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Properties

Label 1-389-389.30-r0-0-0
Degree $1$
Conductor $389$
Sign $-0.0928 - 0.995i$
Analytic cond. $1.80650$
Root an. cond. $1.80650$
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  + (0.868 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $-0.0928 - 0.995i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ -0.0928 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.534092431 - 1.683794966i\)
\(L(\frac12)\) \(\approx\) \(1.534092431 - 1.683794966i\)
\(L(1)\) \(\approx\) \(1.487063977 - 0.9424310964i\)
\(L(1)\) \(\approx\) \(1.487063977 - 0.9424310964i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (0.868 - 0.495i)T \)
3 \( 1 + (-0.590 - 0.807i)T \)
5 \( 1 + (0.898 + 0.438i)T \)
7 \( 1 + (0.948 - 0.318i)T \)
11 \( 1 + (0.393 - 0.919i)T \)
13 \( 1 + (-0.641 + 0.767i)T \)
17 \( 1 + (0.997 - 0.0647i)T \)
19 \( 1 + (-0.777 + 0.628i)T \)
23 \( 1 + (0.898 - 0.438i)T \)
29 \( 1 + (0.712 + 0.701i)T \)
31 \( 1 + (-0.999 - 0.0323i)T \)
37 \( 1 + (-0.816 + 0.577i)T \)
41 \( 1 + (-0.777 - 0.628i)T \)
43 \( 1 + (-0.735 - 0.677i)T \)
47 \( 1 + (-0.777 - 0.628i)T \)
53 \( 1 + (-0.113 + 0.993i)T \)
59 \( 1 + (-0.957 + 0.287i)T \)
61 \( 1 + (-0.641 + 0.767i)T \)
67 \( 1 + (0.271 - 0.962i)T \)
71 \( 1 + (0.509 - 0.860i)T \)
73 \( 1 + (-0.481 + 0.876i)T \)
79 \( 1 + (-0.999 + 0.0323i)T \)
83 \( 1 + (0.271 + 0.962i)T \)
89 \( 1 + (-0.689 - 0.724i)T \)
97 \( 1 + (0.509 + 0.860i)T \)
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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

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

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