Properties

Degree 1
Conductor 41
Sign $-0.228 - 0.973i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.587 − 0.809i)2-s + (0.707 − 0.707i)3-s + (−0.309 − 0.951i)4-s + (0.951 − 0.309i)5-s + (−0.156 − 0.987i)6-s + (−0.156 + 0.987i)7-s + (−0.951 − 0.309i)8-s i·9-s + (0.309 − 0.951i)10-s + (0.453 + 0.891i)11-s + (−0.891 − 0.453i)12-s + (−0.987 + 0.156i)13-s + (0.707 + 0.707i)14-s + (0.453 − 0.891i)15-s + (−0.809 + 0.587i)16-s + (0.891 − 0.453i)17-s + ⋯
L(s,χ)  = 1  + (0.587 − 0.809i)2-s + (0.707 − 0.707i)3-s + (−0.309 − 0.951i)4-s + (0.951 − 0.309i)5-s + (−0.156 − 0.987i)6-s + (−0.156 + 0.987i)7-s + (−0.951 − 0.309i)8-s i·9-s + (0.309 − 0.951i)10-s + (0.453 + 0.891i)11-s + (−0.891 − 0.453i)12-s + (−0.987 + 0.156i)13-s + (0.707 + 0.707i)14-s + (0.453 − 0.891i)15-s + (−0.809 + 0.587i)16-s + (0.891 − 0.453i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.228 - 0.973i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.228 - 0.973i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $-0.228 - 0.973i$
motivic weight  =  \(0\)
character  :  $\chi_{41} (22, \cdot )$
Sato-Tate  :  $\mu(40)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 41,\ (1:\ ),\ -0.228 - 0.973i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.570716065 - 1.982039782i$
$L(\frac12,\chi)$  $\approx$  $1.570716065 - 1.982039782i$
$L(\chi,1)$  $\approx$  1.452804593 - 1.114671296i
$L(1,\chi)$  $\approx$  1.452804593 - 1.114671296i

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

−34.57098790426990201141850914392, −33.522633394099052618542179210216, −32.58469537248153062875023733837, −32.000233759695170456693017947283, −30.377778519836176464964324349162, −29.54096698115388632515263158178, −27.27377209907068440813909275795, −26.43580576020041911315940732984, −25.48558984512545688404173237536, −24.45647919862998073313048741708, −22.839112048069098682824138749379, −21.710317605989105955558236557544, −20.910111691049616261153033230501, −19.241667951987845230366258909078, −17.199880140666461416988891797784, −16.53156006735854602328917409473, −14.76518844354503528328120161563, −14.13073013406655189309846055288, −12.98041660156207099618319267555, −10.6118892983353565584003562852, −9.25479772579750361960605368941, −7.7087857650828650168384003102, −6.10265265287133442535077538189, −4.436559627804492888405206485303, −2.95909149414783302258387143758, 1.66914631617698378760206868839, 2.817653428584656925236350726303, 5.06359892797314647829820109254, 6.63213244811130860440130627372, 8.95352462567187507425350069846, 9.85139962292055437059502507244, 12.1130185448413933660405472985, 12.76726550614957559548941324263, 14.15031711313357200026494310993, 15.05104321582336742796695361485, 17.50907556297056581254913560687, 18.712281595460906208190854616651, 19.77257706605023346748772617837, 21.00082064045830516776659731997, 21.94631223158596377210104728741, 23.47228349927807842680307967447, 24.87345652742755983481181457849, 25.456946076109048328831399745768, 27.51767935216950258284310375231, 28.85816157575666729425208724196, 29.61592123492115326208788431231, 30.79074445437229591162652202513, 31.805001037979794943903604709215, 32.629445376657949226789228676460, 34.197158557060623473772740328618

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