357074909618, 5.02886489750077718800333779695, 5.820100684468119199174727851674, 6.92438872288832434401277482227, 8.20258047398327443852636110711, 9.29787340309986799439642514744, 10.30608148186190346333856511669, 11.38660310625157584201277097153, 12.2940603077592808337971502997, 12.99987222406523792260251917059, 14.42698214872944449111744309499, 14.8298201414350254460070414545, 15.7095065927091673101512785394, 16.9559577958827554593322667527, 17.30104242325337907262427126495, 17.992742439126424116854123675656, 19.94181496590687300733909541578, 20.83892158269557781453370601111, 21.371181608850337429301696265879, 22.00253726463641681440302747762, 22.9655709625585542056243741267, 23.67144316593656513804464348695, 24.77784589317329919794581765173

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