Properties

Degree 1
Conductor 59
Sign $-0.896 - 0.442i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.161 + 0.986i)2-s + (0.267 + 0.963i)3-s + (−0.947 + 0.319i)4-s + (−0.561 + 0.827i)5-s + (−0.907 + 0.419i)6-s + (0.976 + 0.214i)7-s + (−0.468 − 0.883i)8-s + (−0.856 + 0.515i)9-s + (−0.907 − 0.419i)10-s + (−0.796 − 0.605i)11-s + (−0.561 − 0.827i)12-s + (0.856 + 0.515i)13-s + (−0.0541 + 0.998i)14-s + (−0.947 − 0.319i)15-s + (0.796 − 0.605i)16-s + (0.976 − 0.214i)17-s + ⋯
L(s,χ)  = 1  + (0.161 + 0.986i)2-s + (0.267 + 0.963i)3-s + (−0.947 + 0.319i)4-s + (−0.561 + 0.827i)5-s + (−0.907 + 0.419i)6-s + (0.976 + 0.214i)7-s + (−0.468 − 0.883i)8-s + (−0.856 + 0.515i)9-s + (−0.907 − 0.419i)10-s + (−0.796 − 0.605i)11-s + (−0.561 − 0.827i)12-s + (0.856 + 0.515i)13-s + (−0.0541 + 0.998i)14-s + (−0.947 − 0.319i)15-s + (0.796 − 0.605i)16-s + (0.976 − 0.214i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.896 - 0.442i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.896 - 0.442i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(59\)
\( \varepsilon \)  =  $-0.896 - 0.442i$
motivic weight  =  \(0\)
character  :  $\chi_{59} (50, \cdot )$
Sato-Tate  :  $\mu(58)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 59,\ (1:\ ),\ -0.896 - 0.442i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.3045316559 + 1.304709821i$
$L(\frac12,\chi)$  $\approx$  $-0.3045316559 + 1.304709821i$
$L(\chi,1)$  $\approx$  0.5074029858 + 0.9306201209i
$L(1,\chi)$  $\approx$  0.5074029858 + 0.9306201209i

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

−31.48839743219505397915337007473, −30.57547193390976130871261467326, −29.89667007864672495743585493849, −28.368287710721631305929370793259, −27.88562789730615550504208037255, −26.34244656904916923409437644670, −24.80516364129306005289337512729, −23.56422786072751993788245881865, −23.15676971171184889183576168051, −20.9356998601760312877957696480, −20.54276728689143290178783338014, −19.290993726584748441104334852677, −18.23991707957316243290704874176, −17.202598542046517902255498650550, −15.106819623580754912863363209153, −13.774271972085445813430292067586, −12.69711662586715722710422283426, −11.88849600349832764181365542040, −10.50823491171663989865369153462, −8.56959717922100342144640279822, −7.89202426417780616557447216053, −5.50835156535799301907434645518, −4.01167827837642889297214944744, −2.09700577938034295722674632834, −0.70235031663497944756363048938, 3.25768726861814997614714476549, 4.567235827075011257085932917578, 5.948392822550256431415678292170, 7.78380597220530531716202891442, 8.62751814525391353814183192060, 10.33923398394292119306112744664, 11.57073656055675045802818277050, 13.74532004827045252569449861819, 14.66040302615803938134825359898, 15.54558564239617624163850000131, 16.504811155793864654701143361638, 18.013864058520964563925601900304, 19.075586880234454180091380003691, 21.02865747811811356717252686882, 21.7400804545925772933716026043, 23.10652759249588897575629905908, 23.90281202167632979746640379564, 25.56691779428636828459061623520, 26.273823885281081612290533657093, 27.31349751114690846887848891685, 27.97273646045776060998516049818, 30.20469922012916508543621987190, 31.3155667002933255867361435547, 31.90109028044721830133417826300, 33.39795541466962107367165893436

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