Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.838 + 0.544i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.173 − 0.984i)2-s + (0.0581 + 0.998i)3-s + (−0.939 + 0.342i)4-s + (0.802 + 0.597i)5-s + (0.973 − 0.230i)6-s + (0.396 + 0.918i)7-s + (0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (−0.396 − 0.918i)12-s + (−0.396 − 0.918i)13-s + (0.835 − 0.549i)14-s + (−0.549 + 0.835i)15-s + (0.766 − 0.642i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s,χ)  = 1  + (−0.173 − 0.984i)2-s + (0.0581 + 0.998i)3-s + (−0.939 + 0.342i)4-s + (0.802 + 0.597i)5-s + (0.973 − 0.230i)6-s + (0.396 + 0.918i)7-s + (0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (−0.396 − 0.918i)12-s + (−0.396 − 0.918i)13-s + (0.835 − 0.549i)14-s + (−0.549 + 0.835i)15-s + (0.766 − 0.642i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.838 + 0.544i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.838 + 0.544i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.838 + 0.544i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (1229, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4033,\ (0:\ ),\ 0.838 + 0.544i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.496132423 + 0.4434757642i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.496132423 + 0.4434757642i\)
\(L(\chi,1)\)  \(\approx\)  \(1.029423873 + 0.02666609498i\)
\(L(1,\chi)\)  \(\approx\)  \(1.029423873 + 0.02666609498i\)

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

−18.08016698043094018496315574920, −17.472227425312105340186669059190, −17.30357657234391834657431513154, −16.61906142810067288240513452925, −15.82570796236571201147417156013, −14.691198316741618874153083689316, −14.40632418572820660115691667395, −13.60583078498608124236296953779, −13.18521987765161598404694544840, −12.52996501479236589157438235985, −11.76954951747779599334096722034, −10.52562731886707669989802520028, −10.05155795970998328733816946162, −9.1238748448933044154608014181, −8.558311247620951603866510042399, −7.786368015420796515829375422547, −7.1395897645536675108479034286, −6.65864325258706555529113449793, −5.85228313610304483818641675318, −4.97858835273327528055730351002, −4.57822988071511465808592505382, −3.42502397352177767824238445967, −2.00732051506903143914194624109, −1.56250627669556950283831911547, −0.60978080203855128308822641727, 0.7987874746161344997622969130, 2.11984416211384197627612361145, 2.827578359677379475635225511777, 3.045544260733201239522231070125, 4.182995402833270412865736066301, 5.06792655637604141833243967044, 5.58136024966150311792022538363, 6.196026905808489016258582466851, 7.750477479618021043128398573198, 8.36887442705492092716109097038, 8.97109542027867744952352400425, 9.78621432629915458272232140479, 10.27132250705705679840776509481, 10.806750440360285196920851202334, 11.46558194889146616646081495673, 12.232233647688238671498254957607, 12.9705915456330575236026379719, 13.86807364631594714527684352210, 14.40905086909790434190949432713, 14.948565580493827630564609524973, 15.75871860516907372533953739121, 16.76161532038699812049018281212, 17.16013451102896969955630639746, 18.062771388095542863108339027936, 18.59082060037380399948248865235

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