Properties

Degree 1
Conductor $ 11 \cdot 547 $
Sign $0.545 + 0.838i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.498 + 0.867i)2-s + (0.963 − 0.266i)3-s + (−0.503 + 0.863i)4-s + (−0.804 + 0.593i)5-s + (0.710 + 0.703i)6-s + (0.999 + 0.0276i)7-s + (−0.999 − 0.00690i)8-s + (0.858 − 0.512i)9-s + (−0.915 − 0.402i)10-s + (−0.256 + 0.966i)12-s + (−0.134 + 0.990i)13-s + (0.473 + 0.880i)14-s + (−0.618 + 0.786i)15-s + (−0.492 − 0.870i)16-s + (0.436 − 0.899i)17-s + (0.872 + 0.488i)18-s + ⋯
L(s,χ)  = 1  + (0.498 + 0.867i)2-s + (0.963 − 0.266i)3-s + (−0.503 + 0.863i)4-s + (−0.804 + 0.593i)5-s + (0.710 + 0.703i)6-s + (0.999 + 0.0276i)7-s + (−0.999 − 0.00690i)8-s + (0.858 − 0.512i)9-s + (−0.915 − 0.402i)10-s + (−0.256 + 0.966i)12-s + (−0.134 + 0.990i)13-s + (0.473 + 0.880i)14-s + (−0.618 + 0.786i)15-s + (−0.492 − 0.870i)16-s + (0.436 − 0.899i)17-s + (0.872 + 0.488i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.545 + 0.838i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.545 + 0.838i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $0.545 + 0.838i$
motivic weight  =  \(0\)
character  :  $\chi_{6017} (238, \cdot )$
Sato-Tate  :  $\mu(910)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 6017,\ (0:\ ),\ 0.545 + 0.838i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(2.725294524 + 1.478255725i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(2.725294524 + 1.478255725i\)
\(L(\chi,1)\)  \(\approx\)  \(1.545693598 + 0.7588793727i\)
\(L(1,\chi)\)  \(\approx\)  \(1.545693598 + 0.7588793727i\)

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

−17.58795698698995954124811247409, −17.11356891734344654899795947320, −16.0291816032928913377292243270, −15.21562053078294069971199835704, −14.98791041003339226208320937641, −14.44633771971249748273282730145, −13.54559398337222515266983685599, −12.89888605385174923434651626592, −12.56815089965420921221613797494, −11.61079223431770784192059768564, −11.08145307689917526098159582316, −10.34572584336440021028152605273, −9.79027122045270579709596329883, −8.76327975999585250841908203164, −8.43476632473736554302699647180, −7.842026321719796270411643783831, −7.00208884841754726257414131444, −5.67246855531512656446980411131, −5.1087608832349515731648800819, −4.43914600919330574671523619311, −3.769016996585787143215043799771, −3.313305533316416658775183843283, −2.309584082616864159258713578730, −1.6067701559373826222969173529, −0.89331689908771691183449862232, 0.68787204363900846076992400435, 2.12788231826148797146244172352, 2.57216870255987235287447825736, 3.60547142649323057012601962648, 4.13823334828477807987832468402, 4.67654162255842607544704464113, 5.601114185186011276636521575548, 6.77001986133763963972338458127, 6.98466695010034226168782057222, 7.75261433163007013703131654772, 8.208679017757063657814957037953, 8.9282027811574763592290147037, 9.46116606817736958053549507901, 10.64778417752468467637584366748, 11.38565972565452539761348947757, 12.04060260258793470868212602338, 12.653349292377079462004991509162, 13.49917762770150071084500538050, 14.110485753432236285133163143209, 14.69642180530077264319654233869, 14.92268835961330373975895125537, 15.58207040153103915161984188457, 16.4904484433336827835533229089, 16.885914289023794028749286879927, 17.99708961063000505699268170411

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