Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.943 + 0.332i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.939 + 0.342i)2-s + (−0.597 − 0.802i)3-s + (0.766 − 0.642i)4-s + (0.727 + 0.686i)5-s + (0.835 + 0.549i)6-s + (0.973 + 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.286 + 0.957i)9-s + (−0.918 − 0.396i)10-s + (0.918 − 0.396i)11-s + (−0.973 − 0.230i)12-s + (0.973 + 0.230i)13-s + (−0.993 + 0.116i)14-s + (0.116 − 0.993i)15-s + (0.173 − 0.984i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s,χ)  = 1  + (−0.939 + 0.342i)2-s + (−0.597 − 0.802i)3-s + (0.766 − 0.642i)4-s + (0.727 + 0.686i)5-s + (0.835 + 0.549i)6-s + (0.973 + 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.286 + 0.957i)9-s + (−0.918 − 0.396i)10-s + (0.918 − 0.396i)11-s + (−0.973 − 0.230i)12-s + (0.973 + 0.230i)13-s + (−0.993 + 0.116i)14-s + (0.116 − 0.993i)15-s + (0.173 − 0.984i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.943 + 0.332i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.943 + 0.332i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.943 + 0.332i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (51, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.943 + 0.332i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.575404732 + 0.2691708274i$
$L(\frac12,\chi)$  $\approx$  $1.575404732 + 0.2691708274i$
$L(\chi,1)$  $\approx$  0.9241737626 + 0.06598542296i
$L(1,\chi)$  $\approx$  0.9241737626 + 0.06598542296i

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

−18.1164957635844298777642076158, −17.6924798566831439560647341040, −17.13657847645554484643361824929, −16.64500073591091604518314796186, −15.94376170487172086872036617131, −15.34209647333100899955260666641, −14.3195534536507200780900758969, −13.79817758955388699109681957605, −12.49338036253205673914328345728, −12.20428847045650436855536394455, −11.2610115336844020467451825751, −10.886304306983733651510068632834, −10.067473589369872171083530515537, −9.49101059994282938124652624320, −8.820998237504878807307284687465, −8.35512495461235457947248116090, −7.28253873712015775531156661329, −6.43968713223125343453717466479, −5.78227187618924765814486778373, −4.8464648952322836469902160556, −4.250096421825127147644195469928, −3.35529627784460892495613552577, −2.29683597814868000896728597762, −1.18929954930090716166142039738, −0.952450993551833683666620391407, 1.03303736033308772049723098068, 1.40996534329918962309075977452, 2.249434479283955150362735129197, 3.05131432563620123723891488514, 4.5124363107333128640969403787, 5.43047479253220495632678518808, 6.17869986817236135463016705777, 6.49243451783444062246533808972, 7.22323865938470202202697580694, 8.08199459831241448145669973876, 8.68424829216926427483734835703, 9.33579507670776682252350311482, 10.41729309768555339193709401636, 10.972431567939502054945933760959, 11.392728236914893231288178184937, 12.00628207598521953682511474445, 13.18444608011276623344803068503, 13.84627752351376071865714215181, 14.40210249874919681841532843664, 15.18333141916301342389837203084, 15.87093471013782467378327880582, 16.9508228718057991047485494780, 17.27100975954491906862906729788, 17.803062028084794061735236782584, 18.284980388905766993759658205239

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