Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.179 - 0.983i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.766 + 0.642i)2-s + (−0.973 − 0.230i)3-s + (0.173 + 0.984i)4-s + (−0.549 + 0.835i)5-s + (−0.597 − 0.802i)6-s + (−0.0581 + 0.998i)7-s + (−0.5 + 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.957 + 0.286i)10-s + (0.957 + 0.286i)11-s + (0.0581 − 0.998i)12-s + (−0.0581 + 0.998i)13-s + (−0.686 + 0.727i)14-s + (0.727 − 0.686i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s,χ)  = 1  + (0.766 + 0.642i)2-s + (−0.973 − 0.230i)3-s + (0.173 + 0.984i)4-s + (−0.549 + 0.835i)5-s + (−0.597 − 0.802i)6-s + (−0.0581 + 0.998i)7-s + (−0.5 + 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.957 + 0.286i)10-s + (0.957 + 0.286i)11-s + (0.0581 − 0.998i)12-s + (−0.0581 + 0.998i)13-s + (−0.686 + 0.727i)14-s + (0.727 − 0.686i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.179 - 0.983i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.179 - 0.983i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.179 - 0.983i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (14, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.179 - 0.983i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.4483219748 + 0.3739105180i$
$L(\frac12,\chi)$  $\approx$  $-0.4483219748 + 0.3739105180i$
$L(\chi,1)$  $\approx$  0.6535771605 + 0.6787804032i
$L(1,\chi)$  $\approx$  0.6535771605 + 0.6787804032i

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

−17.87485551758106976879751790327, −17.044175707109079781743765787069, −16.635236725918766388373658332427, −15.93360742971461689506852806757, −15.22457413032545126601002412891, −14.47936207670791532732614382855, −13.7133990453712065438744426516, −12.86155973523503006934451323458, −12.37891915346743271324374428236, −11.967259108523280208573786968315, −11.03289214573236192939953476523, −10.55776879274333003572184359291, −10.003299051398236020399911723529, −9.09506774958859620722915029988, −8.18279614111954831514248406805, −7.175256592305788349168922417640, −6.46419281431758516513227474774, −5.745801587652264737703706957812, −4.97859984787012324145527024014, −4.34728134240168128277344823974, −3.817447261059721095479606926330, −3.10848358200601383967122940325, −1.50300837171252634191262940497, −1.05975924976675196196510664592, −0.15561547370487937237975619824, 1.69378027072392504266347774987, 2.41849585641853741623137994167, 3.49295851399787159758954899118, 4.2094219459341385524177304775, 4.80694810968275289176206798293, 5.92742358622965915865409180229, 6.22253651742603822801674327297, 6.811067040372844657849995871093, 7.65471063925941099875335795990, 8.198271638288846168426195550535, 9.343400155176161785241847523915, 10.03242976526714565468852865727, 11.23445886856224308484680251453, 11.68650297040869100278489583519, 12.02718193982936406483353734618, 12.66306713530024986500294304381, 13.66006155092244197912725585921, 14.36050502952856930906045446090, 14.94409778661926184135392320786, 15.59203095412125128086221839504, 16.20385043025020177520817893453, 16.87286596710831694521423097896, 17.49514868486818179601250641752, 18.209917110709907969187181099881, 19.03191568950419686520556939059

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