Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.986 + 0.166i$
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 + (0.973 + 0.230i)3-s + (−0.939 + 0.342i)4-s + (0.893 − 0.448i)5-s + (0.0581 − 0.998i)6-s + (−0.835 − 0.549i)7-s + (0.5 + 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.597 − 0.802i)10-s + (−0.597 + 0.802i)11-s + (−0.993 + 0.116i)12-s + (0.0581 − 0.998i)13-s + (−0.396 + 0.918i)14-s + (0.973 − 0.230i)15-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s,χ)  = 1  + (−0.173 − 0.984i)2-s + (0.973 + 0.230i)3-s + (−0.939 + 0.342i)4-s + (0.893 − 0.448i)5-s + (0.0581 − 0.998i)6-s + (−0.835 − 0.549i)7-s + (0.5 + 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.597 − 0.802i)10-s + (−0.597 + 0.802i)11-s + (−0.993 + 0.116i)12-s + (0.0581 − 0.998i)13-s + (−0.396 + 0.918i)14-s + (0.973 − 0.230i)15-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.986 + 0.166i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.986 + 0.166i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.986 + 0.166i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (2636, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4033,\ (0:\ ),\ -0.986 + 0.166i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.1122868889 - 1.340233087i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.1122868889 - 1.340233087i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9120430915 - 0.6853804771i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9120430915 - 0.6853804771i\)

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

−18.85257421893781658624734874708, −18.167834205795212420530383359536, −17.53823615680860338415826298526, −16.68666619757252554018894456165, −16.01502137799606719592451579581, −15.395949211819477593674935994738, −14.73475976838797228094504393917, −14.13158472506579287612010455416, −13.49460497100189264477721707467, −13.05686800359408786998696589942, −12.39322665511734948772613283081, −11.00442816627747100960164495417, −10.21650952032953290230561097691, −9.589355339567888542724690632503, −8.97678736072725953675411561374, −8.50870669995629705120812224983, −7.6439949672519306227646598559, −6.85466942246727849851831708501, −6.12704552852895111790658834116, −5.91568315295697468482053476798, −4.67089606665571899032141455301, −3.779431448197381600607786508116, −3.03086253497871549980389616490, −2.15092234877084800557327083842, −1.361359744797482946251083692955, 0.32613214053846124502743405651, 1.38841144685324911166636234576, 2.39394999358329720033818917897, 2.676999933461970971409818820405, 3.589792568450934687431862807685, 4.49270983634569655607747264783, 4.9464485772794204097749159295, 6.02490558545593212897044412541, 7.131897354600599169461825519863, 7.7974715906041322256579832980, 8.73550341588742149601583408199, 9.13858060333525950124882720361, 9.98785386388104869279565718104, 10.28290186872743178664082696133, 10.82531864763809585606752299628, 12.29247499189154386909515720603, 12.75946501531403797308341876687, 13.22280164129374551927706916690, 13.75148142679831590716268701889, 14.52507767107145518462696194324, 15.248951528454625783829896035763, 16.27594642566791221486723821503, 16.70947472719468906560310354058, 17.74076210406522654104415810425, 18.16371896031479040615385761748

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