Properties

Degree 1
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $0.919 - 0.393i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.193 − 0.981i)2-s + (−0.925 + 0.379i)4-s + (−0.772 + 0.635i)5-s + (0.550 + 0.834i)8-s + (0.772 + 0.635i)10-s + (−0.0149 + 0.999i)11-s + (−0.753 + 0.657i)13-s + (0.712 − 0.701i)16-s + (0.791 − 0.611i)17-s + (0.104 + 0.994i)19-s + (0.473 − 0.880i)20-s + (0.983 − 0.178i)22-s + (0.999 + 0.0299i)23-s + (0.193 − 0.981i)25-s + (0.791 + 0.611i)26-s + ⋯
L(s,χ)  = 1  + (−0.193 − 0.981i)2-s + (−0.925 + 0.379i)4-s + (−0.772 + 0.635i)5-s + (0.550 + 0.834i)8-s + (0.772 + 0.635i)10-s + (−0.0149 + 0.999i)11-s + (−0.753 + 0.657i)13-s + (0.712 − 0.701i)16-s + (0.791 − 0.611i)17-s + (0.104 + 0.994i)19-s + (0.473 − 0.880i)20-s + (0.983 − 0.178i)22-s + (0.999 + 0.0299i)23-s + (0.193 − 0.981i)25-s + (0.791 + 0.611i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.919 - 0.393i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.919 - 0.393i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $0.919 - 0.393i$
motivic weight  =  \(0\)
character  :  $\chi_{6027} (836, \cdot )$
Sato-Tate  :  $\mu(210)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6027,\ (0:\ ),\ 0.919 - 0.393i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.255355182 - 0.2572162538i$
$L(\frac12,\chi)$  $\approx$  $1.255355182 - 0.2572162538i$
$L(\chi,1)$  $\approx$  0.8205116431 - 0.1775398950i
$L(1,\chi)$  $\approx$  0.8205116431 - 0.1775398950i

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

−17.507027341898871398035070298103, −16.858920392824796791540054075932, −16.64186182234404797445535524649, −15.58415339549202184391639897949, −15.43173055669078335497003351462, −14.661064258663089983535168016, −13.91638488639415184259254961252, −13.15544470135207135518248683707, −12.729550134145807234303525183977, −11.85391460397282364875667396272, −11.158896491608421226085130831528, −10.28606181438127932258015037267, −9.64131049121876809165638051087, −8.793909616397064921396786838784, −8.32959913555981983470448347468, −7.769577817715556035322510679769, −7.10760105615748257577762575081, −6.27624122613121067454905730591, −5.53349462031401111626274758861, −4.91205313578012581256454442651, −4.31274627538465445667897977421, −3.41391676552889462845552736367, −2.68220758452770729677099616843, −1.01206266719427342781802756258, −0.740334799681621016560705704751, 0.62487344892169160842725593506, 1.61356817792454197497732356843, 2.47661833506305301548699405038, 3.06799937262455679323574372942, 3.820146389890213858709348602838, 4.60028952336329577977239913157, 5.03522564103757630593608627274, 6.235798095649654180474882232353, 7.24237000834893633772521289544, 7.54132009600196815367078379729, 8.347684406977639431889454640317, 9.21302805808605384754036150444, 9.87472722118887845183584378262, 10.32261250706044594527180117518, 11.146004136226892867587344809988, 11.73148928018947015605299034018, 12.38474972036544796486560459330, 12.60525117071937073720460442937, 13.91018219893635182383001453025, 14.21660846501817390491742186322, 14.94917089467792660399690977252, 15.59461928362071932015652183108, 16.66251852739178897716823489028, 16.963862169201280757046264996310, 17.97009514548680227733467442975

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